3. Solve the heat equationдиk-for 2 < x < 4, and t >0with the boundary conditionsu(2, t) = 0 and u(4, t) = 0, for t > 0,%3Dand the initial value conditionu(x, 0) = x for x E [2, 4].

Answered: Image /qna-images/answer/0aa54bf5-f2bb-4476-8913-7b9c7d31a0db.jpg

Answered: 3. Solve the heat equation ди k- for 2…

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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Let us consider the following problem for the heat equation ди = 0 in (0,1) × (0,+∞), ốt u(x,0) = u,(x),Vx E (0,1), ди (0,t): (1) ди -(1,t) = 0, Vt > 0. Use the method of separation of variables to compute the solution of problem (1) if 1) и,(х) %3D1, Vx € (0,1). 2) u,(x)= cos77x, Vx e (0,1). 3) u,(x) = 3+2 cos7x,Vx € (0,1). 4) u,(x) = sin x, Vx e (0,1).
Find the solution to the initial-boundary value problem: ди 0 0, д (0, t) — (1, t) %3D0, и(х, 0) — 2л, Ox² , ди (1.t) — 0, ди t>0, dx 0 < x < 1. ||
Given yı (t) = t and y2(t) = t satisfy the corresponding homogeneous equation of t'y" – 2y = 1 – 2t°, t > 0 Then the general solution to the nonhomogeneous equation can be written as y(t) = c1y1(t) + ©2Y2(t) + Yp(t). Use variation of parameters to find a particular solution yp(t). Y,(t) = Tip: Before you use the formula Y29(t) Yı9(t) JW(n, y2) Yp = - Y1 + y2 W (y1, Y2) the ODE should be in the form y'' + a(t)y' + b(t)y = g(t)

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3. Solve the heat equation ди k- for 2 < x < 4, and t > 0 with the boundary conditions u(2, t) = 0 and u(4, t) = 0, for t > 0, %3D and the initial value condition u(x, 0) = x for x E [2, 4|.