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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