vibration analysis of orthotropic mindlin plates with edges elastically restrained against rotation abstract the free vibration analysis of the orthotropic mindlin plates with edges elastically restrained against rotation has been accomplished using the rayleigh ritz method polynomials with properties corresponding to those of the timoshenko beam functions are introduced as trial functions in the spatial representation of the deflection and rotations of cross sections in two directions of the plates the results obtained by using polynomials are nearly the same as the timoshenko beam functions results but were obtained with considerably less numerical efforts

solvability of linear local and nonlocal robin problems over c ω abstract let r n be a bounded lipschitz domain we first consider an elliptic boundary value problem with general robin boundary conditions the boundary conditions can be either local or nonlocal depending on the conditions imposed on the elliptic operator we prove that this boundary value problem is uniquely solvable and moreover we show that such weak solution is holder continuous on we also prove that a realization of the associated differential operator with generalized local or nonlocal robin boundary conditions generates an analytic c 0 semigroup of angle 2 over c we conclude by applying the elliptic regularity theory to solve the corresponding cauchy problem over c

nonreflecting boundary condition in finite element formulation for an elastic wave equation numerical modeling of seismic wavefields using finite difference or finite element methods requires truncation of the model to finite computational domains it is well known that the edges of such truncated models give boundary reflections on the synthetic seismograms an essential step to successful numerical modeling is to eliminate these reflections we present a simple scheme that eliminates such boundary reflections when computing synthetics we also demonstrate the efficiency and robustness of our method on a variety of geological models

on contracting interval iteration for nonlinear problems in rn part ii applications governs classes of evolution type transport processes in the theories of diffusion conduction of heat chemical kinetics etc explicit solutions of 1 1 with boundary conditions are known only in a few exceptional cases e g 4 therefore finite dimensional approximations are of considerable practical interest according to the equivalence theorem e g 7 the convergence of a sequence of such approximations is ensured if the conditions of consistency and stability are satisfied gershgorin s method is a general approach for establishing the stability property in the case of nonlinear bvp boundary value problems this method transfers the property of inverse monotonicity from the bvp to the finite dimensional approximation where it has to be shown that this property holds uniformly as a grid parameter m e n tends to infinity if a is an inverse monotone operator then the operator inequality au d aw making use of a partial ordering of the space u on which a operates implies that u d w for every pair of elements u w e u such that au aw the uniformity of the property of inverse monotonicity can be shown by use of available sufficient conditions if the classes of admissible functions c k and c in 1 1 are rather severely restricted in addition for fixed m the qualitative property of convergence does not yield a quantitative estimate of the relationship between the local and the global discretization error 13 p 131 where local error refers to the order of consistency of the difference scheme and the global error for any fured m is the distance between the restriction to the grid of the solution of the bvp and the solution of the difference scheme rather than verifying the qualitative properties of stability and convergence for fmed m here a quantitative error estimate is derived which relates the local to the global discretization error this estimate depends on both the grid parameter m and the number of the iteration cycle of the numerical method to be employed for the deduction of this relationship it is sufficient that the finite system is individually i e not necessarily uniformly inverse monotone for each value of the grid parameter m e n

discrete singular convolution and its application to the analysis of plates with internal supports part 2 applications part 2 of this series of two papers presents the applications of the discrete singular convolution dsc algorithm the main purpose of this paper is to explore the utility test the accuracy and examine the convergence of the proposed approach for the vibration analysis of rectangular plates with internal supports both partial internal line supports and complex internal supports are considered for 21 square plates of various combinations of edge support conditions the effects of different size shape and topology of the internal supports and different boundary conditions on the vibration response of plates are investigated the partial internal line supports may vary from a central point support to a full range of cross or diagonal line supports several closed loop supports such as ring square and rhombus and their combinations are studied for complex internal supports convergence and comparison studies are carried out to establish the correctness and accuracy of the dsc algorithm the dsc results are compared with those in the available literature obtained by using other methods numerical results indicate that the dsc algorithm exhibits controllable accuracy for plate analysis and shows excellent flexibility in handling complex geometries boundary conditions and support conditions copyright 2002 john wiley sons ltd

an analytical solution for the free vibration analysis of rectangular plates resting on symmetrically distributed point supports abstract a general analytical solution is obtained for the free vibration of rectangular plates supported on their lateral surface by symmetrically distributed point supports highly accurate eigenvalues are provided for a square plate supported at various points along the diagonal results are compared with earlier findings obtained by a finite element method it is shown that the method may be extended to solve various other problems the tabulation of eigenvalues appears to be the most comprehensive yet made for a square plate with point supports on the diagonals

diffusive logistic equations with indefinite weights population models in disrupted environments the dynamics of a population inhabiting a strongly heterogeneous environment are modelledby diffusive logistic equations of the form u t d u m x cu u in 0 where u represents the population density c d 0 are constants describing the limiting effects of crowding and the diffusion rate of the population respectively and m x describes the local growth rate of the population if the environment is bounded and is surrounded by uninhabitable regions then u 0 on 0 the growth rate m x is positive on favourablehabitats and negative on unfavourable ones the object of the analysis is to determine how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled the models are shown to possess a unique stable positive steady state implying persistence for the population provided l d where is the principle positive eigenvalue for the problem m x in 0 on analysis of how depends on m indicates that environments with favourable and unfavourable habitats closely intermingled are worse for the population than those containing large regions of uniformly favourable habitat in the limit as the diffusion rate d 0 the solutions tend toward the positive part of m x c and if m is discontinuous develop interior transition layers the analysis uses bifurcation and continuation methods the variational characterisation of eigenvalues upper and lower solution techniques and singular perturbation theory

a general solution for the free vibration of rectangular plates resting on uniform elastic edge supports abstract in this paper it is shown how the superposition method is employed to obtain accurate solutions for the free vibration frequencies and mode shapes of rectangular plates resting on elastic edge supports the analysis is completely general in that uniform elastic rotational and translational supports of any stiffness magnitudes are considered to act simultaneously along each of the four edges eight stiffness coefficients being required to define the problem it is shown how all of the classical boundary conditions are approached when the coefficients are allowed to take on appropriate limits a limited number of plots of fundamental mode eigenvalues vs stiffness coefficients are presented for square plate problems of special interest to the author s knowledge this represents the first accurate comprehensive treatment of this problem to appear in the literature

the reaction diffusion problem with dynamical boundary condition abstract the aim of this paper is to prove the existence and uniqueness of classical solutions to a coupled system of three parabolic and one ordinary differential equations with the latter determined on the boundary this system describes the receptor toxin antibody interaction problem taking into account diffusion of reacting species

transverse vibrations of rectangular plates with edges elastically restrained against translation and rotation abstract the fundamental frequency coefficient for a rectangular plate with edges elastically restrained against both translation and rotation is calculated by using polynomial coordinate functions and the rayleigh ritz method the approach is simple and straightforward and allows the solution of a rather difficult elastodynamics problem complicating factors orthotropic properties in plane forces concentrated masses etc can also be taken into account without formal difficulties

free vibration analysis of mindlin plates with uniform elastic edge support by the superposition method abstract utilizing the superposition method a solution is obtained for the free vibration eigenvalues of mindlin plates resting on uniform lateral elastic edge support subsequently it is shown how minor modifications to the eigenvalue matrix permit the incorporation of the additional effects of rotational stiffness into the analysis solutions are based on the mindlin thick plate theory convergence is rapid and eigenvalue curves are plotted for square plates and plates of aspect ratio 1 5 for various lateral and rotational spring stiffnesses digital results are tabulated in order that other analysts will have reliable data against which their findings may be compared

an advanced theory of moderately thick plate vibrations in thick plate vibration theory the governing equations are stated with a system of three partial differential equations of motion with total deflection which consists of bending deflection and shear contribution and angles of rotation as fundamental variables most of the methods deal with these three equations some of them with two and recently a solution based on one equation has been offered in the present paper a system of three equations for a moderately thick plate is reduced to a single equation in terms of bending deflection only as a fundamental variable shear deflection and angles of rotation depend on bending deflection as a potential function a simple formula for natural frequencies of a simply supported plate is derived a characteristic equation is also obtained for a plate with simply supported two opposite edges numerical results for a simply supported plate and a plate clamped on the two remaining opposite edges are compared with those known in literature for different aspect ratios and relative thickness and very good agreement is achieved

solvability theorem for a surface reaction type model abstract the purpose of this paper is to investigate the existence and uniqueness of classical solutions to a model of surface reactions between two polyatomic reactants the model is described by a coupled system of four quasilinear parabolic equations where two of them are determined in the domain and the other two are considered on a part of its surface the elliptic operators of the parabolic equations determined on the surface are allowed to be degenerate in the sense that the density dependent diffusion coefficients p i i may have the property p i 1 0 i 1 2

dynamic stiffness formulation and free vibration analysis of specially orthotropic mindlin plates with arbitrary boundary conditions abstract a novel dynamic stiffness formulation which includes the effects of shear deformation and rotatory inertia is proposed to carry out the free vibration analysis of thick rectangular orthotropic plates i e the formulation is based on an extension of the mindlin theory to orthotropic plates in a dynamic stiffness context the modified trigonometric basis is used to construct the general solution for the free vibration problem of the plate in series form permitting the derivation of an infinite system of linear algebraic equations which connect the fourier coefficients of the kinematic boundary conditions of its four edges the boundary conditions are essentially the deflections and angles of rotations constituting the displacement vector and shear forces and bending moments constituting the force vector the force displacement relationship is then obtained by relating the two vectors via the dynamic stiffness matrix essentially the ensuing infinite system forms the fundamental basis of the paper which defines the dynamic stiffness matrix for an orthotropic mindlin plate in an exact sense numerical results are obtained by using the wittrick williams algorithm as solution technique the theory is first validated using published results and then further illustrated by a series of numerical examples the paper concludes with significant conclusions

periodic solutions of quasilinear parabolic systems with nonlinear boundary conditions abstract this paper is concerned with the existence of maximal and minimal periodic solutions of a class of quasilinear parabolic systems with nonlinear boundary conditions our approach to the problem is based on the method of upper and lower solutions and its associated monotone iterations an application is also made to the reaction diffusion system of lotka volterra competition model

a general solution for the free vibration of rectangular plates with arbitrarily distributed lateral and rotational elastic edge support abstract in this paper the superposition method is exploited to obtain a solution for the free vibration of thin rectangular plates resting on elastic edge supports of arbitrarily distributed stiffnesses stiffnesses are represented in series form thereby permitting the specification of any distributions including concentrated stiffnesses and stiffnesses with discontinuities convergence is found to be very rapid for simple stiffness distributions such as the parabolic distribution utilized in illustrative cases this appears to be the first comprehensive treatment of this general problem of plates on elastic edge supports

control of nonlinear vibrations of functionally graded plates using 1 3 piezoelectric composite performance of the vertically obliquely reinforced 1 3 piezoelectric composite material as the constraining layer of the active constrained layer damping treatment for controlling geometrically nonlinear vibrations of the functionally graded plates subjected to a temperature field has been investigated unlike the existing analyses the finite element model uses the induced transverse normal stress in the active constraining layer for the active damping of nonlinear vibrations of the overall functionally graded plates the temperature field is assumed to be uniform over the substrate plate surfaces and varied through the thickness of the host functionally graded plates the temperature dependent material properties of the functionally graded substrate plates are assumed to be graded in the thickness direction of the plates according to a power law distribution and poisson s ratio is assumed to be a constant over the domain of the plate the constrained viscoelastic layer of the active constrained layer damping treatment is modeled using the golla hughes mctavish method the first order shear deformation theory and the green lagrange nonlinear strain displacement relations are used to model the open loop and closed loop nonlinear dynamics of the overall functionally graded substrate plates under a thermal environment the analysis suggests the potential use of the active constrained layer damping treatment with its constraining layer made of the vertically reinforced 1 3 piezoelectric composite material for active control of geometrically nonlinear vibrations of functionally graded plates in the absence or presence of the temperature gradient across the thickness ofthe plates the analysis also reveals that the active constrained layer damping patch is more effective for controlling the nonlinear vibrations of functionally graded plates when it is attached to the softest surface of the functionally graded plates than when it is bonded to the stiffest surface of the plates the effect of piezoelectric fiber orientation in the active constraining layer on the control authority of the active constrained layer damping treatment has also been investigated

positive solutions of quasilinear parabolic systems with nonlinear boundary conditions abstract the aim of this paper is to investigate the existence uniqueness and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions including a system of quasilinear parabolic and ordinary differential equations also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady state solution the elliptic operators in both systems are allowed to be degenerate in the sense that the density dependent diffusion coefficients d i u i may have the property d i 0 0 for some or all i our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations it is shown that the time dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions and if the maximal and minimal solutions coincide then the steady state solution is unique and the time dependent solution converges to the unique solution applications of these results are given to three model problems including a porous medium type of problem a heat transfer problem and a two component competition model in ecology these applications illustrate some very interesting distinctive behavior of the time dependent solutions between density independent and density dependent diffusions

accelerated monotone iterative methods for finite difference equations of reaction diffusion this paper is concerned with numerical methods for a finite difference system of reaction diffusion convection equation under nonlinear boundary condition various monotone iterative methods are presented and each of these methods leads to an existence comparison theorem as well as a computational algorithm for numerical solutions the monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function application is given to a model problem from chemical engineering and some numerical results including a test problem with known analytical solution are presented to illustrate the various rates of convergence of the iterations

a wraparound elimination technique for the pseudospectral wave synthesis using an antiperiodic extension of the wavefield the pseudospectral method e g kosloff and baysal 1982 is an attractive alternative to numerical modeling schemes such as finite difference or finite element because it requires several orders of magnitude less computer memory and computation time in the pseudospectral method the field variables are expanded in terms of fourier interpolation polynomials which impose periodic boundary conditions on the equation of motion numerical differentiation is then implemented via the discrete fourier transform the periodicity condition causes the periodically extended wavefield on either side of the computational domain to propagate in from the sides interfering with the actual solution this phenomenon is called wraparound in many current applications e g cerjan et al 1985 an absorbing boundary condition has been used to damp the wraparound phases through a gradual reduction of the wavefield amplitude in the vicinity of the grid boundary such a method works effectively when the appropriate absorpt

accurate free vibration analysis of point supported mindlin plates by the superposition method the superposition method is exploited to obtain free vibration eigenvalues for thick mindlin plates resting on point supports the mindlin governing differential equations are satisfied exactly and boundary conditions are satisfied to any desired degree of accuracy by taking more terms in the solution series very good agreement is obtained with published results based on thin plate theory when the thickness ratio of the mindlin plates is allowed to take on values characteristic of thin plates eigenvalues are tabulated for a square plate resting on four point supports distributed symmetrically along the plate diagonals significant differences are encountered for higher modes when eigenvalues computed by the mindlin theory are compared with those based on the thin plate theory

vibrations of mindlin rectangular plates with elastically restrained edges using static timoshenko beam functions with the rayleigh ritz method abstract a set of static timoshenko beam functions is developed as the admissible functions for the free vibration analysis of mindlin rectangular plates with uniform elastic translational and or rotational edge constraints by the use of the rayleigh ritz method this set of beam functions is made up of two parts the transverse deflection functions and rotation functions which are the static solutions of a timoshenko beam under the transverse loads of a fourier sinusoidal series distributed along the length of the beam therefore the effect of shear deformation on the admissible functions is taken into account in fact the beam is considered to be a unit width strip taken from the rectangular plate in a direction parallel to the edge of the plate and has the end supports corresponding to the edge conditions of the particular plate under consideration it can be seen that the method is sound theoretically each of the beam functions is a polynomial function of not more than third order plus a sine or cosine function for all cases and the change of the boundary supports only results in a corresponding change of the coefficients of the low order polynomial therefore a unified program with little computational efforts can be easily developed for the vibration analysis of mindlin rectangular plates with arbitrary boundary conditions and thickness span ratios the accuracy and efficiency of the proposed method are demonstrated by comparison and convergency studies finally sets of reasonably accurate natural frequencies are presented which can serve as a supplement to the database of vibration characteristics of moderately thick plates

a nonreflecting boundary condition for discrete acoustic and elastic wave equations one of the nagging problems which arises in application of discrete solution methods for wave propagation calculations is the presence of reflections or wraparound from the boundaries of the numerical mesh these undesired events eventually override the actual seismic signals which propagate in the modeled region the solution to avoiding boundary effects used to be to enlarge the numerical mesh thus delaying the side reflections and wraparound longer than the range of times involved in the modeling obviously this solution considerably increases the expense of computation more recently nonreflecting boundary conditions were introduced for the finite difference method clayton and enquist 1977 reynolds 1978 these boundary conditions are based on replacing the wave equation in the boundary region by one way wave equations which do not permit energy to propagate from the boundaries into the numerical mesh this approach has been relatively successful except that its effectiveness degrades for event

dynamics of food chain models with density dependent diffusion and ratio dependent reaction function abstract this paper is concerned with a 3 species and a 2 species food chain reaction diffusion systems in a bounded domain where the diffusion coefficients may be density dependent and the reaction functions are ratio dependent these equations are quasilinear where the diffusion coefficients may be degenerate on the boundary of the domain three basic types of dirichlet neumann and robin boundary conditions are considered and in each case some very simple conditions are obtained to ensure the dynamical behavior of the time dependent solution in relation to some positive solutions or quasi solutions of the steady state problem including the existence of these solutions this dynamical behavior leads to the coexistence and global attractor of the food chain systems in the case of neumann boundary condition sufficient conditions are given to ensure that the steady state problem has a unique positive constant solution which is a global attractor of the time dependent system

global existence and blow up problems for reaction diffusion model with multiple nonlinearities abstract in this paper we study the following reaction diffusion model u t a u b x c t u g x t f u in d 0 t u n h x t r u on d 0 t u x 0 u 0 x 0 in d where d is a bounded domain in r n with smooth boundary d n 2 this paper deals with interactions among three kinds of nonlinear mechanisms nonlinear reaction nonlinear convection and nonlinear boundary flux the existence theorems of blow up positive solutions upper bound of blow up time upper estimates of blow up rate existence theorems of global positive solutions and upper estimates of global positive solutions are given under suitable assumptions on a b c f g h r and initial data u 0 x

a modified l scheme to solve nonlinear diffusion problems abstract in this work we propose a linearization technique for solving nonlinear elliptic partial differential equations that are obtained from the time discretization of a wide variety of nonlinear parabolic problems the scheme is inspired by the l scheme which gives unconditional convergence of the linear iterations here we take advantage of the fact that at a particular time step the initial guess for the iterations can be taken as the solution of the previous time step first it is shown for quasilinear equations that have linear diffusivity that the scheme always converges irrespective of the time step size the spatial discretization and the degeneracy of the associated functions moreover it is shown that the convergence is linear with convergence rate proportional to the time step size next for the general case it is shown that the scheme converges linearly if the time step size is smaller than a certain threshold which does not depend on the mesh size and the convergence rate is proportional to the square root of the time step size finally numerical results are presented that show that the scheme is at least as fast as the modified picard scheme faster than the l scheme and is more stable than the newton or the picard scheme

dynamics of degenerate quasilinear reaction diffusion systems with nonnegative initial functions abstract this paper is concerned with a system of quasilinear reaction diffusion equations with density dependent diffusion coefficients and mixed quasimonotone reaction functions the equations are allowed to be degenerate and the boundary conditions are of the nonlinear type the main goals are to prove the existence and uniqueness of the weak solution between a pair of coupled upper and lower solutions show that the weak solution evolves into the classical solution and analyze the asymptotic behavior of the solution using quasi solutions of the steady state system the general results are applied to a degenerate lotka volterra competition model conditions are given for the solution to exist globally to evolve into the classical solution and to be attracted into a sector formed by quasi solutions of the elliptic system especially for the neumann problem we give a simple condition for the solution to converge to a unique constant steady state solution which is a global attractor

existence and dynamics of quasilinear parabolic systems with time delays abstract this paper is concerned with a coupled system of quasilinear parabolic equations where the effect of time delays is taken into consideration in the reaction functions of the system the partial differential operators in the system may be degenerate and the reaction functions possess some mixed quasimonotone property including quasimonotone nondecreasing functions the aim of the paper is to show the existence and uniqueness of a global solution to the parabolic system the existence of positive quasisolutions or maximal minimal solutions of the corresponding elliptic system and the asymptotic behavior of the solution of the parabolic system in relation to the quasisolutions or maximal minimal solutions of the elliptic system applications are given to three reaction diffusion models arising from mathematical biology and ecology where the diffusion coefficients are density dependent and are degenerate this degenerate density dependent diffusion leads to some interesting distinct asymptotic behavior of the time dependent solution when compared with density independent diffusion

free vibration analysis by the superposition method of rectangular mindlin plates with internal columns resting on uniform elastic edge supports free vibration analysis of rectangular mindlin plates is carried out via the superposition method and is shown to produce accurate results the analytical method proposed by the authors in this paper solves the problem for the case where the plate has simultaneous elastic edge and internal supports the conditions of edge supports are uniform lateral rotational and torsional elastic supports whereas the internal supports are column supports with finite area compatibility between the plate and column is achieved by requiring that the column and plate rotations be equal the results presented herein are verified through comparison with results presented by others numerical examples presented in this study confirm that the analytical method is able to model the property of the elastic edge supports while simultaneously considering the effect of column restraint an effect which increases with column number and area

some global uniqueness and solvability results for linear complementarity problems over symmetric cones the purpose of this paper is to give a numerical treatment for a class of quasi linear elliptic equations under nonlinear boundary conditions including the three basic types of linear boundary conditions the quasi linear equation is discretized by the finite difference method and the method of upper lower solutions and its associated monotone iteration are used to compute the solutions of the finite difference system this method leads to monotone iterative schemes for the computation of numerical solutions as well as some comparison results among the monotone iterative schemes it also leads to the existence of a maximal and a minimal finite difference solution including the uniqueness of the solution and the convergence of the finite difference solution to the corresponding continuous solution applications are given to two physical problems in heat conduction and combustion theory and numerical results for the heat conduction problem are given and are compared with the known true continuous solution

qualitative properties of discrete nonlinear parabolic operators this paper is devoted to the qualitative properties of discretized parabolic operators such as nonnegativity and nonpositivity preservation maximum minimum principles and maximum norm contractivity in the linear case earlier papers of the authors farago and horvath in siam sci comput 28 2313 2336 2006 ima j numer anal 29 606 631 2009 have established the connections between the above qualitative properties and have given sufficient conditions for their validity the present paper extends the above results to nonlinear discretized parabolic operators also motivated by the authors recent paper farago and horvath in j math anal appl 448 473 497 2017 which has given related results on the continuous pde level a systematic study is presented ranging from general discrete mesh operators to proper finite element applications

continuity of solutions of a singular parabolic equation which is of parabolic divergence form since 3 0 a e however it is the solution u of 0 1 which will be continuous while the solution u of 0 2 need not be in certain physical applications u corresponds to temperature while u is the enthalpy the difficulty in the analysis of this equation written in the form 0 2 stems from the fact that c need not be bounded or bounded away from zero thus the equation may be of degenerate or singular parabolic type as particular cases of the equations covered by our results we mention the three model problems

the method of upper and lower solutions for diffraction problems of quasilinear elliptic reaction diffusion systems abstract the aim of this paper is to show the existence of solutions of the n dimensional diffraction problem for weakly coupled quasilinear elliptic reaction diffusion system the coefficients of the equations under consideration are allowed to be discontinuous we extend the method of upper and lower solutions for reaction diffusion equations with continuous coefficients to the elliptic diffraction problem an application of these results is given to the steady state problem of lotka volterra cooperation model with two cooperating species

positive solutions of quasilinear parabolic systems with dirichlet boundary condition abstract coupled systems for a class of quasilinear parabolic equations and the corresponding elliptic systems including systems of parabolic and ordinary differential equations are investigated the aim of this paper is to show the existence uniqueness and asymptotic behavior of time dependent solutions also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system the elliptic operators in both systems are allowed to be degenerate in the sense that the density dependent diffusion coefficients d i u i may have the property d i 0 0 for some or all i 1 n and the boundary condition is u i 0 using the method of upper and lower solutions we show that a unique global classical time dependent solution exists and converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions and if the maximal and minimal solutions coincide then the steady state solution is unique and the time dependent solution converges to the unique solution applications of these results are given to three model problems including a scalar polynomial growth problem a coupled system of polynomial growth problem and a two component competition model in ecology

some degenerate and quasilinear parabolic systems not in divergence form abstract this paper deals with positive solutions of degenerate and quasilinear parabolic systems not in divergence form u t u p u av v t v q v bu with null dirichlet boundary conditions and positive initial conditions where p q a and b are all positive constants the local existence and uniqueness of classical solution are proved moreover it will be proved that all solutions exist globally if and only if ab 1 2 where 1 is the first eigenvalue of in with homogeneous dirichlet boundary condition

viscosity solutions of a class of degenerate quasilinear parabolic equations with dirichlet boundary condition abstract this paper is concerned with viscosity solutions for a class of degenerate quasilinear parabolic equations in a bounded domain with homogeneous dirichlet boundary condition the equation under consideration arises from a number of practical model problems including reaction diffusion processes in a porous medium the degeneracy of the problem appears on the boundary and possibly in the interior of the domain the goal of this paper is to establish some comparison properties between viscosity upper and lower solutions and to show the existence of a continuous viscosity solution between them an application of the above results is given to a porous medium type of reaction diffusion model which demonstrates some distinctive properties of the solution when compared with the corresponding semilinear problem

forward modeling by a fourier method a fourier or pseudospectral forward modeling algorithm for solving the two dimensional acoustic wave equation is presented the method utilizes a spatial numerical grid to calculate spatial derivatives by the fast fourier transform time derivatives which appear in the wave equation are calculated by second order differcncing the scheme requires fewer grid points than finite diffcrcnce methods to achieve the same accuracy it is therefore believed that the fourier method will prove more efficient than finitedifference methods especially when dealing with threedimensional models the fourier forward modeling method was tested against two problems a single layer problem with a known analytic solution and a wedge problem which was also tested by physical modeling the numerical results agreed with both the analytic and physical model results furthermore the numerical model facilitates the explanation of certain events on the time section of the physical model which otherwise could not easily be taken into account

monotone finite difference schemes for nonlinear systems with mixed quasi monotonicity monotone finite difference schemes are proposed for nonlinear systems with mixed quasi monotonicity two monotone iteration processes for the corresponding discrete problems are presented which converge monotonically to the quasi solutions of the discrete problems the limits are the exact solutions under some conditions a monotone finite difference scheme on uniform mesh with the accuracy of fourth order is constructed the numerical results coincide with theoretical analysis

a semi analytical dqem for free vibration analysis of thick plates with two opposite edges simply supported this paper is introducing a semi analytical differential quadrature element method dqem based on first order shear deformation theory for free vibration analysis of thick plates by decomposing the plate into a series of sub domains or elements any discontinuity in geometry material properties and elastic foundations in one direction can be handled conveniently classical as well as non classical boundary conditions can be imposed one benefits of this semi analytical dqem is reducing the storage requirements and the computational efforts needed to assemble the global equations the method would also increase the computational efficiency in comparison with the finite element method fem or differential quadrature method dqm analysis of such problems the accuracy versatility stability and convergence of the algorithm are certified

coexistence periodic solutions of a doubly nonlinear parabolic system with neumann boundary conditions abstract this paper is concerned with a competitive and cooperative mathematical model for two biological populations which dislike crowding diffuse slowly and live in a common territory under different kind of intra and inter specific interferences the model consists of a system of two doubly nonlinear parabolic equations with nonlocal terms and neumann boundary conditions based on the theory of the leray schauder degree we obtain the coexistence periodic solutions namely the existence of two non trivial non negative periodic solutions representing the densities of the two interacting populations under different intra and inter specific interferences on their natural growth rates

perfectly matched layers for time harmonic elastodynamics of unbounded domains theory and finite element implementation abstract one approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outwards from the bounded domain a perfectly matched layer pml is an unphysical absorbing layer model for linear wave equations that absorbs almost perfectly outgoing waves of all non tangential angles of incidence and of all non zero frequencies this paper develops the pml concept for time harmonic elastodynamics in cartesian coordinates utilising insights obtained with electromagnetics pmls and presents a novel displacement based symmetric finite element implementation of the pml for time harmonic plane strain or three dimensional motion the pml concept is illustrated through the example of a one dimensional rod on elastic foundation and through the anti plane motion of a two dimensional continuum the concept is explored in detail through analytical and numerical results from a pml model of the semi infinite rod on elastic foundation and through numerical results for the anti plane motion of a semi infinite layer on a rigid base numerical results are presented for the classical soil structure interaction problems of a rigid strip footing on a i half plane ii layer on a half plane and iii layer on a rigid base the analytical and numerical results obtained for these canonical problems demonstrate the high accuracy achievable by pml models even with small bounded domains

positive steady state solutions of a competing reaction diffusion system with large cross diffusion coefficients abstract this paper deals with a coupled system of reaction diffusion equations modeling the competition of two species with cross diffusion we discuss the existence of positive steady state solutions in relation to the intrinsic birth rates a 1 a 2 of species and the cross diffusion coefficients 12 21 our results show that positive steady state solutions exist if a 1 and a 2 lie in certain range or if 12 and 21 are sufficiently large

the supplementation of the theory of periodic solutions for a class of nonlinear diffusion equations abstract in this paper we provide supplementation of the theory of periodic solutions for a class of nonlinear diffusion equations u t u m a x t u p with m 0 p 0 by using a new method to deal with the blow up sequence we obtain the l estimate with optimal exponent 1 p m p s where ps is the sobolev critical exponent then we apply topological degree method to prove the existence of positive periodic solutions if p m p s n 3 and is star shaped we establish rellich pohozaev type identity to prove the nonexistence of positive periodic solutions we improve the results of mizoguchi 10 therem 1 by filling the gap m 1 p m 1 2 m n in addition we also give quite complete results about the existence of periodic solutions for the fast diffusion case 0

vibrations of rectangular plates with elastically restrained edges abstract the rayleigh ritz method is used to determined natural frequencies in transverse vibration of rectangular plates with elastically restrained edges by treating an elastically restrained edge as intermediate between an appropriate pair of classical boundary conditions and using the corresponding vibration mode shapes of beams with classical boundary conditions as assumed functions a relatively small number of functions is required consequently only a modest quantity of computation is necessary the good accuracy of the method is demonstrated by solving test problems the method can be applied to a wide range of elastic restraint conditions any aspect ratio and for higher modes in addition to the fundamental the usefulness and accuracy of existing simplified approaches to the problem are assessed the effect of in plane forces on the natural frequencies and the determination of critical loads for plates with these restraint conditions are considered also

determination of an unknown coefficient in a nonlinear heat equation abstract the problem of determining an unknown term k u in the equation k u u t k u u x x is considered in this paper applying tikhonov s regularization approach we develop a procedure to find an approximate stable solution to the unknown coefficient from the overspecified data

postbuckling behavior of bi axially compressed arbitrarily straight sided quadrilateral functionally graded material plates abstract in this paper the problem of the postbuckling behavior of biaxial compressed straight sided functionally graded material fgm plates of quadrilateral shape is studied the plate considered is subjected to in plane loads on all four edges a computational framework based on an improved moving least squares imls approximation for the field variables is developed for the analysis the solution procedures involved a transformation from the physical domain to a computational domain and then discrete the nonlinear governing equation using the imls ritz method the first order shear deformation theory fsdt with the von karman nonlinearity is employed a nonlinear solution to the postbuckling of quadrilateral fgm plates is computed through the modified newton raphson method combined with the arc length iterative algorithm a stabilized conforming nodal integration scheme is employed to improve computational efficiency and eliminate shear and membrane locking the validity and accuracy of the numerical results are established through convergence studies to the best of the authors knowledge the problem has not been attempted in the open literature

singular reaction diffusion equations of porous medium type abstract a class of reaction diffusions equations with nonlinear boundary conditions and porous medium type of diffusion is investigated by the method of upper and lower solutions the diffusion term in the equation includes the so called slow diffusion m 1 fast diffusion 0 m 1 and super diffusion m 0 and the reaction term may involve a singular function the aim of this paper is to show the existence of a global time dependent solution the existence of positive minimal and maximal steady state solutions including the uniqueness of the positive solution and the asymptotic behavior of the time dependent solution in relation to the positive steady state solutions of particular concern is the convergence of the time dependent solution to a unique positive steady state solution and the global attraction property of the reaction diffusion problem applications are given to several specific problems which are of considerable interest in the current literature

numerical methods for semilinear parabolic equations the method of upper lower solutions for continuous parabolic equations is extended to some finite difference system for numerical solutions the idea of this method is that by using the upper or lower solution as the initial iteration one can obtain a monotone sequence that converges to a unique solution of the problem the aim of this paper is to present two iterative schemes for the construction of the monotone sequence and to show that both schemes are numerically stable these two schemes are modified jacobi method and gauss seidel method for nonlinear algebraic equations an advantage of this approach is that each of the two methods yields an error estimate between the true solution and the computed mth iteration on the other hand the standard picard type of iterative scheme is used to show that the finite difference system converges to the continuous parabolic equations

global attractors of some predator prey reaction diffusion systems with density dependent diffusion and time delays abstract this paper deals with a two species and a three species predator prey reaction diffusion systems where the diffusion coefficients are density dependent and time delays are involved in the reaction functions the diffusion terms are of porous medium type which are degenerate and the boundary conditions are of neumann type the aim of the paper is to investigate the asymptotic behavior of the time dependent solution in relation to various steady state solutions of the system this includes the existence of a unique positive classical solution global attraction of a steady state solution and stability or instability of various semitrivial solutions and the positive steady state solution

vibration analysis of rectangular mindlin plates resting on elastic edge supports for the first time to the authors knowledge the problem of free vibration of a moderately thick rectangular plate with edges elastically restrained against transverse and rotational displacements is considered the ritz method combined with a variational formulation and mindlin plate theory is used the admissible functions consist of polynomials and basic functions that impose the required boundary conditions on the mindlin plate the applicability of the formulation is illustrated using three examples of plates with different combinations of elastically restrained edges and classical boundary conditions numerical results are obtained to investigate the effects of elastic spring stiffness relative thickness and aspect ratio upon the natural frequencies of flexural vibration of rectangular mindlin plates

existence theory for one dimensional quasilinear coupled conductive radiative flows abstract the paper deals with the coupled conductive radiative problem with a diffusion coefficient depending on the temperature the technique of upper and lower solutions is used to generate a solution for this nonlinear problem in the space of holder continuous function by pao 1992 2007 1 2 together with certain integral representations given in azevedo et al 2011 3 we also produce numerical results using gfd n method the green function decomposition of the order n coupled with the crank nicolson method and the newton raphson method the gfd n methodology arises from the integral representation involved and does not involve any a priori discretization on the angular variable

quasilinear parabolic and elliptic equations with nonlinear boundary conditions abstract this paper is concerned with a class of quasilinear parabolic and elliptic equations in a bounded domain with both dirichlet and nonlinear neumann boundary conditions the equation under consideration may be degenerate or singular depending on the property of the diffusion coefficient the consideration of the class of equations is motivated by some heat transfer problems where the heat capacity and thermal conductivity are both temperature dependent the aim of the paper is to show the existence and uniqueness of a global time dependent solution of the parabolic problem existence of maximal and minimal steady state solutions of the elliptic problem including conditions for the uniqueness of a solution and the asymptotic behavior of the time dependent solution in relation to the steady state solutions applications are given to some heat transfer problems and an extended logistic reaction diffusion equation

absorbing boundary conditions and surface waves one of the major problems in numerically simulating waves traveling in the earth is that an artificial boundary must be introduced to produce unique solutions to eliminate the spurious reflections introduced by this artificial boundary we use a damping expression based on analogies to shock absorbers this method can reduce the amplitude of the reflected wave to any pre specified value and is successful for waves at any angle of incidence the method can eliminate unwanted reflections from the surface reflections at the corners of the model and waves reflected off an interface that strike the artificial boundary many of the boundary conditions currently used in the numerical solution of waves are approximations to perfectly absorbing boundary conditions and depend upon the angle of incidence of the incoming wave at the artificial boundary stability problems often occur with these boundary conditions the method we use at the artificial boundary allows use of stable dirichlet or von neumann condition

quasilinear parabolic and elliptic systems with mixed quasimonotone functions abstract this paper deals with a class of quasilinear parabolic and elliptic systems with mixed quasimonotone reaction functions the boundary condition in the system may be dirichlet nonlinear or a combination of these two types the elliptic operators in the system are allowed to be degenerate the aim is to show the existence and uniqueness of a classical solution to the parabolic system the existence of maximal and minimal solutions or quasisolutions of the elliptic system and the asymptotic behavior of the solution of the parabolic system this consideration leads to a global attractor of the parabolic system as well as an one sided stability of the maximal and minimal solutions applications of these results are given to three models arising from biology and ecology where diffusion coefficients are density dependent and are degenerate these applications exhibit quite distinct dynamical behavior of the population species between degenerate density dependent diffusion and constant diffusion

coexistence and optimal control problems for a degenerate predator prey model abstract in this paper we present a predator prey mathematical model for two biological populations which dislike crowding the model consists of a system of two degenerate parabolic equations with nonlocal terms and drifts we provide conditions on the system ensuring the periodic coexistence namely the existence of two non trivial non negative periodic solutions representing the densities of the two populations we assume that the predator species is harvested if its density exceeds a given threshold a minimization problem for a cost functional associated with this process and with some other significant parameters of the model is also considered

strongly coupled elliptic systems and applications to lotka volterra models with cross diffusion abstract the aim of this paper is to investigate the existence and method of construction of solutions for a general class of strongly coupled elliptic systems by the method of upper and lower solutions and its associated monotone iterations the existence problem is for nonquasimonotone functions arising in the system while the monotone iterations require some mixed monotone property of these functions applications are given to three lotka volterra model problems with cross diffusion and self diffusion which are some extensions of the classical competition prey predator and cooperating ecological systems the monotone iterative schemes lead to some true positive solutions of the competition system and to quasisolutions of the prey predator and cooperating systems also given are some sufficient conditions for the existence of a unique positive solution to each of the three model problems

accurate free vibration analysis of shear deformable plates with torsional elastic edge support abstract the modified superposition galerkin method is utilized to solve the free vibration problem of shear deformable plates resting on uniform elastic foundations lateral displacement of the plate is forbidden at the boundaries due to elasticity in the support edge rotation is opposed by bending moments proportional to the degree of rotation results of a study conducted on thick mindlin plates are presented the analytical procedure is found to be much more efficient than that involved in the traditional superposition method and good convergence is encountered

finite difference reaction diffusion equations with nonlinear diffusion coefficients a monotone iterative method for numerical solutions of a class of finite difference reaction diffusion equations with nonlinear diffusion coefficient is presented it is shown that by using an upper solution or a lower solution as the initial iteration the corresponding sequence converges monotonically to a unique solution of the finite difference system it is also shown that the solution of the finite difference system converges to the solution of the continuous equation as the mesh size decreases to zero

the flexural vibration of rectangular plates with point supports abstract orthogonally generated polynomial functions are used in the lagrangian multiplier method to study the free flexural vibration problem of point supported thin flat rectangular plates the analysis applies to isotropic and specially orthotropic plates having any combination of clamped simply supported or free edges with arbitrarily located point supports and to plates which are continuous over line supports parallel to the plate edges numerical results are presented for a number of specific problems illustrating the accuracy and versatility of the approach and which include natural frequencies and nodal patterns for a point supported plate which is continuous over two perpendicular line supports

a class of quasi linear parabolic and elliptic equations with nonlocal robin boundary conditions abstract let p 1 n r n a bounded w 1 p extension domain and let be an upper d ahlfors measure on with d n p n we show in the first part that for every p 2 n n 2 n 1 n a realization of the p laplace operator with nonlinear generalized nonlocal robin boundary conditions generates a nonlinear strongly continuous submarkovian semigroup on l 2 and hence the associated first order cauchy problem is well posed on l q for every q 1 in the second part we investigate existence uniqueness and regularity of weak solutions to the associated quasi linear elliptic equation more precisely global a priori estimates of weak solutions are obtained

convergence of finite difference schemes for nonlinear complex reaction diffusion processes this paper is devoted to the proof of the convergence properties of a class of finite difference schemes applied to nonlinear complex reaction diffusion equations we investigate the accuracy of the numerical solution considering implicit and semi implicit discretizations to illustrate the theoretical results we present some numerical examples computed with a semi implicit scheme applied to a nonlinear equation

free vibration analysis of rectangular plates on elastic intermediate supports a procedure using the finite strip element method in combination with a spring system is proposed to treat the free vibration analysis of plates on elastic intermediate supports results indicate that the spring system can successfully simulate elastic intermediate supports such as point supports line supports local uniformly distributed supports and mixed edge supports from the results it is also evident that support stiffness and support areas have significant influence on the free vibration response of plates on line supports and local uniformly distributed supports

existence of positive solutions for elliptic systems degenerate and nondegenerate ecological models with homogeneous dirichlet conditions w o on dd the function s satisfies the conditions rl e c o oo q o and s o for s o problems of this nature are of interest in reaction diffusion processes in biology and chemistry for example the case for u urn m 1 or m e 0 1 for single parabolic equations i e u du f x u has been studied recently for porous medium analysis and population dynamics cf 2 7 15 18 191 as t cc these solutions tend to a solution of the corresponding elliptic scalar equation studies have also been carried out in these and other papers e g 3 4 16 171 with urn replaced by u satisfying the conditions described above in this article however the hypotheses on f are quite different from those in the papers mentioned above for example f is not necessarily lipschitz in u and may depend discontinuously on x thus even for the scalar case the results here are not covered by the other papers although there are some overlaps moreover our emphasis here is on degenerate systems section 3 and their applications section 4 these results are quite different and generalize the papers mentioned above to practical interesting cases other relevant results for the nondegenerate cases can be found in s 9 111 in section 2 we discuss some existence and uniqueness theorems for

global existence for nonlinear diffusion equations solutions of nonlinear possibly degenerate reaction diffusion models are known to exist for all time if the asymptotic growth of the reaction term is not greater than that of the diffusion term via concavity methods it is also known that negating such a condition results in solutions which blow up in finite time on the other hand two recent studies regarding the case of linear diffusion have reported on the ability for convective terms in a model to create global existence of solutions where no such result is true in their absence in this paper we develop a theory of global existence for a general class of reaction diffusion convection models which only requires a similar balance of diffusion and reaction our result extends previous work to models including convection but it does not reveal the exact role of convection in yielding global solutions further analysis of a slightly simplified model establishes the global existence of all solutions if the reaction term grows asymptotically at a rate less than that of either diffusion or convection

a lotka volterra cooperating reaction diffusion system with degenerate density dependent diffusion abstract in the lotka volterra cooperating reaction diffusion system if the diffusion coefficients are constants then for a certain set of reaction rates in the reaction function the solution of the system blows up in finite time and for another set of reaction rates a unique global solution exists and converges to the trivial solution however if the diffusion coefficients are density dependent then the dynamic behavior of the solution can be quite different the aim of this paper is to investigate the global existence and the asymptotic behavior of the solution for a class of density dependent cooperating reaction diffusion systems where the diffusion coefficients are degenerate it is shown that the time dependent problem has a unique bounded global solution and in addition to the trivial and semi trivial solutions the corresponding steady state problem has a positive maximal solution and a positive minimal solution moreover the time dependent solution converges to the maximal solution for one class of initial functions and to the minimal solution for another class of initial functions the above convergence property holds true for any reaction rates in the reaction function applications of the above results are given to a porous medium type of reaction diffusion problem as well as other types of diffusion coefficients including the finite sum and products of these functions

free vibration analysis of rectangular mindlin plates with elastic restraints uniformly distributed along the edges abstract free vibration analysis of isotropic mindlin plates with edges elastically restrained against rotation and translation is carried out by a variational method the elastic restraints are assumed to be uniformly distributed along the edges elastically restrained timoshenko beam functions are used as constitutive shape functions and the boundary conditions not satisfied remain as boundary terms in the final variational energy expression the results for reduced problems are compared with those available in the literature and found to be in good agreement results showing the effect of elastic restraints on the natural frequencies of a thick plate indicate that the combined variation of rotational and translational restraint is more significant than their independent variations variations in mode shapes caused by the changes in the boundary conditions are also presented

numerical analysis of coupled systems of nonlinear parabolic equations this paper is concerned with numerical solutions of a general class of coupled nonlinear parabolic equations by the finite difference method three monotone iteration processes for the finite difference system are presented and the sequences of iterations are shown to converge monotonically to a unique solution of the system including an existence uniqueness comparison theorem a theoretical comparison result for the various monotone sequences and an error analysis of the three monotone iterative schemes are given also given is the convergence of the finite difference solution to the continuous solution of the parabolic boundary value problem an application to a reaction diffusion model in chemical engineering and combustion theory is given

the heat equation with nonlinear generalized robin boundary conditions abstract let r n be a bounded domain and let be an admissible measure on we show in the first part that if has the h 1 extension property then a realization of the laplace operator with generalized nonlinear robin boundary conditions formally given by u d x u d 0 on generates a strongly continuous nonlinear submarkovian semigroup s b s b t t 0 on l 2 we also obtain that this semigroup is ultracontractive in the sense that for every u v l p p 2 and every t 0 one has s b t u s b t v c 1 e c 2 t t n 2 p u v p for some constants c 1 c 2 0 in the second part we prove that if is a bounded lipschitz domain one can also define a realization of the laplacian with nonlinear robin boundary conditions on c and this operator generates a strongly continuous and contractive nonlinear semigroup

free vibration analysis of doubly curved shallow shells using the superposition galerkin method abstract this paper demonstrates the applicability of the superposition method for free vibration analysis of doubly curved thin shallow shells of rectangular planform with any possible combination of simply supported and clamped edges the same building block yields the natural frequencies for 55 combinations of edge conditions the natural frequency parameters of the shells were obtained using the superposition galerkin method sgm for seven sets of boundary conditions several different curvature ratios and two aspect ratios the sgm uses approximate steady state solutions as building blocks but the method proves to be accurate and efficient it has also been shown that even with approximate building blocks the monotonic nature of convergence of the natural frequencies with respect to the number of driving coefficients holds as long as the number of admissible functions in the steady state solution is kept constant the results for natural frequencies of the seven boundary conditions may be considered as benchmarks

mixed covolume methods for quasi linear second order elliptic problems we consider covolume methods for the mixed formulations of quasi linear second order elliptic problems covolume methods for the mixed formulations of linear elliptic problem was first considered by russell rigorous block centered discretizations on irregular grids improved simulation of complex reservoir systems tech report 3 project report reservoir simulation research corporation tulsa ok 1995 and tested extensively in cai et al comput geosci 1 1997 pp 289 315 jones a mixed finite volume element method for accurate computation of fluid velocities in porous media ph d thesis university of colorado denver 1995 the analysis was carried out by chou and kwak siam j numer anal 37 2000 pp 758 771 for linear symmetric problems where they showed optimal error estimates in l2 norm for the pressure and in h div norm for the velocity in this paper we extend their results to quasi linear problems by following milner s argument math comp 44 1985 pp 303 320 through an adaptation of the duality argument of douglas and roberts math comp 44 1985 pp 39 52 for mixed covolume methods