initial boundary value problems for a class of pseudoparabolic equations with integral boundary conditions in this paper we deal with a class of pseudoparabolic problems with integral boundary conditions we will first establish an a priori estimate then we prove the existence uniqueness and continuous dependence of the solution upon the data finally some extensions of the problem are given

on a comparison principle for delay coupled systems with nonlocal and nonlinear boundary conditions abstract this paper is concerned with delay coupled systems of parabolic equations with nonlocal and nonlinear boundary conditions for them a new and general comparison principle is established which is more general and useful than the existing results

convergence analysis of compact difference schemes for diffusion equation with nonlocal boundary conditions difference schemes for solving the diffusion equation with nonlocal boundary conditions are considered compact difference is used and the integrals are approximated by the composite simpson scheme convergence of the two compact schemes are derived using the energy method compact difference schemes for solving the diffusion equation with nonlocal boundary conditions are considered in this paper fourth order compact difference is used to approximate the second order spatial derivative and the integrals in the boundary conditions are approximated by the composite simpson quadrature formula the backward euler and crank nicolson schemes are presented as the fully discrete schemes error estimates in the discrete h1 and l norms are given by the energy method showing both schemes are fourth order accurate in space and they have first order and second order accuracy in time respectively numerical results are provided to confirm the theoretical results

geometry of the modulational instability iii homoclinic orbits for the periodic sine gordon equation in this paper the homoclinic geometric structure of the integrable sine gordon equation under periodic boundary conditions is developed specifically focus is given to orbits homoclinic to n tori simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given a labeling is provided which identifies and catalogues all such orbits these orbits are related in a one to one manner to linearized instabilities explicit formulas for all homoclinic orbits are given in terms of backlund transformations

numerical algorithms for diffusion reaction problems with non classical conditions abstract parabolic equations with nonlocal boundary conditions have been given considerable attention in recent years in this paper new high order algorithms for the linear diffusion reaction problem are derived the convergence of the new schemes is studied and numerical examples are given to show the efficiency of the new methods to solve linear and nonlinear diffusion reaction equations with these non classical conditions

nonlocal problems involving spatial structure for coupled reaction diffusion systems the purpose of this work is to investigate the uniqueness and existence of solutions for a system of nonlinear parabolic equations weakly coupled with ordinary differential equations in bounded domains subject to nonlocal boundary conditions that involve values of the unknown functions inside of the spatial domain the solutions of such problems allow for considering not only physical chemical or biological properties but also spatial structure of the modelled system the uniqueness of classical solutions of the nonlocal boundary problems for coupled reaction diffusion systems is proved by means of comparison principles for differential inequalities the existence of the unique solution is obtained via a monotone iterative method applications are given to some model problems in epidemiology and ecology with the nonlocal boundary conditions based on the inverse distance weighting interpolation and cokriging

nonlinear fourth order elliptic equations with nonlocal boundary conditions this paper is concerned with a class of fourth order nonlinear elliptic equations with nonlocal boundary conditions including a multi point boundary condition in a bounded domain of rn also considered is a second order elliptic equation with nonlocal boundary condition and the usual multi point boundary problem in ordinary differential equations the aim of the paper is to show the existence of maximal and minimal solutions the uniqueness of a positive solution and the method of construction for these solutions our approach to the above problems is by the method of upper and lower solutions and its associated monotone iterations the monotone iterative schemes can be developed into computational algorithms for numerical solutions of the problem by either the finite difference method or the finite element method

numerical methods for fourth order elliptic equations with nonlocal boundary conditions this paper is concerned with some numerical methods for a fourth order semilinear elliptic boundary value problem with nonlocal boundary condition the fourth order equation is formulated as a coupled system of two second order equations which are discretized by the finite difference method three monotone iterative schemes are presented for the coupled finite difference system using either an upper solution or a lower solution as the initial iteration these sequences of monotone iterations called maximal sequence and minimal sequence respectively yield not only useful computational algorithms but also the existence of a maximal solution and a minimal solution of the finite difference system also given is a sufficient condition for the uniqueness of the solution this uniqueness property and the monotone convergence of the maximal and minimal sequences lead to a reliable and easy to use error estimate for the computed solution moreover the monotone convergence property of the maximal and minimal sequences is used to show the convergence of the maximal and minimal finite difference solutions to the corresponding maximal and minimal solutions of the original continuous system as the mesh size tends to zero three numerical examples with different types of nonlinear reaction functions are given in each example the true continuous solution is constructed and is used to compare with the computed solution to demonstrate the accuracy and reliability of the monotone iterative schemes

stability of the weighted splitting finite difference scheme for a two dimensional parabolic equation with two nonlocal integral conditions nonlocal conditions arise in mathematical models of various physical chemical or biological processes therefore interest in developing computational techniques for the numerical solution of partial differential equations pdes with various types of nonlocal conditions has been growing fast we construct and analyse a weighted splitting finite difference scheme for a two dimensional parabolic equation with nonlocal integral conditions the main attention is paid to the stability of the method we apply the stability analysis technique which is based on the investigation of the spectral structure of the transition matrix of a finite difference scheme we demonstrate that depending on the parameters of the finite difference scheme and nonlocal conditions the proposed method can be stable or unstable the results of numerical experiments with several test problems are also presented and they validate theoretical results

an application of symmetry groups to nonlocal continuum mechanics in this study the symmetry group properties of the one dimensional elastodynamics problem in nonlocal continuum mechanics is discussed by using an approach developed for symmetry group analysis of integro differential equations with general form this approach is based on the modification of the invariance criterion of the differential equations which include nonlocal variables and integro differential operators lie point symmetries of the nonlocal elasticity equation are obtained based on solving nonlocal determining equations by using a new approach the symmetry groups for different types of kernel function and the free term including the classical linear elasticity case are presented

new method based on the hpm and rkhsm for solving forced duffing equations with integral boundary conditions this paper investigates the forced duffing equation with integral boundary conditions its approximate solution is developed by combining the homotopy perturbation method hpm and the reproducing kernel hilbert space method rkhsm hpm is based on the use of the traditional perturbation method and the homotopy technique the hpm can reduce nonlinear problems to some linear problems and generate a rapid convergent series solution in most cases rkhsm is also an analytical technique which can solve powerfully linear boundary value problems therefore the forced duffing equation with integral boundary conditions can be solved using advantages of these two methods two numerical examples are presented to illustrate the strength of the method

influence of solitons in the initial state on chaos in the driven damped sine gordon system abstract the appearance of chaos in the a c driven damped sine gordon equation is studied numerically several transitions from periodic to chaotic behaviour are investigated in detail for flat initial conditions spatial structures breather kink in the initial conditions smooth out many of these transitions and give rise to an interesting symbiosis of time and spatial intermittency this symbiosis appears to be due to the competition between the background tendency towards chaos and the system s preference to maintain a spatial pattern the way that this competition is relieved is also found to depend very strongly on symmetry in the initial conditions

weak solutions for nonlocal boundary value problems with low regularity data abstract this paper concerns the existence and uniqueness of weak solutions for elliptic and parabolic equations under nonlocal boundary conditions based on maximal regularity it also gives the positivity of solutions which can be used in monotone iteration methods as an application the results are used to discuss some specific nonlocal problems

reaction diffusion equations with nonlocal boundary and nonlocal initial conditions in this paper we investigate the existence multiplicity and asymptotic behavior of solutions for a reaction diffusion equation under nonlocal boundary and nonlocal initial conditions a distinctive feature of this type of boundary and initial conditions is that the parabolic equation may possess multiple solutions even if the reaction function is smooth including linear functions of special concern is the asymptotic behavior of these multiple solutions in relation to the solutions of the corresponding steady state problem which may also possess multiple solutions in some special cases both the time dependent and the steady state problems possess an infinite number of solutions and each of the time dependent solutions converges to one of the steady state solutions

spatially structured activity in synaptically coupled neuronal networks ii lateral inhibition and standing pulses we consider the existence and stability of standing pulse solutions to a system of integro differential equations used to describe the activity of synaptically coupled networks of excitatory and inhibitory neurons in a single spatial domain assuming an arrangement of synaptic connections described by lateral inhibition previous formal arguments have demonstrated the existence of both stable and unstable standing pulses s amari biol cybern 27 1977 pp 77 87 these results have formed the basis for several recent hypotheses regarding the generation of sustained activity patterns in prefrontal cortex and other brain regions implicit in the lateral inhibition arrangement however is the assumption that the dynamics of inhibition are instantaneous here we present two arguments demonstrating the loss of stability of standing pulse solutions through a hopf bifurcation when more realistic inhibitory dynamics are considered the first argument parallels amari s formal presentation while the seco

stability of difference schemes for two dimensional parabolic equations with non local boundary conditions the stability of difference schemes for one dimensional and two dimensional parabolic equations subject to non local bitsadze samarskii type boundary conditions is dealt with to analyze the stability of difference schemes the structure of the spectrum of the matrix that defines the linear system of difference equations for a respective stationary problem is studied depending on the values of parameters in non local conditions this matrix can have one zero one negative or complex eigenvalues the stepwise stability is proved and the domain of stability of difference schemes is found

finite difference methods for solution of nonlocal boundary value problems the article is devoted to the construction and investigation of various types of difference schemes relevant to nonlocal boundary value problems stated for multidimensional elliptic equations suitable tools for performing the investigation are developed including difference analogue of green s formula and inequalities corresponding to nonlocal boundary conditions

on a singular two dimensional nonlinear evolution equation with nonlocal conditions abstract this paper deals with a nonclassical initial boundary value problem for a two dimensional parabolic equation with bessel operator we prove the existence and uniqueness of the weak solution of the given nonlinear problem we start by solving the associated linear problem after writing this latter in its operator form we establish an a priori bound from which we deduce the uniqueness of the strong solution for the solvability of the associated linear problem we prove that the range of the operator generated by the considered problem is dense on the basis of the obtained results of the linear problem we apply an iterative process to establish the existence and uniqueness of the nonlinear problem

updating the inverse of a matrix the sherman morrison woodbury formulas relate the inverse of a matrix after a small rank perturbation to the inverse of the original matrix the history of these fomulas is presented and various applications to statistics networks structural analysis asymptotic analysis optimization and partial differential equations are discussed the sherman morrison woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix this paper surveys the history of these formulas and we examine some applications where these formulas are helpful

numerical solution of the heat equation with nonlocal boundary conditions abstract the method for the heat equation with nonlocal boundary conditions is discussed in this paper the unconditional stability is proved for 1 2 subject to a condition that is much weaker than the one assumed in a paper by ekolin due to the nonlocal boundary conditions the systems of linear equations generated by the method have a coefficient matrix that is tridiagonal except its first and last rows three efficient algorithms for solving this kind of linear systems are presented a simple numerical example is given to compare their efficiency

on large toeplitz band matrices with an uncertain block abstract this paper investigates the possible spectra of large finite dimensional toeplitz band matrices with perturbations impurities uncertainties in the upper left m m block the main result shows that the asymptotic spectrum of such a matrix is not affected by these perturbations provided they have sufficiently small norm this follows from analysis of structured pseudospectra structured spectral value sets in contrast for typical non hermitian toeplitz matrices there exist certain rank one perturbations of arbitrarily small norm that move an eigenvalue away from the asymptotic spectrum in the large dimensional limit

muntz systems and orthogonal muntz legendre polynomials the muntz legendre polynomials arise by orthogonalizing the muntz system xio x l with respect to lebesgue measure on 0 1 in this paper differential and integral recurrence formulae for the muntz legendre polynomials are obtained interlacing and lexicographical properties of their zeros are studied and the smallest and largest zeros are universally estimated via the zeros of laguerre polynomials the uniform convergence of the christoffel functions is proved equivalent to the nondenseness of the muntz space on 0 1 which implies that in this case the orthogonal muntz legendre polynomials tend to 0 uniformly on closed subintervals of 0 1 some inequalities for muntz polynomials are also investigated most notably a sharp l2 markov inequality is proved

on the solutions of a boundary value problem of linear thermoelasticity system with nonlocal conditions abstract this paper studies a boundary value problem with nonlocal conditions for a coupled system of linear thermoelasticity in one dimensional case using an a priori estimate we prove the uniqueness of the solution also some explicit solutions are obtained by using the separation method

space like loxodromes on rotational surfaces in minkowski 3 space abstract we find all space like loxodromes on rotational surfaces which have space like meridians or time like meridians respectively by using a relevant lorentzian angle in minkowski 3 space to understand loxodromes better we draw some pictures of them via mathematica computer program

eigenvalues of tridiagonal pseudo toeplitz matrices abstract in this article we determine the eigenvalues of sequences of tridiagonal matrices that contain a toeplitz matrix in the upper left block

blowup of solutions to a porous medium equation with nonlocal boundary condition in this paper we investigate the blowup properties of the positive solutions to a porous medium equation with nonlocal boundary we obtain the blowup condition and its blowup rate estimate

asymptotic behavior of solutions of reaction diffusion equations with nonlocal boundary conditions abstract this paper is concerned with some dynamical property of a reaction diffusion equation with nonlocal boundary condition under some conditions on the kernel in the boundary condition and suitable conditions on the reaction function the asymptotic behavior of the time dependent solution is characterized in relation to a finite or an infinite set of constant steady state solutions this characterization is determined solely by the initial function and it leads to the stability and instability of the various steady state solutions in the case of finite constant steady state solutions the time dependent solution blows up in finite time when the initial function in greater than the largest constant solution also discussed is the decay property of the solution when the kernel function in the boundary condition prossesses alternating sign in its domain

asymptotic behavior of solutions for finite difference equations of reaction diffusion abstract the method of upper lower solutions and the associated existence comparison theorem for continuous reaction diffusion equations are extended to a finite difference system which is a discrete version of the continuous problem of either parabolic type or elliptic type under appropriate boundary condition it is shown by using the existence comparison theorem for the discrete system that for a certain class of initial functions the corresponding time dependent solution converges monotonically to a steady state solution this monotone convergence leads to the stability property of the steady state solution as well as an estimate of the stability region in the framework of the finite difference system numerical results are given for an enzyme substrate reaction model where multiple steady state solutions exist for a certain range of physical parameters in this situation the time dependent solution converges either to the maximal solution or to the minimal solution depending on the corresponding initial function

local variability of non smooth functions abstract with the help of muntz powers a formula of the taylor type for non smooth functions is presented the approximation provides a local study for the variability of some curves which do not have a derivative the approach includes the classical case but at the same time other non analytical and non differentiable mappings in the first place a muntz curve representing the local variability of a function is defined the coefficient and exponent of the model allow a numerical characterization of the relative extremes and differentiability of the map the introduction of exponents of higher order provides a generalization of the taylor s formula including some cases of non differentiability in the last part a series expansion of non necessarily integer powers representing the function is presented several properties of convergence continuity integrability and density are studied

on the solvability of a class of nonlinear singular parabolic equation with integral boundary condition abstract in this paper the existence and uniqueness of a weak solution for nonlinear singular parabolic equation with integral boundary conditions is proved first the associated linear problem is solved after writing the linear problem on its operatorial form we establish an a priori bound from which we deduce the uniqueness of the strong solution for the solvability of the associated linear problem we prove that the range of the operator generated by the considered problem is dense on the basis of the obtained results of the linear problem we apply an iterative process to establish the existence and uniqueness of the weak solution of the nonlinear problem

on some invariant solutions of 2 1 dimensional korteweg de vries equations abstract in the present research similarity transformation method is proposed to obtain some more general invariant solutions of 2 1 dimensional korteweg de vries equations this system of equations describes nonlinear waves propagation on the surface of shallow water the method reduces the number of independent variables by one using invariance property of lie group theory thus korteweg de vries equations are reduced into a system of ordinary differential equations employing twice of similarity transformation method this system of ordinary differential equations is solved under some parametric restrictions and provides invariant solutions the obtained results are supplemented by numerical simulation taking suitable choice of arbitrary constants and functions eventually the elastic behavior of multisoliton compacton negaton positon kink wave solution and dromion annihilation profiles are shown to make this research more admirable

roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition in this paper we investigates the blow up properties of the positive solutions to a porous medium equation with nonlocal reaction source and with nonlocal boundary condition we obtain the blow up condition and its blow up rate estimate

a nonlinear parabolic equation with a nonlocal boundary term a nonlinear parabolic problem with a nonlocal boundary condition is studied we prove the existence of a solution for a monotonically increasing and lipschitz continuous nonlinearity the approximation method is based on rothe s method the solution on each time step is obtained by iterations convergence of which is shown using a fixed point argument the space discretization relies on fem theoretical results are supported by numerical experiments

some new operational matrices and its application to fractional order poisson equations with integral type boundary constrains abstract enormous application of fractional order partial differential equations fpdes subjected to some constrains in the form of nonlocal boundary conditions motivated the interest of many scientists around the world the prime objective of this article is to find approximate solution of a general fpdes subject to nonlocal integral type boundary conditions on both ends of the domain the proposed method is based on spectral method we construct some new operational matrices which have the ability to handle integral type non local boundary constrains these operational matrices can be effectively applied to convert the fpdes together with its integral types boundary conditions to easily solvable matrix equation the accuracy and efficiency of proposed method are demonstrated by solving some bench mark problems the proposed method has the ability to solve non local fpdes with high accuracy and low computational cost different aspects of presented approach are compared with two other recently developed methods haar wavelets collocation method and a family of collocation methods which are based on radial base functions it is observed that the proposed method is highly accurate robust efficient and stable as compared to these methods

numerical solutions of reaction diffusion equations with nonlocal boundary conditions abstract the purpose of this paper is to present some iterative methods for numerical solutions of a class of nonlinear reaction diffusion equations with nonlocal boundary conditions using the finite difference method and the method of upper and lower solutions we present some monotone iterative schemes for both the time dependent and the steady state finite difference systems each monotone iterative scheme gives a computational algorithm for numerical solutions and an existence comparison theorem for the corresponding finite difference system the existence comparison theorems are used to investigate the asymptotic behavior of the discrete time dependent solution in relation to the discrete maximal and minimal solutions of the steady state problem numerical results are given to a model problem where the solution of the continuous problem is explicitly known and its values at the mesh points are used to compare with the numerical solutions obtained by the monotone iterative schemes

the utilization of total mass to determine the switching points in the symmetric boundary control problem with a linear reaction term abstract the authors study the problem u t u x x u 0 x 1 t 0 u x 0 0 and u 0 t u 1 t t where t u 0 for t 2 k t t 2 k 1 and t 0 for t 2 k 1 t t 2 k 2 k 0 1 2 with t 0 0 and the sequence t k is determined by the equations 0 1 u x t k d x m for k 1 3 5 and 0 1 u x t k d x m for k 2 4 6 and where 0 m m note that the switching points t k k 1 2 3 are unknown existence and uniqueness are demonstrated theoretical estimates of the t k and t k 1 t k are obtained and numerical verifications of the estimates are presented the case of u x 0 t u x 1 t t is also considered and analyzed

influence of complex coefficients on the stability of difference scheme for parabolic equations with non local conditions abstract the stability of a finite difference scheme for schrodinger kuramoto tsuzuki and parabolic equations subject to non local conditions with complex coefficients is dealt with the stability conditions which have to be met by complex coefficients in non local conditions have been determined the main result of this study is that complex coefficients together with non local conditions cause new effects on the stability of difference scheme numerical experiment has revealed additional regularities in the stability conditions

the two level finite difference schemes for the heat equation with nonlocal initial condition abstract in this paper the two level finite difference schemes for the one dimensional heat equation with a nonlocal initial condition are analyzed as the main result we obtain conditions for the numerical stability of the schemes in addition we revise the stability conditions obtained in 21 for the crank nicolson scheme we present several numerical examples that confirm the theoretical results within linear as well as nonlinear problems in some particular cases it is shown that for small regions of the time step size values the explicit ftcs scheme is stable while certain implicit methods such as crank nicolson scheme are not

numerical spectral analysis of a difference operator with non local boundary conditions abstract the spectrum of a finite difference operator subject to non local robin type boundary conditions is dealt with we analyse the spectral properties that relate to the stability of finite difference schemes for parabolic equations the impact of functions and parameters defining non local conditions on a spectral structure is examined and theoretical study is supported by numerical experiments also for a difference scheme applied to a parabolic equation with non local conditions a sufficient stability criterion based on spectral properties of the difference operator is discussed numerical evidence suggests that such a criterion is not only sufficient for stability but necessary too

numerical algorithm for parabolic problems with non classical conditions the parabolic problems with non classical conditions are discussed in a reproducing kernel space in this paper a reproducing kernel space is constructed in which the non classical conditions of the parabolic problems are satisfied based on the reproducing kernel space a new technique for solving the non classical parabolic problems is presented some examples are displayed to demonstrate the validity and applicability of the proposed method

blow up phenomena for a parabolic system with gradient nonlinearity under nonlinear boundary conditions abstract in this paper we consider the following reaction diffusion systems with gradient nonlinearity under nonlinear boundary condition u t u u p v q u x t 0 t v t v v r u s v x t 0 t u g u v h v x t 0 t u x 0 u 0 x v x 0 v 0 x x where r n n 1 is a bounded region with smooth boundary p q r s 0 1 t is a possible blow up time when blow up occurs by constructing an appropriate auxiliary functions and by means of payne weinberger or scott s method a lower bound on blow up time when blow up occurs is derived

local approximation of functions with several variables abstract this paper presents the problem of local approximation of scalar functions with several variables including points of non differentiability the procedure considers that the mapping displays rates of change of power type with respect to linear changes in the coordinate domain and the exponents are not necessarily integer the approach provides a formula describing the local variability of scalar fields which contains and generalizes taylor s formula of first order the functions giving the contact are muntz polynomials the knowledge of their coefficients and exponents enables the finding of local extremes including cases of non smoothness sufficient conditions for the existence of global maxima and minima of concave convex functions are obtained as well

a high order composite scheme for the second order elliptic problem with nonlocal boundary and its fast algorithm the elliptic problem with nonlocal boundary condition is widely applied in the field of science and engineering firstly we construct a linear finite element scheme for the nonlocal boundary problem and derive the optimal l 2 error estimate then based on the quadratic finite element and the extrapolation linear finite element methods we present a composite scheme and prove that it is convergent order three furthermore we design an upper triangular preconditioning algorithm for the linear finite element discrete system finally numerical results not only validate that the new algorithm is efficient but also show that the new scheme is convergent order three furthermore order four on uniform grids

generalized quasilinearization method for reaction diffusion equations under nonlinear and nonlocal flux conditions abstract in this paper we consider an initial boundary value problem for a reaction diffusion equation under nonlinear and nonlocal robin type boundary condition assuming the existence of an ordered pair of upper and lower solutions we establish a generalized quasilinearization method for the problem under consideration whose characteristic feature consists in the construction of monotone sequences converging to the unique solution within the interval of upper and lower solutions and whose convergence rate is quadratic thus this method provides an efficient iteration technique that produces not only improved approximations due to the monotonicity of its iterates but yields also a measure of the convergence rate

a new adi technique for two dimensional parabolic equation with an integral condition abstract nonclassical parabolic initial boundary value problems arise in the study of several important physical phenomena this paper presents a new approach to treat complicated boundary conditions appearing in the parabolic partial differential equations with nonclassical boundary conditions a new fourth order finite difference technique based upon the noye and hayman n h alternating direction implicit adi scheme is used as the basis to solve the two dimensional time dependent diffusion equation with an integral condition replacing one boundary condition this scheme uses less central processor time cpu than a second order fully implicit scheme based on the classical backward time centered space btcs method for two dimensional diffusion it also has a larger range of stability than a second order fully explicit scheme based on the classical forward time centered space ftcs method the basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach developed from the 1974 work of warming and hyeet this allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference methods the results of numerical experiments for the new method are presented the central processor times needed are also reported error estimates derived in the maximum norm are tabulated

the exact solution of system of linear operator equations in reproducing kernel spaces abstract in this paper we discuss the problem of how to solve the system of linear operator equations in hilbert spaces and applied the result to reproducing kernel spaces under the assumption that the operator equations has a unique solution its formal solution is given as a result when h h 1 are both reproducing kernel spaces the formal solution turned into analytic solution truncating the series where formal solution and the analytic solution are denoted by a series the approximate solution of the operator equations is obtained when increasing the number of the nodes the error of the approximate solution is monotone decreasing in the sense of the norm of the hilbert space the final numerical experiment shows the efficiency of our method

roles of weight functions in a nonlinear nonlocal parabolic system abstract this paper deals with a semilinear parabolic system with coupled nonlinear nonlocal sources subject to weighted nonlocal dirichlet boundary conditions we establish the conditions for global and non global solutions it is interesting to observe that the weight functions for the nonlocal dirichlet boundary conditions play substantial roles in determining not only whether the solutions are global or non global but also whether for the non global solutions the blowing up occurs for any positive initial data or just for large ones

on a class of parabolic equations with nonlocal boundary conditions in this paper we study a class of parabolic equations subject to a nonlocal boundary condition the problem is a generalized model for a theory of ion diffusion in channels by using energy method we first derive some a priori estimates for solutions and then prove that the problem has a unique global solution moreover under some assumptions on the nonlinear boundary condition it is shown that the solution blows up in finite time finally the long time behavior of solution to a linear problem is also studied in the paper

global existence and blow up for a porous medium system with nonlocal boundary conditions and nonlocal sources abstract this paper is devoted to a porous medium system subject to nonlocal boundary conditions and with nonlocal sources we investigate the global existence and blow up in finite time of a nonnegative solution by using a sub super solution method moreover under appropriate hypotheses for the conditions we obtain asymptotic properties of the solutions

smoothing of crank nicolson scheme for the two dimensional diffusion with an integral condition the parabolic partial differential equations pdes with nonlocal boundary conditions model various physical phenomena e g chemical diffusion thermoelasticity heat conduction process control theory and medicine science this paper deals with the smoothing of the crank nicolson numerical scheme for two dimensional parabolic pdes with nonlocal boundary conditions we use the numerical scheme based on pade approximations of the matrix exponential the graphs of numerical results demonstrate the successful smoothing of the crank nicolson numerical scheme

blow up problem for semilinear heat equation with absorption and a nonlocal boundary condition abstract in this paper we consider a semilinear parabolic equation u t u c x t u p for x t 0 with nonlinear and nonlocal boundary condition u 0 k x y t u l d y and nonnegative initial data where p 0 and l 0 we prove some global existence results criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data are also given

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

the method of approximate particular solutions for the time fractional diffusion equation with a non local boundary condition in this paper we consider the numerical solution of the time fractional diffusion equation with a non local boundary condition the method of approximate particular solutions maps using multiquadric radial basis function mq rbf is employed for this equation due to the accuracy of the mq based meshless methods is severely influenced by the shape parameter we adopt a leave one out cross validation loocv algorithm proposed by rippa 34 to enhance the performance of the maps the numerical results obtained show that the proposed numerical algorithm is accurate and computationally efficient for solving time fractional diffusion equation with a non local boundary condition

a numerical approach for a semilinear parabolic equation with a nonlocal boundary condition a semilinear reaction diffusion problem with a nonlocal boundary condition is studied this paper presents a new and very easy implementable numerical algorithm for computations this is based on a suitable linearization in time and on the principle of linear superposition any method for the space discretization fem was taken in this analysis can be chosen the derived algorithm is implicit and it does not need any iteration scheme to get a solution with the nonlocal boundary condition stability analysis has been performed and the optimal error estimates have been derived numerical results have been compared with other known techniques

spatially structured activity in synaptically coupled neuronal networks i traveling fronts and pulses we consider traveling front and pulse solutions to a system of integro differential equations used to describe the activity of synaptically coupled neuronal networks in a single spatial dimension our first goal is to establish a series of direct links between the abstract nature of the equations and their interpretation in terms of experimental findings in the cortex and other brain regions this is accomplished first by presenting a biophysically motivated derivation of the system and then by establishing a framework for comparison between numerical and experimental measures of activity propagation speed our second goal is to establish the existence of traveling pulse solutions using more rigorous methods two techniques are presented the first a shooting argument reduces the problem from finding a specific solution to an integro differential equation system to finding any solution to an ode system the second a singular perturbation argument provides a construction of traveling pulse solutions un

a galerkin procedure for the diffusion equation subject to the specification of mass continuous and discrete galerkin procedures are derived and analyzed for the numerical solution of the diffusion equation u 1 u xx 0 x 1 0 t leqq t u x 0 f x u x 1 t g t and the specification of the mass int 0 b u x t dx m t 0 b 1 some numerical experiments are also presented

remarks on a semilinear heat equation with integral boundary conditions abstract for a semilinear heat equation we consider a nonlocal boundary problem on the basis of the solution of a dirichlet problem for a parabolic equation and volterra integral equation we establish the well posedness for the nonlocal problem which generalizes some recent results

a finite difference method for the singularly perturbed problem with nonlocal boundary condition the purpose of this paper is to present a finite difference method for numerical solutions of singularly perturbed boundary value problem for second order ordinary differential equation with nonlocal boundary condition by the method of integral identities with the use exponential basis functions and interpolating quadrature rules with the weight and remainder term in integral form an exponentially fitted difference scheme on an uniform mesh is developed which is shown to be original e uniformly first order accurate in the discrete maximum norm for original problem numerical results are presented which illustrate the theoretical results

stepwise stability for the heat equation with a nonlocal constraint the authors prove the stepwise stability for a finite difference scheme for the heat equation with an integral constraint the resulting matrix is nonsymmetric and does not have the usual band structure the proof is based on the method of matrix analysis the eigenvalues of several matrices are found explicitly or their location described precisely this method relies upon the relationship of the characteristic polynomials of these matrices with orthogonal polynomials

a galerkin method for two dimensional hyperbolic integro differential equation with purely integral conditions the present paper is devoted to the solution for two dimensional hyperbolic integro differential equations subject to purely integral conditions first it is demonstrated the existence and uniqueness of the solution under certain conditions for the numerical approach a galerkin method based on least squares is proposed the numerical examples illustrate the technique and show the efficiency of the proposed method

fisher equation for anisotropic diffusion simulating south american human dispersals the fisher equation is commonly used to model population dynamics this equation allows describing reaction diffusion processes considering both population growth and diffusion mechanism some results have been reported about modeling human dispersion always assuming isotropic diffusion nevertheless it is well known that dispersion depends not only on the characteristics of the habitats where individuals are but also on the properties of the places where they intend to move then isotropic approaches cannot adequately reproduce the evolution of the wave of advance of populations solutions to a fisher equation are difficult to obtain for complex geometries moreover when anisotropy has to be considered and so few studies have been conducted in this direction with this scope in mind we present in this paper a solution for a fisher equation introducing anisotropy we apply a finite difference method using the crank nicholson approximation and analyze the results as a function of the characteristic parameters finally this methodology is applied to model south american human dispersal

on group invariant solutions of the boltzmann equation after reviewing the complete lie group for the full boltzmann equation it is shown that projective transformations play a special role in the general case and the class of invariant solutions giving rise to homoenergetic affine flows is presented homoenergetic affine flows in the two dimensional case potential u r r 2 are considered in detail it is shown that the general solution of this problem can be essentially simplified by projective transformations

soliton solutions of blmp equation by lie symmetry approach abstract shallow water wave equations are usually described by korteweg de vries kdv type equations in this paper we have used lie transformation group theory to solve 2 1 dimensional boiti leon manna pempinelli blmp equation we have obtained some exact solutions of blmp equation in the explicit form through similarity reduction all the reported results are expressed in closed form and analysed physically through their evolution profiles the physical analysis reveals that the nature of solutions is parabolic quasi periodic multisoliton and asymptotic

blow up of solutions for nonlinear parabolic equation with nonlocal source and nonlocal boundary condition this paper is devoted to a nonlinear parabolic equation subject to nonlocal boundary condition and with nonlocal reaction source some criteria on this problem which determine whether the solutions blow up in a finite time or exist globally are given moreover the blow up rate estimates are also obtained

asymptotic behavior for nonlocal dispersal equations abstract this paper is concerned with the existence and asymptotic behavior of solutions of a nonlocal dispersal equation by means of super subsolution method and monotone iteration we first study the existence and asymptotic behavior of solutions for a general nonlocal dispersal equation then we apply these results to our equation and show that the nonnegative solution is unique and the behavior of this solution depends on parameter in equation for 1 the solution decays to zero as t while for 1 the solution converges to the unique positive stationary solution as t in addition we show that the solution blows up under some conditions

global blow up for a localized nonlinear parabolic equation with a nonlocal boundary condition abstract this paper deals with the blow up properties of positive solutions to a nonlinear parabolic equation with a localized reaction source and a nonlocal boundary condition under certain conditions the blowup criteria is established furthermore when f u u p 0 p 1 the global blowup behavior is shown and the blowup rate estimates are also obtained

adomian method for a class of hyperbolic equations with an integral condition abstract in this paper adomian decomposition method adm is applied to the solvability of a boundary problem with a nonlocal condition for a class of hyperbolic equations and illustrated with a few simple examples

solving nonlocal initial boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations by the adomian decomposition method in this paper we present a new approach to solve nonlocal initial boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations subject to initial and nonlocal boundary conditions of integral type we first transform the given nonlocal initial boundary value problems of integral type for the linear and nonlinear parabolic and hyperbolic partial differential equations into local dirichlet initial boundary value problems and then use a relatively new modified adomian decomposition method adm furthermore we investigate the fourier adomian method which also does not require any a priori assumptions on the solution for the solution of nonlocal initial boundary value problems combined with our new approach several examples are presented to demonstrate the efficiency of the adm