similarity solutions for unsteady free convection flow from a continuous moving vertical surface the transformation group theoretic approach is applied to present an analysis of the problem of unsteady free convection flow over a continuous moving vertical sheet in an ambient fluid the thermal boundary layer induced within a vertical semi infinite layer of boussinseq fluid by a constant heated bounding plate the application of two parameter groups reduces the number of independent variables by two and consequently the system of governing partial differential equations with the boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions the obtained ordinary differential equations are solved analytically for the temperature and numerically for the velocity using the shooting method effect of prandtl number on the thermal boundary layer and velocity boundary layer are studied and plotted in curves

toward a universal adaptive finite element strategy part 3 design of meshes abstract in this third paper in our series the issue of designing h p meshes which are optimal in some sense is addressed criteria for h p meshes are derived which are based on the idea of minimizing the estimated error over a mesh with a fixed number of degrees of freedom an optimization algorithm is developed based on these criteria and is applied to several model one dimensional and two dimensional elliptic boundary value problems numerical results indicate that the approach can lead to exponential rates of convergence

first passage and risk evaluation under stochastic volatility we solve the first passage problem for the heston random diffusion model we obtain exact analytical expressions for the survival and the hitting probabilities to a given level of return we study several asymptotic behaviors and obtain approximate forms of these probabilities which prove among other interesting properties the nonexistence of a mean first passage time one significant result is the evidence of extreme deviations which implies a high risk of default when certain dimensionless parameter related to the strength of the volatility fluctuations increases we confront the model with empirical daily data and we observe that it is able to capture a very broad domain of the hitting probability we believe that this may provide an effective tool for risk control which can be readily applicable to real markets both for portfolio management and trading strategies

from a discrete to a continuum model of cell dynamics in one dimension multiscale modeling is emerging as one of the key challenges in mathematical biology however the recent rapid increase in the number of modeling methodologies being used to describe cell populations has raised a number of interesting questions for example at the cellular scale how can the appropriate discrete cell level model be identified in a given context additionally how can the many phenomenological assumptions used in the derivation of models at the continuum scale be related to individual cell behavior in order to begin to address such questions we consider a discrete one dimensional cell based model in which cells are assumed to interact via linear springs from the discrete equations of motion the continuous rouse p e rouse j chem phys 21 1272 1953 model is obtained this formalism readily allows the definition of a cell number density for which a nonlinear fast diffusion equation is derived excellent agreement is demonstrated between the continuum and discrete models subsequently via the incorporation of cell division we demonstrate that the derived nonlinear diffusion model is robust to the inclusion of more realistic biological detail in the limit of stiff springs where cells can be considered to be incompressible we show that cell velocity can be directly related to cell production this assumption is frequently made in the literature but our derivation places limits on its validity finally the model is compared with a model of a similar form recently derived for a different discrete cell based model and it is shown how the different diffusion coefficients can be understood in terms of the underlying assumptions about cell behavior in the respective discrete models

new basis functions and computational procedures for p version finite element analysis new basis functions and solution procedures for p version finite element analysis are described they are used in a highly efficient p version finite element solver for linear elastostatics and dynamics which has been used in an industrial environment for over two years using two sample applications it is shown that using the techniques proposed here p version finite element analysis can have a substantially lower computational cost for given accuracy than standard finite element methods this makes the industrial applicability of p version finite element analysis much wider than is commonly believed

painleve test and exact similarity solutions of a class of nonlinear diffusion equations for 1 1 dimensional diffusion equations where the diffusion coefficient is an arbitrary power of the dependent variable a painleve test is performed on both the partial differential equation and the similarity reductions the invariance of the painleve property under a large class of similarity reductions is shown for the reductions with the painleve property the explicit similarity solutions are given it is demonstrated how results of a painleve analysis can be used to obtain further exact similarity solutions even in cases without the painleve property

exact similarity solutions to some nonlinear diffusion equations the author gives new closed form similarity solutions to n dimensional radially symmetric nonlinear diffusion equations with power law and exponential diffusivities these solutions are generalisations of known similarity solutions

new variable separation approach application to nonlinear diffusion equations the concept of the derivative dependent functional separable solution ddfss as a generalization to the functional separable solution is proposed as an application it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach as a consequence a complete list of canonical forms for such equations which admit the ddfss is obtained and some exact solutions to the resulting equations are described

integration by table look up for p version finite elements on curved tetrahedra abstract we develop a method of evaluating element level matrices and vectors associated with p version finite element computations on tetrahedral meshes with curved boundaries the procedure uses optimal interpolation formula q chen i babuska approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle comput methods appl mech engrg 128 1995 405 417 q chen i babuska approximate optimal polynomial interpolation points in the tetrahedron comput methods appl mech engrg 137 1997 89 94 for non polynomial portions of integrands followed by exact integration exact integrals of product combinations of polynomial interpolants and shape functions are stored so that different integrals may be efficiently evaluated by a table look up we present error analysis for our scheme applied to curvilinear tetrahedral meshes with exact geometric representation of the boundary of the domain as defined in a geometric modelling system numerical results are presented for problems involving the helmholtz equation

default risk modeling beyond the first passage approximation extended black cox model we develop a generalization of the black cox structural model of default risk the extended model captures uncertainty related to firm s ability to avoid default even if company s liabilities momentarily exceeding its assets diffusion in a linear potential with the radiation boundary condition is used to mimic a company s default process the exact solution of the corresponding fokker planck equation allows for derivation of analytical expressions for the cumulative probability of default and the relevant hazard rate obtained closed formulas fit well the historical data on global corporate defaults and demonstrate the split behavior of credit spreads for bonds of companies in different categories of speculative grade ratings with varying time to maturity introduction of the finite rate of default at the boundary improves valuation of credit risk for short time horizons which is the key advantage of the proposed model we also consider the influence of uncertainty in the initial distance to the default barrier on the outcome of the model and demonstrate that this additional source of incomplete information may be responsible for nonzero credit spreads for bonds with very short time to maturity

potential symmetries of a porous medium equation potential symmetries which are not local symmetries are carried out for the porous medium equation where when it can be written in a conserved form these symmetries are realized as local symmetries of a related auxiliary system and lead to the construction of corresponding invariant solutions as well as to the linearization of the equation by non invertible mappings

exact polynomial solutions to some nonlinear diffusion equations abstract a class of non group invariant exact solutions to a number of nonlinear diffusion equations is constructed expressed in the appropriate variables these solutions are polynomials in the spatial variable the method is mainly applicable to equations which may be written in an appropriate quadratically nonlinear form a number of equivalence transformations between various nonlinear diffusion equations are also derived

a straightforward structure to construct shape functions for variable p order meshes this paper describes a method for specifying variable p order meshes based on the topological hierarchy of mesh entities defining the finite elements a decomposition of p version mesh entity shape functions into a product of a function defined over the mesh entity and a blending function defined over the elements that the mesh entity bounds is presented the decomposed shape functions allow the effective construction of the shape functions and derivatives needed to construct element matrices for variable p enrichment

group method analysis of steady free convective laminar boundary layer flow on a nonisothermal vertical circular cylinder the transformation group theoretic approach is applied to present an analysis of the problem of unsteady free convection from the outer surface of a vertical circular cylinder the application of two parameter group reduces the number of independent variables by two and consequently the system of the governing partial differential equations with the boundary conditions reduces to a system of ordinary differential equations with the appropriate boundary conditions the ordinary differential equations are solved numerically using a fourth order runge kutta scheme and the gradient method numerical results are obtained for the study of the boundary layer characteristics the general analysis developed in this study corresponds to the case of surface temperature that varies exponentially with time and uniform with respect to the axial coordinate i e in the formt w ae bt wherea andb are constants the effect of prandtl number pr andb on the boundary layer characteristics and the maximum value of the vertical component of the velocity are studied

default risk modeling with position dependent killing abstract diffusion in a linear potential in the presence of position dependent killing is used to mimic a default process different assumptions regarding transport coefficients initial conditions and elasticity of the killing measure lead to diverse models of bankruptcy one stylized fact is fundamental for our consideration empirically default is a rather rare event especially in the investment grade categories of credit ratings hence the action of killing may be considered as a small parameter in a number of special cases we derive closed form expressions for the entire term structure of the cumulative probability of default its hazard rate and intensity comparison with historical data on aggregate global corporate defaults confirms the validity of the perturbation method for estimations of long term probability of default for companies with high credit quality on a single company level we implement the derived formulas to estimate the one year likelihood of default of enron on a daily basis from august 2000 to august 2001 three months before its default and compare the obtained results with forecasts of traditional structural models

the h p and h p versions of the finite element methods in 1 dimension part iii the adaptive h p version abstract the paper is the third and final part in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimension the first part analyzed the p version and the second part concentrated on the h and h p version this paper analyzes a theoretical frame of the adaptive h p version and based on it the authors provide concrete algorithm for the one dimensional problem it is proven that in the case that the solution has x sub alpha type singularity the adaptive algorithm give an exponential rate of convergence very close to the optimal one analyzed in the second part of the paper additional keyword error analysis

on point transformations of generalized nonlinear diffusion equations this paper classifies all finite point transformations of a general class between generalized diffusion equations of the form ut x1 m xn 1f u ux x these transformations may be divided into three cases depending on the functional form of f u i f arbitrary ii f un and iii f eu in particular these transformations include all the invariant infinitesimal transformations and in addition they include a number of point transformations which relate different equations of the above form many exact solutions are already known and the transformations which are derived may be used to obtain new solutions from these

the non classical solution of the inhomogeneous non linear diffusion equation a complete classification of the symmetry reductions of the inhomogeneous nonlinear diffusion inld equation using the non classical method leads to one family of symmetry generators the resulting ordinary differential equation is of second order and non linear new solutions are obtained for some physical processes corresponding to particular values of n and p m

toward a universal h p adaptive finite element strategy part 1 constrained approximation and data structure abstract this is the first of a series of papers devoted to the development of an adaptive finite element method to solve various boundary value problems arising in solid and fluid mechanics the paper deals with the general concepts of h p finite element methods constrained approximation and data structure and related issues several numerical examples illustrate the main ideas of the method

symmetries and integrability of generalized diffusion reaction equations using lie group methods and the painleve test the authors analyse nonlinear diffusion reaction equations ut d u u f u with power law diffusion coefficients d approximately uv and arbitrary nonlinear reaction terms f u which have a wide spectrum of applications in many areas of science the lie group based similarity method leads to a classification of the reaction terms according to its symmetry properties with the help of the adjoint representation the optimal system of similarity reductions is calculated to check the integrability of the partial differential equation the existence of generalized lie backlund symmetries is investigated apart from three known cases no further cases with third order symmetries exist examining the integrability of the second order ordinary differential equations resulting from the reductions only a few parameter combinations can be found for which the painleve property is given however they are able to construct unknown integrals of motion for a much larger range of parameter values from the integrals exact similarity solutions may be derived this is demonstrated by examples corresponding to the important moving wave reduction

hierarchical basis for stabilized finite element methods for compressible flows the quest for more accurate numerical simulations of the navier stokes equations has motivated the study of high order piecewise polynomial basis functions also known as k version finite element methods as a means to attain this accuracy in the most cost effective manner stabilized finite element methods e g streamline upwind petrov galerkin have been proven to attain optimal rates of convergence for any polynomial order basis as well as perform well on a variety of flows including turbulence applications we present a stabilized finite element formulation for fluid dynamics using mesh entity based hierarchical basis functions the implementation is shown to attain theoretical convergence results for the linear model problem as well as the navier stokes equations

h p adaptive finite element methods in computational fluid dynamics the principal ideas of h p adaptive finite element methods for fluid dynamics problems are discussed applications include acoustics compressible euler and both compressible and incompressible navier stokes equations several numerical examples illustrate the presented concepts

group transformations and the nonlinear heat diffusion equation a nonlinear diffusion equation with a power law diffusion coefficient is studied self similar and partially invariant solutions are shown to be identical the theory of homology is introduced and used to generate classes of equations the solutions of which are related through backlund transformations an example of an application is given which generalizes the solution of storm j appl phys 22 1951 pp 940 951 on heat conduction in metals

further transformation properties of generalised inhomogeneous nonlinear diffusion equations with variable coefficients abstract we consider the variable coefficient inhomogeneous nonlinear diffusion equations of the form x p u t x q u n u x x we present further transformation properties for this nonlinear class of equations that do not appear in the literature in particular we map this class of variable coefficient into constant coefficient evolution equations we also introduce hodograph and generalised hodograph transformation for a specific form of this class we derive an auto hodograph transformation new nonlocal symmetries known as potential symmetries are found finally examples of exact solutions are given

geometry representation issues associated with p version finite element computations this paper addresses issues related to accurate geometry representation for p version finite elements on curved three dimensional domains specific options to account for domain geometry information during element level computation are identified accuracy requirements on the geometry related approximations to preserve the optimal rate of finite element error convergence for second order elliptic boundary value problems are given an element geometric mapping scheme based on blending the exact shape of the domain boundary is described that can either be used directly during element integrations or used to construct element level geometric approximations of required accuracy smoothness issues of the rational blends on simplex topologies are discussed and a numerical example based on the solution of poisson s equation in three dimensions is presented to illustrate the impact of the rational blends on the optimal rate of finite element error convergence

a new triangular and tetrahedral basis for high order hp finite element methods in this paper we describe the foundations of a new hierarchical modal basis suitable for high order hp finite element discretizations on unstructured meshes it is based on a generalized tensor product of mixed weight jacobi polynomials the generalized tensor product property leads to a low operation count with the use of sum factorization techniques variable p order expansions in each element are readily implemented which is a crucial property for efficient adaptive discretizations numerical examples demonstrate the exponential convergence for smooth solutions and the ability of this formulation to handle easily very complex two and three dmensional computational domains employing standard meshes

the problem of selecting the shape functions for a p type finite element abstract the paper addresses the question of the optimal selection of the shape functions for p type finite elements and discusses the effectivity of the conjugate gradient and multilevel iteration method for solving the corresponding linear system the selection of the shape functions is of major importance for the performance of the solver based on iterative methods neither the theory nor practice of the optimal selection of the shape functions is available yet we have seen that the condensation approach which has obvious advantages from the point of parallel computations is a very effective tool for keeping the condition number under the control and is especially advantageous for the conjugate gradient method

a class of fokker planck equations with logarithmic factors in diffusion and drift terms the x2 ln x dependence of the diffusion coefficient in the fokker planck equation is retrieved by means of symmetry arguments exact solutions of the equation with logarithmic factors in coefficients are presented algebraic and log algebraic solutions are found for some values of exponents they seem to be analogues on an interval of log normal distributions

multilevel preconditioned qmr methods for unstructured mesh computation we present a variant of the quasi minimal residual qmr algorithm of freund and nachtigal with a preconditioner based on the algebraic multilevel amli algorithm of axelsson and vassilevski this combination provides an effective solution method for indefinite algebraic systems and is tested by application to finite element discretizations of the helmholtz equation the implementation is applicable with both h and p refinements and easily extends to a parallel environment

separation of variables of a generalized porous medium equation with nonlinear source this paper considers a general form of the porous medium equation with nonlinear source term ut d u u n x f u n 1 the functional separation of variables of this equation is studied by using the generalized conditional symmetry approach we obtain a complete list of canonical forms for such equations which admit the functional separable solutions as a consequence some exact solutions to the resulting equations are constructed and their behavior are also investigated 2002 elsevier science usa all rights reserved

using group theoretic method to solve multi dimensional diffusion equation the nonlinear diffusion equation arises in many important areas of science and technology such as modeling of dopant diffusion in semiconductors we give analytical solution to n dimensional radially symmetric nonlinear diffusion equation of the form r d c c r n 1 r d c c r c t where c r t is the concentration and d c is diffusion coefficient the transformation group theoretic approach is applied to present an analysis of the nonlinear diffusion equation the one parameter group transformation reduces the number of independent variables by one and the governing partial differential equation with the boundary conditions reduce to an ordinary differential equation with the appropriate boundary onditions effect of the time t and the number of dimension n on the concentration diffusion function c r t has been studied and the results are plotted

symmetries and form preserving transformations of generalised inhomogeneous nonlinear diffusion equations we consider the variable coefficient inhomogeneous nonlinear diffusion equations of the form f x ut g x unux x we present a complete classification of lie symmetries and form preserving point transformations in the case where f x 1 which is equivalent to the original equation we also introduce certain nonlocal transformations when f x xp and g x xq we have the most known form of this class of equations if certain conditions are satisfied then this latter equation can be transformed into a constant coefficient equation it is also proved that the only equations from this class of partial differential equations that admit lie backlund symmetries is the well known nonlinear equation ut u 2ux x and an equivalent equation finally two examples of new exact solutions are given

systematic and generic construction of shape functions for p adaptive meshes of multidimensional finite elements this paper presents a methodology for generating high order shape functions for the complete family of finite elements the geometric entities presented are the point line triangle quadrilateral tetrahedron pyramid prism and hexahedron the shape functions constructed are hierarchical and generate continuous approximation spaces the order of interpolation of the shape function can be determined independently for each edge face and volume the space of interpolation of order p for each element is complete this definition allows for the construction of hybrid meshes with any combination of these elements including elements of distinct dimension and or interpolation order the reader will observe that our systematic way of constructing shape functions can be extended to elements of higher dimension

a stabilized finite element method for computing turbulence abstract for many flows reynold averaged navier stokes simulation ranss techniques do not give an acceptable description of the flow in these cases a more detailed simulation of the turbulence is required one such detailed simulation technique large eddy simulation les has matured to the point of application to complex flows historically these simulations have been carried out with structured grids which suffer from two major difficulties the extension to higher reynolds numbers leads to an impractical number of grid points and most real world flows are rather difficult to represent geometrically with structured grids unstructured grid methods offer a release from both of these constraints herein we describe the development and validation of a massively parallel stabilized finite element method for les

a parallel hp fem infrastructure for three dimensional structural acoustics this paper describes a parallel three dimensional numerical infrastructure for the solution of a wide range of time harmonic problems in structural acoustics and vibration high accuracy and rate of error convergence in the mid frequency regime is achieved by the use of hp finite and infinite element approximations the infrastructure supports parallel computation in both single and multi frequency settings multi frequency solves utilize concurrent factoring of the frequency dependent linear algebraic systems and are naturally scalable scalability of large scale single frequency problems is realized by using feti dp an iterative domain decomposition scheme numerical examples are presented to cover applications in vibratory response of fluid filled elastic structures as well as radiation and scattering from elastic structures submerged in an infinite acoustic medium we demonstrate both the numerical accuracy as well as parallel scalability of the infrastructure in terms of problem parameters that include wavenumber and number of frequencies polynomial degree of finite infinite element approximations as well as the number of processors scalability and accuracy is evaluated for both single and multiple frequency sweeps on four high performance parallel computing platforms sgi altix sgi origin ibm p690 sp and linux cluster results show good performance on shared as well as distributed memory architecture copyright 2006 john wiley sons ltd

waiting time solutions of a nonlinear diffusion equation similarity solutions of the equation partial c partial t partial partial x c n partial c partial x n 0 are studied for both t 0 and t 0 and there is a stationary free boundary at x 0 and for t 0 have support in x at beta where beta 1 a 0 i e there is a moving free boundary x at beta by prescribing appropriate initial data c c 0 x x 0 at t t 0 these solutions demonstrate the existence of a waiting time t 0 before the free boundary moves a comparison theorem then enables a number of results to be obtained and conjectures to be made about the existence and length of waiting times for general initial data a three parameter family of solutions which exhibit blow up at t 0 is also constructed applications of these results to models for the spreading of a viscous blob on a hor

symmetric gaussian quadrature formulae for tetrahedronal regions abstract quadrature formulae of degrees 2 to 6 are presented for the numerical integration of a function over tetrahedronal regions the formulae presented are of gaussian type and fully symmetric with respect to the four vertices of the tetrahedron

the h p and h p versions of the finite element method in 1 dimension this paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting the main emphasis is placed on the analysis when the exact solution has singularity of x type the first part analyzes thep version the second theh version and generalh p version and the final third part addresses the problems of the adaptiveh p version

symmetry groups and separation of variables of a class of nonlinear diffusion convection equations in this paper a class of nonlinear diffusion convection equations ut d u uxn x p u ux which has quite a large number of physical applications is analysed by using symmetry group methods which include the classical method the potential symmetry method and the generalized conditional symmetry method a complete classification of the functional forms of the diffusion and convection coefficients is presented when the equation admits lie s point symmetry groups and potential symmetry groups the separation of variables for the equation is investigated using the generalized conditional symmetry approach for some interesting cases exact solutions using the method of separation of variables are discussed in detail

the h p and h p versions of the finite element method in 1 dimension part ii the error analysis of the h and h p versions the paper is the second in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimensional setting the first part 1 analyzed thep version the present one concentrates on theh andh p versions

group theoretic methods applied to burgers equation abstract in this study the group theoretic methods for calculating the solution of burgers equation with appropriate boundary and initial conditions is presented the application of a one parameter group reduces the number of independent variables by one and consequently the governing partial differential equation with the boundary and initial conditions to an ordinary differential equation with the appropriate corresponding conditions the obtained differential equation is solved analytically and the solution obtained in closed form for a specific choice of boundary condition

some self similar solutions in river morphodynamics 1 aggradation and degradation in one dimensional channels are often modeled with a simplified nonlinear diffusion equation different degrees of nonlinearity are obtained using the chezy and manning gauckler strickler laws for the friction coefficient combined with a sediment transport equation having a generalized form of the meyer peter and muller formula analytical self similar solutions for the dam break and the base level lowering are presented while the linear case corresponds to the classic diffusion equation the main effect of the nonlinearity appears to be the presence of singularities in the self similar solutions related to the finite speed of propagation of perturbations

derivative dependent functional separable solutions for the kdv type equations abstract a complete list of canonical forms for a special kdv type of equations which admit derivative dependent functional separable solutions is obtained by using the generalized conditional symmetry approach some exact solutions to the resulting equations are explicitly obtained

on symmetries of radially symmetric nonlinear diffusion equations in this paper symmetries are sought for radially symmetric nonlinear diffusion equations of the type ut r1 n rn 1f u ur r the nonlinear forms of this equation which admit lie classical symmetries are completely classified and are shown to exist when f un eu also a trivial symmetry exists for arbitrary f lie backlund symmetries are also considered when f un and shown to exist when n 0 linear case and when n 2 and n 1

q gaussian distributions of leverage returns first stopping times and default risk valuations a b s t r a c t we study the probability distributions of daily leverage returns of 520 north american industrial companies that survive de listing during the financial crisis 2006 2012 we provide evidence that distributions of unbiased leverage returns of all individual firms belong to the class of q gaussian distributions with the tsallis entropic parameter within the interval 1 q 2 the fat tails of the observed distributions imply a much higher probability of extreme movements in a company s leverage ratio than forecasted by the normal distribution q 1 motivated by these findings we develop a q gaussian generalization of traditional structural models of default derived exact analytical expressions for the probability distribution of a first stopping time and its intensity forecast significantly higher probability of default and much wider credit spreads at short time horizons our findings are broadly consistent with the results of empirical studies in equity markets and are essential for single name default forecasting as well as valuations of portfolio credit risk and economic capital which might be underestimated by a classic theory of diversified portfolio optimization

some non local transformations between nonlinear diffusion equations generalized backlund transformations are applied to derive links between a large number of different types of nonlinear diffusion equations including many which are of physical significance some new exactly linearizable forms are determined

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

p version fem for structural acoustics with a posteriori error estimation we demonstrate the advantages of using p version finite element approximations for structural acoustics problems in the mid to high frequency regime we then present a sub domain based a posteriori error estimation procedure to quantify the errors in the setting of a 3d interior acoustics problem with resonances and give numerical results effectivity indices show robust behavior of the error estimator away from the resonant frequencies

nonclassical symmetries of a porous medium equation with absorption in this paper new symmetry reductions and exact solutions are presented for the porous medium equation with absorption those spatial forms for which the equation can be reduced to an ordinary differential equation are studied the symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by bluman and cole and are unobtainable by lie classical method

new classes of similarity solutions of the inhomogeneous nonlinear diffusion equations the lie similarity method has been used to extend the similarity solutions of the one dimensional inhomogeneous nonlinear diffusion equations we determine the lie point symmetry vector fields and calculate the similarity ansatz then we discuss the resulting nonlinear ordinary differential equations exact solutions are found and their relations to some real physical situations are discussed

similarity solutions to three dimensional nonlinear diffusion equations the similarity transformation method has been used to solve three dimensional nonlinear diffusion equations the general lie group is calculated exact solutions to the cylindrical and spherical symmetry cases are found and their relations to some real physical processes are discussed

the diffusion reaction d r hamiltonian and the solutions of certain types of linear and nonlinear d r equations in one dimension with a view to having further insight into the mathematical content of the non hermitian hamiltonian associated with the diffusion reaction d r equation in one dimension we investigate a the solitary wave solutions of certain types of its nonlinear versions and b the problem of real eigenvalue spectrum associated with its linear version or with this class of non hermitian hamiltonians for case a we use the standard techniques to handle the quadratic and cubic nonlinearities in the d r equation whereas for case b a newly proposed method based on an extended complex phase space is employed for a particular class of solutions an ermakov system of equations is also found for the linear case further corresponding to the classical version of the above one dimensional complex hamiltonian an equivalent integrable system of two two dimensional real hamiltonians is suggested

a differential quadrature hierarchical finite element method using fekete points for triangles and tetrahedrons and its applications to structural vibration abstract a differential quadrature hierarchical finite element method dqhfem using fekete points was formulated for triangles and tetrahedrons and applied to structural vibration analyses first orthogonal polynomials on triangles and tetrahedrons that can be used as bases of the hierarchical finite element method hfem were derived and simple formulas of transforming one dimensional non uniform nodes to simplexes were presented then the non uniform nodes were used as initial guesses to solve the fekete points on simplexes through newton raphson s method together with the orthogonal polynomials new differential quadrature dq rules on simplexes were formulated using the hfem bases and the fekete points the numbers of nodes or bases on different edges and faces and inside the body of the new dq elements do not relate with each other like the hfem that can freely assign different numbers of bases on different edges and faces and inside the body so the new dq method was named as a differential quadrature hierarchical dqh method that uses either interpolation functions or orthogonal polynomials as bases inside the element its weak form was named as the dqhfem besides the dqh method and its weak form a simple method of generating high quality linear and high order triangular and tetrahedral meshes from a single nurbs patch was presented numerical tests of the dqhfem through structural vibration analyses showed that high accuracy results can be obtained using only a few nodes even on curvilinear domains using the dqh bases on both physical and geometric fields it was concluded that wide applications of the dqh method and the dqhfem to science and engineering are possible and commercial codes based on them are deserved to be developed