. So far, we have seem how to solve various partial differential equations through separation of...

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. So far, we have seem how to solve various partial differential equations through separation of variables. It is not difficult to extend this to inhomogeneous equations. Consider, for the equation (1) 2² 2² + = 1, əx² Əy² with boundary conditions V = 0 on the four sides of the square. (0<x<T, 0<y < π), (a) Express the right hand side of Eq. (1) as a Fourier sine series in z (we use a sine series because of the zero boundary conditions): and evaluate the bn's. (b) Evaluate V(x, y) as 1 = 2b, sin(nz), bn n=1 ∞ (2) V(x, y) = Vn(y) sin(n.x). (3) n=1 Substitute into Eq. (1), and also use Eq. (2). Equating the coefficients of sin(nx) on both sides obtain an inhomogeneous ordinary differential equation for each Vn(y). (c) Verify that the constant solution, Vn (y) Un, satisfies the differential equation for Vn(y), and find Un. Hence verify that the general solution to the differential equation is Vn(u) =un + B, sinh(ng) + An sinh(n(# – y)), and find An and B, using the boundary conditions. (d) Write down the solution for V(x, y). . So far, we have seem how to solve various partial differential equations through separation of variables. It is not difficult to extend this to inhomogeneous equations. Consider, for the equation (1) 2² 2² + = 1, əx² Əy² with boundary conditions V = 0 on the four sides of the square. (0<x<T, 0<y < π), (a) Express the right hand side of Eq. (1) as a Fourier sine series in z (we use a sine series because of the zero boundary conditions): and evaluate the bn's. (b) Evaluate V(x, y) as 1 = 2b, sin(nz), bn n=1 ∞ (2) V(x, y) = Vn(y) sin(n.x). (3) n=1 Substitute into Eq. (1), and also use Eq. (2). Equating the coefficients of sin(nx) on both sides obtain an inhomogeneous ordinary differential equation for each Vn(y). (c) Verify that the constant solution, Vn (y) Un, satisfies the differential equation for Vn(y), and find Un. Hence verify that the general solution to the differential equation is Vn(u) =un + B, sinh(ng) + An sinh(n(# – y)), and find An and B, using the boundary conditions. (d) Write down the solution for V(x, y).

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