98 - Diagonalization of a Linear System of PDEs

Abstract

This chapter describes the diagonalization of a linear system of partial differential equations (PDEs). This technique is applicable to a linear system of partial differential equations in two independent variables of the form ut + Aux = 0, where A is a constant matrix, it yields a set of uncoupled equations. By diagonalizing the coefficient matrix, the equations can be uncoupled and then solved. Given the linear system of differential equations ut + Aux = 0, the dependent variables are changed to decouple the system. If the matrix A is n × n and has the eigenvectors {v1, v2,…, vn} (which are assumed to be linearly independent), then the matrix S is defined by S = (v1, v2, . vn). Changing variables in ut + Aux = 0 by u = Sw results in Swt + ASwx = 0, or wt + Awx = 0, where A = S-1 AS is a diagonal matrix. The equations in wt + Awx = 0 are now decoupled and can be solved separately for {w1(x, t), w2(x, t),…,wn(x, t)}. After they have been found, u may be determined from u = Sw.