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Solve the following initial boundary value problem for the wave equation: Utt = 16uz, u,(0, t) = u_(4, t) = 0, u(z,0) = 0, u,(z,0) = 2z + 1, 00 (1)| t>0 (2) 0

Solve the following initial boundary value problem for the wave equation: Utt = 16uz, u,(0, t) = u_(4, t) = 0, u(z,0) = 0, u,(z,0) = 2z + 1, 00 (1)| t>0 (2) 0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve the following initial boundary value problem for the wave equation: 16uzz, u, (0, t) = u,(4, t) = 0, u(x, 0) = 0, u(z, 0) = 2x +1, 0<z< 4, t >0 (1)| t>0 (2) 0 <z<4 (3) Derive an expression for all nontrivial product solutions u(z, t) = X(z)T(t)| of (1) satisfying boundary conditions (2), and solve the corresponding eigenvalue problem. The eigenvalues are ,n = 1,2,...| The corresponding eigenfunctions are Xp(x) = n = 1,2,...| F(2)"X All product solutions of equation (1) satisfying boundary conditions (2) are uo(z, t) = } Ao + Bd| %3D U,(z, t) = (A|| +B ,n = 1,2,...| where An, Bn, n = 0,1, 2, ... | are arbitrary constants. Use Fourier series representation to find all constants An, Bn, n = 0, 1, 2,.. such that u(z, t) = u9(x, t) + 4„(z,t) is a solution of problem (1)-(3). Ag = Bo = A, = n = 1,2, ...| B = n = 1,2, ...|
Transcribed Image Text:Solve the following initial boundary value problem for the wave equation: 16uzz, u, (0, t) = u,(4, t) = 0, u(x, 0) = 0, u(z, 0) = 2x +1, 0<z< 4, t >0 (1)| t>0 (2) 0 <z<4 (3) Derive an expression for all nontrivial product solutions u(z, t) = X(z)T(t)| of (1) satisfying boundary conditions (2), and solve the corresponding eigenvalue problem. The eigenvalues are ,n = 1,2,...| The corresponding eigenfunctions are Xp(x) = n = 1,2,...| F(2)"X All product solutions of equation (1) satisfying boundary conditions (2) are uo(z, t) = } Ao + Bd| %3D U,(z, t) = (A|| +B ,n = 1,2,...| where An, Bn, n = 0,1, 2, ... | are arbitrary constants. Use Fourier series representation to find all constants An, Bn, n = 0, 1, 2,.. such that u(z, t) = u9(x, t) + 4„(z,t) is a solution of problem (1)-(3). Ag = Bo = A, = n = 1,2, ...| B = n = 1,2, ...|
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