- 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

(1) f³ √√4-9x² dx, (5) sint-cost dt (6) x³ - 2x²+2x-5 x4 +4x²+3 (8) f² (2) 2) 1 √ ₁²√7² = 16 dt, dx 2π cos sin cos 0 + sin 0 1+x² (3) [*²√x-x²dx, (4) (4) √(x² - 2x + 2)² dx, de, (7) [² x ² + x³+4x²+x-1 x³ + x² dx,

2- yy'= √√√y³ - y² [y (0)=16]

+) Solve Poisson problem for a=lı bal ру u(x₁b) = f₂(x) Du= f(x,y) u(0₁9)=9₁ (9) b 0 u(a.y) = 9₂ (y) >x a u(x₁) = f(x) Figure 1. f(x,y) = sin2TX₁ f₁ = £₂=0, 9₁=9₂=0 using one dimensional eigen function expansion.