108 - Separation of Variables

Abstract

This chapter examines the yield and application of separation of variables. It is applicable to most often, linear homogeneous partial differential equations. It yields an exact solution, generally in the form of an infinite series. A solution to a partial differential equation by separating the solution into pieces, where each piece deals with a single dependent variable is found. It is suggested for linear homogeneous partial differential equations to represent the solution as a sum of terms where each term factors into a product of expressions, each expression dealing with a single independent variable. For nonlinear equations, the solution may be represented as a sum of such expressions. In all cases, not only must the equation admit a solution of the proposed form, but the boundary conditions must also have the right form. It is found that since superposition can be used in linear equations, any number of terms will also be a solution of the original equation. It is observed that if each of these terms is multiplied by some constant, and then added together, the resulting expression will also be a solution.