INITIAL BOUNDARY VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL AND DIFFERENCE EQUATIONS IN ONE SPACE DIMENSION

Abstract

This chapter presents initial boundary value problems for partial differential and difference equations in one space dimension. It highlights a system of partial differential equations. There are a number of ways to define when the initial boundary value problem is well-posed. One way is to use essentially the same definition as for the Cauchy problem. The chapter discusses strictly hyperbolic systems in more than one space dimension. For hyperbolic and parabolic systems, it is easy to generalize the estimates to equations with variable coefficients because the matrix H depends smoothly on the coefficients of the differential equations. The chapter presents difference approximations for hyperbolic systems and highlights a first-order system of partial differential equations. It also explores the Ryabenkii–Gudonow condition.