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*Suppose U := (0, 1) c R and a smooth enough function u : U × R, → R satisfies Un – Uxx = f(x, t) u(0, t) = g1(t), u(1,t) = g2(t) t>0 и(х, 0) %3D Ф(х) и,(х,0) %3 W(х) x € U, t > 0 χεU. x € U, for f e C(U × R,), ø, Y E C²(U) and g1,92 € C(R,). Prove, using the energy method, that there is at most one solution to the above system. (Hint: Suppose there are two distinct solutions u and v and take the inner product of the PDE for w := u – v with w, to reach a contradiction.)

*Suppose U := (0, 1) c R and a smooth enough function u : U × R, → R satisfies Un – Uxx = f(x, t) u(0, t) = g1(t), u(1,t) = g2(t) t>0 и(х, 0) %3D Ф(х) и,(х,0) %3 W(х) x € U, t > 0 χεU. x € U, for f e C(U × R,), ø, Y E C²(U) and g1,92 € C(R,). Prove, using the energy method, that there is at most one solution to the above system. (Hint: Suppose there are two distinct solutions u and v and take the inner product of the PDE for w := u – v with w, to reach a contradiction.)

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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1.* Suppose U := (0, 1) C R and a smooth enough function u : U x R, → R satisfies χε U t> 0 Un – Uxx = f(x, t) u(0, 1) %3D 9:(), и(1,1) %3 92(1) 1>0 u(х,0) %3 Ф(х) u,(x, 0) = µ(x) χεU x € U, for f e C(U × R,), ø, Y e C²(U) and g1, 92 € C(R,). Prove, using the energy method, that there is at most one solution to the above system. (Hint: Suppose there are two distinct solutions u and v and take the inner product of the PDE for w := u – v with w, to reach a contradiction.)
Transcribed Image Text:1.* Suppose U := (0, 1) C R and a smooth enough function u : U x R, → R satisfies χε U t> 0 Un – Uxx = f(x, t) u(0, 1) %3D 9:(), и(1,1) %3 92(1) 1>0 u(х,0) %3 Ф(х) u,(x, 0) = µ(x) χεU x € U, for f e C(U × R,), ø, Y e C²(U) and g1, 92 € C(R,). Prove, using the energy method, that there is at most one solution to the above system. (Hint: Suppose there are two distinct solutions u and v and take the inner product of the PDE for w := u – v with w, to reach a contradiction.)
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