Equations in one space variable

Abstract

Equations in One Space Variable INTRODUCTIONIn Chapt~r 1 we discussed methods for solving IVPs, whereas in Chapters 2 and 3 boundary-value problems were treated. This chapter combines the techniques from these chapters to solve parabolic partial differential equations in one space variable. CLASSifiCATION Of PARTIAL DiffERENTIAL EQUATIONSConsider the most general linear partial differential equation of the second order in two independent variables: To each of the equations (4.1) we must adjoin appropriate subsidiary relations, called boundary and/or initial conditions, which serve to complete the formulation of a "meaningful problem." These conditions are related to the domain in which (4.1) is to be solved. METHOD OF LINESConsider the diffusion equation: Discretize the spatial derivative in (4.4) using finite differences to obtain the following system of ordinary differential equations: In the MOL, the boundary conditions are incorporated into the discretization in the x-direction while the initial condition is used to start the associated IVP.