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3 0 0, subject to the following boundary and initial conditions t > 0, t> 0, v(0, t) = 0, %3D v(3, t) = 0, %3D v(z,0) 7 sin() - 12 sin(r2), 0

3 0 0, subject to the following boundary and initial conditions t > 0, t> 0, v(0, t) = 0, %3D v(3, t) = 0, %3D v(z,0) 7 sin() - 12 sin(r2), 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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partC D

7. Consider the partial differential equation du 0, 0 <z < 3, t> 0, subject to the following boundary and initial conditions u(0, t) 2, t> 0, %3! u(3, t) 5, t>0, u(r, 0) = r+2+7sin() - 12 sin(rr), 0<rく3. The solution has the form u(r, t) = v(x,t) + ü(x). (a) Find the steady state solution ü(x). (b) Show that v(x,t) satisfies the PDE = 3- 0 <r< 3, t > 0, subject to the following boundary and initial conditions v(0, t) = 0, t>0, v(3, t) = 0. t> 0, v(x,0) = 7 sin()- 12 sin(r2), 0<I< 3. 3 (c) Solve the system in (b) for v(x, t) using separation of variables. (d) Find u(r, t).
Transcribed Image Text:7. Consider the partial differential equation du 0, 0 <z < 3, t> 0, subject to the following boundary and initial conditions u(0, t) 2, t> 0, %3! u(3, t) 5, t>0, u(r, 0) = r+2+7sin() - 12 sin(rr), 0<rく3. The solution has the form u(r, t) = v(x,t) + ü(x). (a) Find the steady state solution ü(x). (b) Show that v(x,t) satisfies the PDE = 3- 0 <r< 3, t > 0, subject to the following boundary and initial conditions v(0, t) = 0, t>0, v(3, t) = 0. t> 0, v(x,0) = 7 sin()- 12 sin(r2), 0<I< 3. 3 (c) Solve the system in (b) for v(x, t) using separation of variables. (d) Find u(r, t).
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