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2. Consider a problem we have already solved in class, the 1D wave equation with Dirichlet boundary conditions: Utt - c²Uxx = 0, u(x,0) = f(x), ut(x,0) = g(x), u(0, t) = u(L, t) = 0 In this problem you will show that solutions u(x, t) to this wave equation are unique. (a) Let u₁(x, t) and u₂(x, t) be two solutions to the above problem with the same initial data f and g. Define w(x, t) = u₁(x, t) - u₂(x, t). What initial boundary value problem does w solve? (b) Define the energy of the system by 0 < x

2. Consider a problem we have already solved in class, the 1D wave equation with Dirichlet boundary conditions: Utt - c²Uxx = 0, u(x,0) = f(x), ut(x,0) = g(x), u(0, t) = u(L, t) = 0 In this problem you will show that solutions u(x, t) to this wave equation are unique. (a) Let u₁(x, t) and u₂(x, t) be two solutions to the above problem with the same initial data f and g. Define w(x, t) = u₁(x, t) - u₂(x, t). What initial boundary value problem does w solve? (b) Define the energy of the system by 0 < x

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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2. Consider a problem we have already solved in class, the 1D wave equation with Dirichlet boundary conditions: Utt - c²Uxx = 0, u(x,0) = f(x), ut(x,0) = g(x), u(0, t) = u(L, t) = 0 In this problem you will show that solutions u(x, t) to this wave equation are unique. (a) Let u₁(x, t) and u₂(x, t) be two solutions to the above problem with the same initial data f and g. Define w(x, t) = u₁(x, t) - u₂(x, t). What initial boundary value problem does w solve? (b) Define the energy of the system by 0 < x <L 0 < x <L 0 < x <L B (t) = 1/2 √² (w² + 2 w2²) de What is E(0)? Show by passing the derivative inside the integral, i.e. by com- puting E'(t), that E'(t) = 0. HINT: if a function g(x, t) satisfies g(0,t) = 0 = g(L,t)=0 for all time, this necessarily implies that gt(0, t) = 0 = gt(L,t).
Transcribed Image Text:2. Consider a problem we have already solved in class, the 1D wave equation with Dirichlet boundary conditions: Utt - c²Uxx = 0, u(x,0) = f(x), ut(x,0) = g(x), u(0, t) = u(L, t) = 0 In this problem you will show that solutions u(x, t) to this wave equation are unique. (a) Let u₁(x, t) and u₂(x, t) be two solutions to the above problem with the same initial data f and g. Define w(x, t) = u₁(x, t) - u₂(x, t). What initial boundary value problem does w solve? (b) Define the energy of the system by 0 < x <L 0 < x <L 0 < x <L B (t) = 1/2 √² (w² + 2 w2²) de What is E(0)? Show by passing the derivative inside the integral, i.e. by com- puting E'(t), that E'(t) = 0. HINT: if a function g(x, t) satisfies g(0,t) = 0 = g(L,t)=0 for all time, this necessarily implies that gt(0, t) = 0 = gt(L,t).
(c) By part (b), E(t) is constant, so E(t) = E(0) = 0. But since w is a solution, then we conclude that wx(x, t) = 0 = wt(x, t). Thus w(x, t) = C for some constant C. Determine the constant and conclude uniqueness.
Transcribed Image Text:(c) By part (b), E(t) is constant, so E(t) = E(0) = 0. But since w is a solution, then we conclude that wx(x, t) = 0 = wt(x, t). Thus w(x, t) = C for some constant C. Determine the constant and conclude uniqueness.
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