ABSTRACT
Partial differential equations are widely used in describing various physical phenomena such as sound wave propagation, heat and mass transport, electrostatics and electrodynamics, elasticity of solids, and fluid dynamics. Analytical solutions of partial differential equations are mainly possible to obtain for linear equations and in canonical domains, which makes numerical solution an important option. In this chapter, in addition to summarizing various schemes employed to solve linear partial differential equations, an iterative scheme known as generalized minimal residual method (GMRES) will be discussed which approximates the solution of discretized linear equations by a vector in Krylov subspace with minimum residual. As an illustration, this scheme has been used here to solve a transient 2D heat equation with Dirichlet boundary conditions. Finite difference method has been used to discretize the heat equation in space and time, resulting in a linear system of equation, Ax = b, where A is a sparse matrix.
