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Throughout this question, consider the following problen Question 4, for the Laplace equation on a rectangle nC R2,N = {0

Throughout this question, consider the following problen Question 4, for the Laplace equation on a rectangle nC R2,N = {0

PREALGEBRA
PREALGEBRA
15th Edition
ISBN:9781938168994
Author:OpenStax
Publisher:OpenStax
Chapter11: Graphs
Section11.1: Use The Rectangular Coordinate System
Problem 11.1TI: use the map in Figure 11.2. a Find the grid section of Taylor Hall. b. What is located in section...
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Throughout this question, consider the following probler Question 4, for the Laplace equation on a rectangle nC R?, n = {0 < x <a,0 <y < b} Ux+ Uyy = 0, U(x,0) = 0, U(0, y) = g(y), (x,y) En U(x,b) = 0 U(a, y) = 0. %3D (a) Following the method of separation of variables consider solutions of the form U(x, y) = X(x)Y(y), where X and Y are functions of a single argument. Show that X and Y satisfy the ordinary differential equations X" = kX y" = -kY for some constant k. Moreover, show that Y(0) = Y(b) = X(a) = 0. (b) Show that the constant k obtained in (a) must be positive if Y (y) is not identically 0 for y e [0,b). (c) Find the general solution to the ordinary differential equations in (a). (d) Use the conditions Y(0) = Y(b) = 0 to determine the value of the constant k and show that the solutions Y obtained in (c) must be of the form (nny Y(y) = sin (), n = 1,2,3, ... Moreover, show that if X(a) = 0, then (ηπ(x- a) X(x) = sinh n = 1,2,3,... (e) Use the Principle of Superposition to find the general solution to the Laplace equation on the rectangle N with the prescribed boundary conditions. (f) Briefly explain how would you solve the general problem Ux+Uyy = 0, U(x,0) = fi(x), u(0, y) = g1(y), (x,y) En U(x, b) = f2(x) U(a, y) = 82(y). You may use a diagram to explain your idea.
Transcribed Image Text:Throughout this question, consider the following probler Question 4, for the Laplace equation on a rectangle nC R?, n = {0 < x <a,0 <y < b} Ux+ Uyy = 0, U(x,0) = 0, U(0, y) = g(y), (x,y) En U(x,b) = 0 U(a, y) = 0. %3D (a) Following the method of separation of variables consider solutions of the form U(x, y) = X(x)Y(y), where X and Y are functions of a single argument. Show that X and Y satisfy the ordinary differential equations X" = kX y" = -kY for some constant k. Moreover, show that Y(0) = Y(b) = X(a) = 0. (b) Show that the constant k obtained in (a) must be positive if Y (y) is not identically 0 for y e [0,b). (c) Find the general solution to the ordinary differential equations in (a). (d) Use the conditions Y(0) = Y(b) = 0 to determine the value of the constant k and show that the solutions Y obtained in (c) must be of the form (nny Y(y) = sin (), n = 1,2,3, ... Moreover, show that if X(a) = 0, then (ηπ(x- a) X(x) = sinh n = 1,2,3,... (e) Use the Principle of Superposition to find the general solution to the Laplace equation on the rectangle N with the prescribed boundary conditions. (f) Briefly explain how would you solve the general problem Ux+Uyy = 0, U(x,0) = fi(x), u(0, y) = g1(y), (x,y) En U(x, b) = f2(x) U(a, y) = 82(y). You may use a diagram to explain your idea.
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