Partial Differential Equations

Abstract

Partial differential equations (PDEs) are multivariate different equations where derivatives of more than one dependent variable occur. That is, the derivatives in the equation are partial derivatives. As such they are generalizations of ordinary differentials equations, which were covered in Chapter 9. Conceptually, the difference between ordinary and partial differential equations is not that big, but the computational techniques required to deal with ODEs and PDEs are very different, and solving PDEs is typically much more computationally demanding. Most techniques for solving PDEs numerically are based on the idea of discretizing the problem in each independent variable that occurs in the PDE, and thereby recasting the problem into an algebraic form. This usually results in very large-scale linear algebra problems. Two common techniques for recasting PDEs into algebraic form is the finite-difference methods (FDMs), where the derivatives in the problem are approximated with their finite-difference formula; and the finite-element methods (FEMs), where the unknown function is written as linear combination of simple basis functions that can be differentiated and integrated easily. The unknown function is described by a set of coefficients for the basis functions in this representation, and by a suitable rewriting of the PDEs we can obtain algebraic equations for these coefficients.