Evolutionary Equations
D.S. Jones, Michael Plank, B.D. Sleeman · 2009
In addition to the techniques described in Chapter 10, the method of separation of variables is a powerful tool with which to solve linear partial differential equations. To begin with, we seek solutions of (11.1.2) in the form u(x, t) = X(x)T (t). (11.1.3) Substitution of this into (11.1.2) leads to the identity X dT dt = T d2X dx2 or 1 T dT dt = X d2X dx2 . (11.1.4) Now 1T dT dt is a function of t only, while d2X dx2 is a function of x only. Con- sequently, both sides of (11.1.4) must be equal to a constant, say λ. Thus X and T must satisfy the ordinary differential equations dT dt − λT = 0, d2X dx2 − λX = 0. (11.1.5) X(x) = exp± √ λx, T (t) = expλt.