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Question 3. Separation of variables. Consider Laplace's Equation in two dimensions:-+-- (a) Write φ(z y) F(x)G(y) and use separation of variables to get ordinary differential equa- tions for F and CG (b) Consider the rectangular region ((r,y) E R:0 conditions on Φ < a, 0 y b with three boundary 0(x, 0) = 0, D(x, b) = 0, (0,y) = 0 Obtain conditions on F and G on those boundaries where conditions on Φ are given (c) (i) Solve the differential equations found in (a), subject to the conditions found in (b) (You must consider all three cases of the separation constant. Solve for G(y) first, and then only find F() for the case(s) where G(y) is not trivial.) (ii) Assemble the general solution for Ф(x,y) (d) Consider now the semi-infinite region {(r,y) E R2 : x > 0, 0 < y < b}. Two boundaries are earthed: Φ(x,0) = 0 and Φ(x, b) 0. Furthermore, as x → oo, φ → 0 (i) Obtain the corresponding conditions on F and G for this problenm (ii) Solve the differential equations found in (a) subject to these boundary conditions (Hint: You already know G(y) and the possible values of the separation constant) (iii) Assemble the general solution for Φ(x,y)

Answer 1

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