![]() Second, MathPDE can produce a standalone executable that runs independently of Mathematica, providing portability. First, MathPDE is able to handle irregular spatial domains, not just rectangular or cubic domains. So a natural question that arises is why another solver is needed. Mathematica has a built-in function NDSolve that can numerically solve a variety of PDEs and offers the user a very simple and quick route to solving PDEs numerically. We first review, very briefly, the extant PDE packages that we are familiar with that have been developed over the years. ![]() Before doing so, we would like to put our work in perspective. We give examples of the way in which symbolic programming lends a degree of generality to MathPDE. In this article, we present the Mathematica package MathPDE that implements the finite-difference method for time-dependent PDE problems. ![]() The numerical problem in both methods therefore is one of solving a system of algebraic equations. A weighted residual method is then used to set up a system of algebraic equations for these coefficients. The interpolation polynomial is determined by a set of coefficients in each element the small size of the elements generally ensures that a low-degree polynomial is sufficient. The domain is subdivided into a set of non-overlapping regions that completely cover it each such region is called a finite element. In the finite-element method, the dependent variable is approximated by an interpolation polynomial. This identification helps us transform the PDE problem to a system of coupled algebraic equations. Any derivative then automatically acquires the meaning of a certain kind of difference between dependent variable values at the grid points. In finite-difference methods, the domain of the independent variables is approximated by a discrete set of points called a grid, and the dependent variables are defined only at these points. Roughly speaking, both transform a PDE problem to the problem of solving a system of coupled algebraic equations. These fall into two broad categories: the finite-difference methods and the finite-element methods. The numerical task is made difficult by the dimensionality and geometry of the independent variables, the nonlinearities in the system, sensitivity to boundary conditions, a lack of formal understanding of the kind of solution method to be employed for a particular problem, and so on.Ī number of methods have been developed to deal with the numerical solution of PDEs. ![]() However, few PDEs have closed-form analytical solutions, making numerical methods necessary. ![]() Examples range from the simple (but very common) diffusion equation, through the wave and Laplace equations, to the nonlinear equations of fluid mechanics, elasticity, and chaos theory. Mathematical problems described by partial differential equations (PDEs) are ubiquitous in science and engineering. Since a standalone C++ program is generated to compute the numerical solution, the package offers portability. This article discusses the wide range of PDEs that can be handled by MathPDE, the accuracy of the finite-difference schemes used, and importantly, the ability to handle both regular and irregular spatial domains. When the algebraic system is nonlinear, the Newton-Raphson method is used and SuperLU, a library for sparse systems, is used for matrix operations. MathPDE then internally calls MathCode, a Mathematica-to-C++ code generator, to generate a C++ program for solving the algebraic problem, and compiles it into an executable that can be run via MathLink. After making a sequence of symbolic transformations on the PDE and its initial and boundary conditions, MathPDE automatically generates a problem-specific set of Mathematica functions to solve the numerical problem, which is essentially a system of algebraic equations. A package for solving time-dependent partial differential equations (PDEs), MathPDE, is presented. ![]()
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