# Parabolic Equation (parabolic + equation)

Distribution by Scientific Domains

Kinds of Parabolic Equation

 degenerate parabolic equation semilinear parabolic equation

## Selected Abstracts

### On the modified Crank,Nicholson difference schemes for parabolic equation with non-smooth data arising in biomechanics

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 5 2010
Allaberen Ashyralyev
Abstract In the present paper, we consider the mixed problem for one-dimensional parabolic equation with non-smooth data generated by the blood flow through glycocalyx on the endothelial cells. Stable numerical method is developed and solved by using the r-modified Crank,Nicholson schemes. Numerical analysis is given for a constructed problem. Copyright © 2008 John Wiley & Sons, Ltd. [source]

### Localization for a doubly degenerate parabolic equation with strongly nonlinear sources

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2010
Zhaoyin Xiang
Abstract In this paper, we study the strict localization for the doubly degenerate parabolic equation with strongly nonlinear sources, We prove that, for non-negative compactly supported initial data, the strict localization occurs if and only if q,m(p,1). Copyright © 2009 John Wiley & Sons, Ltd. [source]

### Finite-dimensional attractors and exponential attractors for degenerate doubly nonlinear equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2009
M. Efendiev
Abstract We consider the following doubly nonlinear parabolic equation in a bounded domain ,,,3: where the nonlinearity f is allowed to have a degeneracy with respect to ,tu of the form ,tu|,tu|p at some points x,,. Under some natural assumptions on the nonlinearities f and g, we prove the existence and uniqueness of a solution of that problem and establish the finite-dimensionality of global and exponential attractors of the semigroup associated with this equation in the appropriate phase space. Copyright © 2009 John Wiley & Sons, Ltd. [source]

### On an initial-boundary value problem for a wide-angle parabolic equation in a waveguide with a variable bottom

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2009
V. A. Dougalis
Abstract We consider the third-order Claerbout-type wide-angle parabolic equation (PE) of underwater acoustics in a cylindrically symmetric medium consisting of water over a soft bottom B of range-dependent topography. There is strong indication that the initial-boundary value problem for this equation with just a homogeneous Dirichlet boundary condition posed on B may not be well-posed, for example when B is downsloping. We impose, in addition to the above, another homogeneous, second-order boundary condition, derived by assuming that the standard (narrow-angle) PE holds on B, and establish a priori H2 estimates for the solution of the resulting initial-boundary value problem for any bottom topography. After a change of the depth variable that makes B horizontal, we discretize the transformed problem by a second-order accurate finite difference scheme and show, in the case of upsloping and downsloping wedge-type domains, that the new model gives stable and accurate results. We also present an alternative set of boundary conditions that make the problem exactly energy conserving; one of these conditions may be viewed as a generalization of the Abrahamsson,Kreiss boundary condition in the wide-angle case. Copyright © 2008 John Wiley & Sons, Ltd. [source]

### Heat transfer in composite materials with Stefan,Boltzmann interface conditions

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2008
Yang Gufan
Abstract In this paper, we discuss nonstationary heat transfer problems in composite materials. This problem can be formulated as the parabolic equation with Stefan,Boltzmann interface conditions. It is proved that there exists a unique global classical solution to one-dimensional problems. Moreover, we propose a numerical algorithm by the finite difference method for this nonlinear transmission problem. Copyright © 2007 John Wiley & Sons, Ltd. [source]

### A fourth-order parabolic equation in two space dimensions

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2007
Changchun Liu
Abstract In this paper, we consider an initial-boundary problem for a fourth-order nonlinear parabolic equations. The problem as a model arises in epitaxial growth of nanoscale thin films. Based on the Lp type estimates and Schauder type estimates, we prove the global existence of classical solutions for the problem in two space dimensions. Copyright © 2007 John Wiley & Sons, Ltd. [source]

### Semigroup approach for identification of the unknown diffusion coefficient in a quasi-linear parabolic equation

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2007
Ali Demir
Abstract This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u(x,t)) in the quasi-linear parabolic equation ut(x,t)=(k(u(x,t))ux(x,t))x, with Dirichlet boundary conditions u(0,t)=,0, u(1,t)=,1. The main purpose of this paper is to investigate the distinguishability of the input,output mappings ,[,]:,, ,C1[0,T], ,[,]:,,,C1[0,T] via semigroup theory. In this paper, it is shown that if the null space of the semigroup T(t) consists of only zero function, then the input,output mappings ,[,] and ,[,] have the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of these mappings. Moreover, under the light of measured output data (boundary observations) f(t):=k(u(0,t))ux(0,t) or/and h(t):=k(u(1,t))ux(1,t), the values k(,0) and k(,1) of the unknown diffusion coefficient k(u(x,t)) at (x,t)=(0,0) and (x,t)=(1,0), respectively, can be determined explicitly. In addition to these, the values ku(,0) and ku(,1) of the unknown coefficient k(u(x,t)) at (x,t)=(0,0) and (x,t)=(1,0), respectively, are also determined via the input data. Furthermore, it is shown that measured output dataf(t) and h(t) can be determined analytically by an integral representation. Hence the input,output mappings ,[,]:,,, C1[0,T], ,[,]:,,,C1[0,T] are given explicitly in terms of the semigroup. Copyright © 2007 John Wiley & Sons, Ltd. [source]

### A critical exponent in a degenerate parabolic equation

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2002
Michael Winkler
We consider positive solutions of the Cauchy problem in for the equation $$u_t=u^p\,\Delta u+u^q,\quad p\geq1,\; q\geq 1$$\nopagenumbers\end and show that concerning global solvability, the number q = p + 1 appears as a critical growth exponent. Copyright © 2002 John Wiley & Sons, Ltd. [source]

### On the numerical computation of blowing-up solutions for semilinear parabolic equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2001
Theoretical aspects related to the approximation of the semilinear parabolic equation: $u_t=\Delta u+f(u)$\nopagenumbers\end, with a finite unknown ,blow-up' time Tb have been studied in a previous work. Specifically, for , a small positive number, we have considered coupled systems of semilinear parabolic equations, with positive solutions and ,mass control' property, such that: \def\ve{^\varepsilon}$$u_t\ve=\Delta u\ve+f(u\ve)v\ve\qquad v_t\ve=\Delta v\ve-\varepsilon f(u\ve)v\ve$$\nopagenumbers\end The solution \def\ve{^\varepsilon}$$\{u\ve,v\ve\}$$\nopagenumbers\end of such systems is known to be global. It is shown that $$\|(u^\varepsilon-u)(\, .\, ,t)\|_\infty\leq C(M_T)\varepsilon$$\nopagenumbers\end, \def\lt{\char'74}$t\leq T \lt T_b$\nopagenumbers\end where $M_T=\|u(\, .\, ,T)\|_\infty$\nopagenumbers\end and $C(M_T)$\nopagenumbers\end is given by (6). In this paper, we suggest a numerical procedure for approaching the value of the blow-up time Tb and the blow-up solution u. For this purpose, we construct a sequence $\{M_\eta\}$\nopagenumbers\end, with $\lim_{\eta\rightarrow 0}M_\eta=\infty$\nopagenumbers\end. Correspondingly, for $\varepsilon\leq1/2C(M_\eta+1)=\eta^\alpha$\nopagenumbers\end and \def\lt{\char'74}$0\lt\alpha\lt\,\!1$\nopagenumbers\end, we associate a specific sequence of times $\{T_\varepsilon\}$\nopagenumbers\end, defined by $\|u^\varepsilon(\, .\, ,T_\varepsilon)\|_\infty=M_\eta$\nopagenumbers\end. In particular, when $\varepsilon=\eta\leq\eta^\alpha$\nopagenumbers\end, the resulting sequence $\{T_\varepsilon\equiv T_\eta\}$\nopagenumbers\end, verifies, $\|(u-u^\eta)(\, .\, ,t)\|_\infty\leq{1\over2}(\eta)^{1-\alpha}$\nopagenumbers\end, \def\lt{\char'74}$0\leq t\leq T_\eta\lt T_{\rm b}$\nopagenumbers\end with $\lim_{\eta\rightarrow 0}T_\eta=T_{\rm b}$\nopagenumbers\end. The two special cases of a single-point blow-up where $f(u)=\lambda{\rm e}^u$\nopagenumbers\end and $f(u)=u^p$\nopagenumbers\end are then studied, yielding respectively sequences $\{M_\eta\}$\nopagenumbers\end of order $O(\ln|\ln(\eta)|)$\nopagenumbers\end and $O(\{|\ln(\eta)|\}^{1/p-1})$\nopagenumbers\end. The estimate $|T_\eta-T_{\rm b}|/T_{\rm b}=O(1/|\ln(\eta)|)$\nopagenumbers\end is proven to be valid in both cases. We conduct numerical simulations that confirm our theoretical results. Copyright © 2001 John Wiley & Sons, Ltd. [source]

### Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem

MATHEMATISCHE NACHRICHTEN, Issue 4 2010
Angelo Favini
Abstract We prove a very general form of the Angle Concavity Theorem, which says that if (T (t)) defines a one parameter semigroup acting over various Lp spaces (over a fixed measure space), which is analytic in a sector of opening angle ,p, then the maximal choice for ,p is a concave function of 1 , 1/p. This and related results are applied to give improved estimates on the optimal Lp angle of ellipticity for a parabolic equation of the form ,u /,t = Au, where A is a uniformly elliptic second order partial differential operator with Wentzell or dynamic boundary conditions. Similar results are obtained for the higher order equation ,u /,t = (,1)m +lAmu, for all positive integers m. [source]

### The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients

MATHEMATISCHE NACHRICHTEN, Issue 9 2009
Abstract We consider the Dirichlet problem for non-divergence parabolic equation with discontinuous in t coefficients in a half space. The main result is weighted coercive estimates of solutions in anisotropic Sobolev spaces. We give an application of this result to linear and quasi-linear parabolic equations in a bounded domain. In particular, if the boundary is of class C1,,, , , [0, 1], then we present a coercive estimate of solutions in weighted anisotropic Sobolev spaces, where the weight is a power of the distance to the boundary (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials

MATHEMATISCHE NACHRICHTEN, Issue 13-14 2007
Maurizio Grasselli
Abstract In this article, we study the long time behavior of a parabolic-hyperbolic system arising from the theory of phase transitions. This system consists of a parabolic equation governing the (relative) temperature which is nonlinearly coupled with a weakly damped semilinear hyperbolic equation ruling the evolution of the order parameter. The latter is a singular perturbation through an inertial term of the parabolic Allen,Cahn equation and it is characterized by the presence of a singular potential, e.g., of logarithmic type, instead of the classical double-well potential. We first prove the existence and uniqueness of strong solutions when the inertial coefficient , is small enough. Then, we construct a robust family of exponential attractors (as , goes to 0). (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### The radial basis functions method for identifying an unknown parameter in a parabolic equation with overspecified data

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2007
Mehdi Dehghan
Abstract Parabolic partial differential equations with overspecified data play a crucial role in applied mathematics and engineering, as they appear in various engineering models. In this work, the radial basis functions method is used for finding an unknown parameter p(t) in the inverse linear parabolic partial differential equation ut = uxx + p(t)u + ,, in [0,1] × (0,T], where u is unknown while the initial condition and boundary conditions are given. Also an additional condition ,01k(x)u(x,t)dx = E(t), 0 , t , T, for known functions E(t), k(x), is given as the integral overspecification over the spatial domain. The main approach is using the radial basis functions method. In this technique the exact solution is found without any mesh generation on the domain of the problem. We also discuss on the case that the overspecified condition is in the form ,0s(t)u(x,t)dx = E(t), 0 < t , T, 0 < s(t) < 1, where s and E are known functions. Some illustrative examples are presented to show efficiency of the proposed method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source]

### Anisotropic a posteriori error estimate for an optimal control problem governed by the heat equation

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2006
Marco Picasso
Abstract The abstract framework of Becker et al. is considered to solve an optimal control problem governed by a parabolic equation. Existence and uniqueness of a solution are proved using the inf-sup framework and space-time functional spaces. A Crank-Nicolson time discretization is proposed, together with continuous, piecewise linear finite elements in space. Existence and uniqueness of a solution to the discretized problem is also proved using the inf-sup framework. An a posteriori error estimate is proposed, the goal being to control the error between the true and computed cost functional. The error estimate remains valid on strongly anisotropic meshes and an anisotropic error indicator is proposed when the time step is small. Finally, the quality of this error indicator is studied numerically on isotropic and anisotropic meshes. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 [source]

### On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation,

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2005
Mehdi Dehghan
Abstract Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second-order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one-dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 [source]

### Numerical solution of the three-dimensional parabolic equation with an integral condition ,

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2002
Mehdi Dehghan
Abstract Developement of numerical methods for obtaining approximate solutions to the three dimensional diffusion equation with an integral condition will be carried out. The numerical techniques discussed are based on the fully explicit (1,7) finite difference technique and the fully implicit (7,1) finite difference method and the (7,7) Crank-Nicolson type finite difference formula. The new developed methods are tested on a problem. Truncation error analysis and numerical examples are used to illustrate the accuracy of the new algorithms. The results of numerical testing show that the numerical methods based on the finite difference techniques discussed in the present article produce good results. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 193,202, 2002; DOI 10.1002/num.1040 [source]

### Analysis of a block red-black preconditioner applied to the Hermite collocation discretization of a model parabolic equation

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2001
Stephen H. Brill
Abstract We are concerned with the numerical solution of a model parabolic partial differential equation (PDE) in two spatial dimensions, discretized by Hermite collocation. In order to efficiently solve the resulting systems of linear algebraic equations, we choose the Bi-CGSTAB method of van der Vorst (1992) with block Red-Black Gauss-Seidel (RBGS) preconditioner. In this article, we give analytic formulae for the eigenvalues that control the rate at which Bi-CGSTAB/RBGS converges. These formulae, which depend on the location of the collocation points, can be utilized to determine where the collocation points should be placed in order to make the Bi-CGSTAB/RBGS method converge as quickly as possible. Along these lines, we discuss issues of choice of time-step size in the context of rapid convergence. A complete stability analysis is also included. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:584,606, 2001 [source]

### Thermoelastic rolling contact problem with temperature dependent friction

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2008
Andrzej Chudzikiewicz
The paper is concerned with the numerical solution of a thermoelastic rolling contact problem with wear. The friction between the bodies is governed by Coulomb law. A frictional heat generation and heat transfer across the contact surface as well as Archard's law of wear in contact zone are assumed. The friction coefficient is assumed to depend on temperature. In the paper quasistatic approach to solve this contact problem is employed. This approach is based on the assumption that for the observer moving with the rolling body the displacement of the supporting foundation is independent on time. The original thermoelastic contact problem described by the hyperbolic inequality governing the displacement and the parabolic equation governing the heat flow is transformed into elliptic inequality and elliptic equation, respectively. In order to solve numerically this system we decouple it into mechanical and thermal parts. Finite element method is used as a discretization method. Numerical examples showing the influence of the temperature dependent friction coefficient on the temperature distribution and the length of the contact zone are provided. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### A deterministic-control-based approach motion by curvature,

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 3 2006
Robert Kohn
The level-set formulation of motion by mean curvature is a degenerate parabolic equation. We show that its solution can be interpreted as the value function of a deterministic two-person game. More precisely, we give a family of discrete-time, two-person games whose value functions converge in the continuous-time limit to the solution of the motion-by-curvature PDE. For a convex domain, the boundary's "first arrival time" solves a degenerate elliptic equation; this corresponds, in our game-theoretic setting, to a minimum-exit-time problem. For a nonconvex domain the two-person game still makes sense; we draw a connection between its minimum exit time and the evolution of curves with velocity equal to the "positive part of the curvature." These results are unexpected, because the value function of a deterministic control problem is normally the solution of a first-order Hamilton-Jacobi equation. Our situation is different because the usual first-order calculation is singular. © 2005 Wiley Periodicals, Inc. [source]

### Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6-7 2009
F. Nobile
Abstract We consider the problem of numerically approximating statistical moments of the solution of a time-dependent linear parabolic partial differential equation (PDE), whose coefficients and/or forcing terms are spatially correlated random fields. The stochastic coefficients of the PDE are approximated by truncated Karhunen,Loève expansions driven by a finite number of uncorrelated random variables. After approximating the stochastic coefficients, the original stochastic PDE turns into a new deterministic parametric PDE of the same type, the dimension of the parameter set being equal to the number of random variables introduced. After proving that the solution of the parametric PDE problem is analytic with respect to the parameters, we consider global polynomial approximations based on tensor product, total degree or sparse polynomial spaces and constructed by either a Stochastic Galerkin or a Stochastic Collocation approach. We derive convergence rates for the different cases and present numerical results that show how these approaches are a valid alternative to the more traditional Monte Carlo Method for this class of problems. Copyright © 2009 John Wiley & Sons, Ltd. [source]

### A fourth-order parabolic equation in two space dimensions

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2007
Changchun Liu
Abstract In this paper, we consider an initial-boundary problem for a fourth-order nonlinear parabolic equations. The problem as a model arises in epitaxial growth of nanoscale thin films. Based on the Lp type estimates and Schauder type estimates, we prove the global existence of classical solutions for the problem in two space dimensions. Copyright © 2007 John Wiley & Sons, Ltd. [source]

### Asymptotic behaviour of solutions of quasilinear evolutionary partial differential equations of parabolic type on unbounded spatial intervals

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2006
Poul
Abstract We study the long-time behaviour of solutions to a quasilinear parabolic problem on a half-line. The main result lies in showing the existence of a positive solution that converges to the travelling wave of solution to the stationary problem on the whole line. The main tools used here are the zero number theory and the concentration compactness principle. This result is a generalization of a result know for semilinear parabolic equations. Copyright © 2006 John Wiley & Sons, Ltd. [source]

### On the numerical computation of blowing-up solutions for semilinear parabolic equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2001
Theoretical aspects related to the approximation of the semilinear parabolic equation: $u_t=\Delta u+f(u)$\nopagenumbers\end, with a finite unknown ,blow-up' time Tb have been studied in a previous work. Specifically, for , a small positive number, we have considered coupled systems of semilinear parabolic equations, with positive solutions and ,mass control' property, such that: \def\ve{^\varepsilon}$$u_t\ve=\Delta u\ve+f(u\ve)v\ve\qquad v_t\ve=\Delta v\ve-\varepsilon f(u\ve)v\ve$$\nopagenumbers\end The solution \def\ve{^\varepsilon}$$\{u\ve,v\ve\}$$\nopagenumbers\end of such systems is known to be global. It is shown that $$\|(u^\varepsilon-u)(\, .\, ,t)\|_\infty\leq C(M_T)\varepsilon$$\nopagenumbers\end, \def\lt{\char'74}$t\leq T \lt T_b$\nopagenumbers\end where $M_T=\|u(\, .\, ,T)\|_\infty$\nopagenumbers\end and $C(M_T)$\nopagenumbers\end is given by (6). In this paper, we suggest a numerical procedure for approaching the value of the blow-up time Tb and the blow-up solution u. For this purpose, we construct a sequence $\{M_\eta\}$\nopagenumbers\end, with $\lim_{\eta\rightarrow 0}M_\eta=\infty$\nopagenumbers\end. Correspondingly, for $\varepsilon\leq1/2C(M_\eta+1)=\eta^\alpha$\nopagenumbers\end and \def\lt{\char'74}$0\lt\alpha\lt\,\!1$\nopagenumbers\end, we associate a specific sequence of times $\{T_\varepsilon\}$\nopagenumbers\end, defined by $\|u^\varepsilon(\, .\, ,T_\varepsilon)\|_\infty=M_\eta$\nopagenumbers\end. In particular, when $\varepsilon=\eta\leq\eta^\alpha$\nopagenumbers\end, the resulting sequence $\{T_\varepsilon\equiv T_\eta\}$\nopagenumbers\end, verifies, $\|(u-u^\eta)(\, .\, ,t)\|_\infty\leq{1\over2}(\eta)^{1-\alpha}$\nopagenumbers\end, \def\lt{\char'74}$0\leq t\leq T_\eta\lt T_{\rm b}$\nopagenumbers\end with $\lim_{\eta\rightarrow 0}T_\eta=T_{\rm b}$\nopagenumbers\end. The two special cases of a single-point blow-up where $f(u)=\lambda{\rm e}^u$\nopagenumbers\end and $f(u)=u^p$\nopagenumbers\end are then studied, yielding respectively sequences $\{M_\eta\}$\nopagenumbers\end of order $O(\ln|\ln(\eta)|)$\nopagenumbers\end and $O(\{|\ln(\eta)|\}^{1/p-1})$\nopagenumbers\end. The estimate $|T_\eta-T_{\rm b}|/T_{\rm b}=O(1/|\ln(\eta)|)$\nopagenumbers\end is proven to be valid in both cases. We conduct numerical simulations that confirm our theoretical results. Copyright © 2001 John Wiley & Sons, Ltd. [source]

### The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients

MATHEMATISCHE NACHRICHTEN, Issue 9 2009
Abstract We consider the Dirichlet problem for non-divergence parabolic equation with discontinuous in t coefficients in a half space. The main result is weighted coercive estimates of solutions in anisotropic Sobolev spaces. We give an application of this result to linear and quasi-linear parabolic equations in a bounded domain. In particular, if the boundary is of class C1,,, , , [0, 1], then we present a coercive estimate of solutions in weighted anisotropic Sobolev spaces, where the weight is a power of the distance to the boundary (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### On the decay of the solutions of second order parabolic equations with Dirichlet conditions

MATHEMATISCHE NACHRICHTEN, Issue 8 2007
Brice FrankeArticle first published online: 8 MAY 200
Abstract We use rearrangement techniques to investigate the decay of the parabolic Dirichlet problem in a bounded domain. The coefficients of the second order term are used to introduce an isoperimetric problem. The resulting isoperimetric function together with the divergence of the first order coefficients and the value distribution of the zero order part are then used to construct a symmetric comparison equation having a slower heat-flow than the original equation. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2010
Hong-Lin Liao
Abstract Alternating direction implicit (ADI) schemes are computationally efficient and widely utilized for numerical approximation of the multidimensional parabolic equations. By using the discrete energy method, it is shown that the ADI solution is unconditionally convergent with the convergence order of two in the maximum norm. Considering an asymptotic expansion of the difference solution, we obtain a fourth-order, in both time and space, approximation by one Richardson extrapolation. Extension of our technique to the higher-order compact ADI schemes also yields the maximum norm error estimate of the discrete solution. And by one extrapolation, we obtain a sixth order accurate approximation when the time step is proportional to the squares of the spatial size. An numerical example is presented to support our theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 [source]

### On stable implicit difference scheme for hyperbolic,parabolic equations in a Hilbert space

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2009
Allaberen Ashyralyev
Abstract The first-order of accuracy difference scheme for approximately solving the multipoint nonlocal boundary value problem for the differential equation in a Hilbert space H, with self-adjoint positive definite operator A is presented. The stability estimates for the solution of this difference scheme are established. In applications, the stability estimates for the solution of difference schemes of the mixed type boundary value problems for hyperbolic,parabolic equations are obtained. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source]

### On the maximum principle and its application to diffusion equations

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2007
T. Stys
Abstract In this article, an analog of the maximum principle has been established for an ordinary differential operator associated with a semi-discrete approximation of parabolic equations. In applications, the maximum principle is used to prove O(h2) and O(h4) uniform convergence of the method of lines for the diffusion Equation (1). The system of ordinary differential equations obtained by the method of lines is solved by an implicit predictor corrector method. The method is tested by examples with the use of the enclosed Mathematica module solveDiffusion. The module solveDiffusion gives the solution by O(h2) uniformly convergent discrete scheme or by O(h4) uniformly convergent discrete scheme. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source]

### The probability approach to numerical solution of nonlinear parabolic equations

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2002
G. N. Milstein
Abstract A number of new layer methods for solving semilinear parabolic equations and reaction-diffusion systems is derived by using probabilistic representations of their solutions. These methods exploit the ideas of weak sense numerical integration of stochastic differential equations. In spite of the probabilistic nature these methods are nevertheless deterministic. A convergence theorem is proved. Some numerical tests are presented. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 490,522, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10020 [source]

### An unconditionally stable and O(,2 + h4) order L, convergent difference scheme for linear parabolic equations with variable coefficients

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2001
Zhi-Zhong Sun
Abstract M. K. Jain, R. K. Jain, and R. K. Mohanty presented a finite difference scheme of O(,2 + ,h2 + h4) for solving the one-dimensional quasilinear parabolic partial differential equation, uxx = f(x, t, u, ut, ux), with Dirichlet boundary conditions. The method, when applied to a linear constant coefficient case, was shown to be unconditionally stable by the Von Neumann method. In this article, we prove that the method, when applied to a linear variable coefficient case, is unconditionally stable and convergent with the convergence order O(,2 + h4) in the L, -norm. In addition, we obtain an asymptotic expansion of the difference solution, with which we obtain an O(,4 + ,2h4 + h6) order accuracy approximation after extrapolation. And last, we point out that the analysis method in this article is efficacious for complex equations. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:619,631, 2001 [source]