1- Solve the problemuz = Uxx + sin(nx) + sin(2nx)fu(0, t) = 0lu(1,t) = 0u(x, 0) = 0P.D.E.0

A partial differential equation is given by, ut = uxx+sinπx+sin2πx with boundary conditions u0, t =…

Answered: 1- Solve the problem P.D.E. Uz = Uxx +…

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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When solving the y" - Ay = 0, 0
Consider the equation (Vi-2)+ =0, -1 <*<1. d -1 < x < 1. dx dx V1 – x2 a) Decide if a Sturm-Liouville problem consisting of this equation would be regular or singular. b) Show that To(x) = 1, T1(x) = x, and T2(x) = 2x2 – 1 are eigenfunctions of such Sturm-Liouville problem, and find the corresponding eigenvalues 1o, 11, and å2. c) Using the appropriate weight function, show that T1 is orthogonal to T2 on the interval (-1, 1).
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6. Consider the eigenvalue problem y" + ày = 0; y'(0) = 0, y(1) + y'(1) = 0. All the eigenvalues are nonnegative, so write à = a² where a 2 0. (a) Show that A = 0 is not an eigen- value. (b) Show that y = Acos ax + B sin ax satis- fies the endpoint conditions if and only if B = 0 and a is a positive root of the equation tan z = 1/z. These roots {an}° are the abscissas of the points of intersection of the curves y = tan z and y = 1/z, as indicated in Fig. 3.8.13. Thus the eigenvalues and eigenfunctions of this problem are the numbers {a;}° and the functions {cos an x}9°, re- spectively. y = 2n Зл I ly = tan z
Q3(a) Using Parseval's theorem, establish 1 1 1 + + + 1² 3² 52 ... = where Euler-Fourier coefficient of a periodic ao = 3, an = 0 and bn and f(x) = {1, 0 2π 8 function f(x) is given by [(-1)" - 1] = NI and f(x + 2n) = f(x).
The eigenfunctions for y" + 2y' + Ay = y(0) = 1, y(1) = 0 are {e" sin(nπa)O {e2r sin(프플프) 1O n=1 {e-"sin(뿌)EO od In=1 4 II {e-2" sin(nπa)이 {e" sin ()O 2 n=1 .2r -:.. ( po
Question 9 Find the solution of the integro-differential equation using Laplace Transform: y" + 4y' + 3y = 2 e-tdt;y(0) = 0,y'(0) = 0 %3D Oy = (4 – 3e(1+ 2t) – e-3*] Oy = (4+ 3e-(1– 2) – e-3«] Oy = (4+ 3e (1 – 2t) – e*] Oy = }(4 – 3eʻ (1 + 2t) – e*] %3D
5). (3 Prove that K(x, y) = sin²(x - y) is an NDS kernel.
b) Find L[e4t (-cos 2) dt] and L-1 cos 2LY 4s+5 (s-1)3(s+2).
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1- Solve the problem P.D.E. Uz = Uxx + sin(nx) + sin(2nx) 0