Consider Laplaces equation uxx +uyy = 0 in the parallelogram whose vertices are (0, 0)

Chapter 11, Problem 1

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Consider Laplaces equation uxx +uyy = 0 in the parallelogram whose vertices are (0, 0), (2, 0), (3, 2), and (1, 2). Suppose that the boundary condition u( x, 2) = f ( x) is imposed on the side y = 2 for 1 x 3, and that on the other three sides u = 0 (see Figure 11.5.1). a. Show that there are no nontrivial solutions of the partial differential equation of the form u( x, y) = X( x)Y ( y) that also satisfy the homogeneous boundary conditions. b. Let = x 1 2 y, = y. Show that the given parallelogram in the xy-plane transforms into the square 0 2, 0 2 in the -plane. Show that the differential equation transforms into 5 4 u u + u = 0. (21) How are the boundary conditions transformed? c. Show that in the -plane, the differential equation (21) possesses no solution of the form u(, ) = U()V( ). Thus in the xy-plane, the shape of the boundary precludes a solution by the method of the separation of variables, while in the -plane, the region is acceptable but the variables in the differential equation can no longer be separated.