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- Consider an eigenvalue problem -u" - Au = 0; uES = C₂ (0,00) and VvE S, v'(0) = 0 & v(co) = ok. (a) Determine the Green's function G(x,; 2)for the problem. (b) Derive the spectral representation by utilizing the identity. 1 R→∞ 2πi 8(x - 5) = - lim (c) Obtain the Fourier cosine transform pair & G(x, 5; 2) da CR 8 f(x) = f f(x) coskx dk -8

- Consider an eigenvalue problem -u" - Au = 0; uES = C₂ (0,00) and VvE S, v'(0) = 0 & v(co) = ok. (a) Determine the Green's function G(x,; 2)for the problem. (b) Derive the spectral representation by utilizing the identity. 1 R→∞ 2πi 8(x - 5) = - lim (c) Obtain the Fourier cosine transform pair & G(x, 5; 2) da CR 8 f(x) = f f(x) coskx dk -8

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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4. Consider an eigenvalue problem -u" - Au = 0; u ES = C₂ (0, ∞o) and VvE S, v'(0) = 0 & v(0) = ok. (a) Determine the Green's function G(x, 5; 2)for the problem. (b) Derive the spectral representation by utilizing the identity. 1 8(x - 5) = - lim ZTri Per if R→∞⁰0 2πi (c) Obtain the Fourier cosine transform pair CR G(x, }; λ)dλ f(x) = [ f(x) coskx dk
Transcribed Image Text:4. Consider an eigenvalue problem -u" - Au = 0; u ES = C₂ (0, ∞o) and VvE S, v'(0) = 0 & v(0) = ok. (a) Determine the Green's function G(x, 5; 2)for the problem. (b) Derive the spectral representation by utilizing the identity. 1 8(x - 5) = - lim ZTri Per if R→∞⁰0 2πi (c) Obtain the Fourier cosine transform pair CR G(x, }; λ)dλ f(x) = [ f(x) coskx dk
ƒ (k): = 2 TT J₁ f(x) coskx dx. 88 (d) Solve the following initial value problem for u(t,x): ² u əx² ди at Satisfying the boundary and initial conditions ди əx u is ok as x→∞0&t> 0 = = 1 at x = 0 &t> 0 u= 0 at t=0&x > 0. Is this solution uniformly convergent?
Transcribed Image Text:ƒ (k): = 2 TT J₁ f(x) coskx dx. 88 (d) Solve the following initial value problem for u(t,x): ² u əx² ди at Satisfying the boundary and initial conditions ди əx u is ok as x→∞0&t> 0 = = 1 at x = 0 &t> 0 u= 0 at t=0&x > 0. Is this solution uniformly convergent?
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