89 - Superposition*

Abstract

This chapter discusses the concept of superposition applicable to linear differential equations. Superposition yields a set of linear differential equations with “easier” initial conditions or boundary conditions. The sum of the solutions to these new equations will produce the solution to the original equation. By use of superposition, the solution to an inhomogeneous linear differential equation may be determined in terms of simpler systems. Given a linear differential equation with a forcing term, inhomogeneous initial conditions, or inhomogeneous boundary conditions, a set of equations is constructed with each equation having more homogeneous parts than the original system. Each of these parts is solved separately, and then combined for the final solution. In fluid dynamics, the influence of an obstacle in a flow can be simulated by a continuous superposition of sources. There also exist superposition principles for nonlinear equations. These are relations that allow new solutions, with arbitrary constants in them, to be calculated from other solutions.