Basics of Forward Problem

Correct formulation of the forward problem is essential to obtain a meaningful solution of the associated inverse problem. Since the partial differential equation (PDE) is a suitable mathematical tool to describe most physical phenomena, this chapter reviews the different kinds of PDEs commonly used in physical science. In constructing a mathematical model that transforms physical phenomena into a collection of mathematical expressions and data, the chapter considers the following three properties: existence, uniqueness and continuity or stability. The chapter introduces the heat equation used to describe a heat conduction process within an object. A wave is a spatial disturbance of a medium that propagates at a wave speed depending on the substance of the medium. The wave equation describes wave propagation in the medium. There are many physical boundary value problems expressed as Laplace or Poisson equations. The chapter provides several examples to obtain some insights into their solutions. Controlled Vocabulary Terms Laplace equations; Poisson equation; wave equations