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a²u ô²u 2du Ət² Əx² r dx u(0, x) = 0, u¿(0, x) = g(x), where g(x) = g(-x) is an even function. Hint: Set w = xu. = 0. Solve the initial value problem 2.4.18. The Poisson-DarboUL equation is

a²u ô²u 2du Ət² Əx² r dx u(0, x) = 0, u¿(0, x) = g(x), where g(x) = g(-x) is an even function. Hint: Set w = xu. = 0. Solve the initial value problem 2.4.18. The Poisson-DarboUL equation is

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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Course: Elementary Partial Differential Equations

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#4 (a variant of #2.4.18 on page 62). Consider the initial value problem for the Poisson- Darbour equation 2 uz = 0, u(0, x) = 0, u;(0, x) = g(x), --. Assuming that g(x) = g(-x) E C® (R'), show that the solution u(t, r) E C® (R²). Hint. Set w := xu. You may need to use the representation e=1 1 1 - [g(z + x) – g(z – x)] = = g(z + 0x) = - . 10=-1 1 d 09(2 + 0x) d0 = | g'(2 + 0x) d0. -1 -1 Despite the fact that the left side is formally not defined for r = 0, it is reduced to the form which is a smooth function of (x, z) E R², including the values on the line r = 0.
Transcribed Image Text:#4 (a variant of #2.4.18 on page 62). Consider the initial value problem for the Poisson- Darbour equation 2 uz = 0, u(0, x) = 0, u;(0, x) = g(x), --. Assuming that g(x) = g(-x) E C® (R'), show that the solution u(t, r) E C® (R²). Hint. Set w := xu. You may need to use the representation e=1 1 1 - [g(z + x) – g(z – x)] = = g(z + 0x) = - . 10=-1 1 d 09(2 + 0x) d0 = | g'(2 + 0x) d0. -1 -1 Despite the fact that the left side is formally not defined for r = 0, it is reduced to the form which is a smooth function of (x, z) E R², including the values on the line r = 0.
2 ди = 0. Solve the initial value problem %3D 2.4.18. The Poisson-DarboUx equation is u(0, x) = 0, u,(0, x) = g(x), where g(x) = g(-æ) is an even function. Hint: Set w = xu.
Transcribed Image Text:2 ди = 0. Solve the initial value problem %3D 2.4.18. The Poisson-DarboUx equation is u(0, x) = 0, u,(0, x) = g(x), where g(x) = g(-æ) is an even function. Hint: Set w = xu.
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