This book talks about the symbolic generation of finite difference schemes, especially so-called compact finite difference schemes, and their numerical applications to elliptic equations, integro-differential equations, the American option pricing problem, and cardiac tissue models. We take as a base Corless and Rokicki’s 1995 work on automatic generation of finite difference formulae and numerical integration formulae of univariate and bivariate problems. We then extend this methodology to any dimension. The new Maple routine FINDIF allows for automatic, symbolic discretization of various finite difference formulae, integration formulae, and computes formulae for truncation errors. Compact finite difference schemes are given for boundary value problems and elliptic partial differential equations. Compact finite difference methods are also used to solve efficiently integro-differential equations (IDE’s). Furthermore, we apply the method to solve the famous American option pricing problem and simulate the action potential propagation through two dimensional cardiac tissues. All simulation results demonstrate the compact finite difference method is a promising approach.

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