Advanced Engineering Mathematics, 7th Edition

16.1 Laplace’s Equation

INTRODUCTION

Recall from Section 13.1 that linear second-order PDEs in two independent variables are classified as elliptic, parabolic, and hyperbolic. Roughly, elliptic PDEs involve partial derivatives with respect to spatial variables only and as a consequence solutions of such equations are determined by boundary conditions alone. Parabolic and hyperbolic equations involve partial derivatives with respect to both spatial and time variables, and so solutions of such equations generally are determined from boundary and initial conditions. A solution of an elliptic PDE (such as Laplace’s equation) can describe a physical system whose state is in equilibrium (steady state), a solution of a parabolic PDE (such as the heat ...