### 13.5 Laplace’s Equation

## INTRODUCTION

Suppose we wish to find the steady-state temperature *u*(*x*, *y*) in a rectangular plate whose vertical edges *x* = 0 and *x* = *a* are insulated, and whose upper and lower edges *y* = *b* and *y* = 0 are maintained at temperatures *f*(*x*) and 0, respectively. See FIGURE 13.5.1. When no heat escapes from the lateral faces of the plate, we solve the following boundary-value problem:

(1)

(2)

*u*(*x*, 0) = 0, *u*(*x*, *b*) = *f*(*x*), 0 < *x* < *a*.(3)

### Solution of the BVP

With ...

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