Question

This is a Numerical Methods question.

The original question is an exercise question 3.6 from Parviz Moin. Fundamentals of Engineering Numerical Analysis, Cambridge University Press; 2nd edition, 2010.

Thank you for your help!

6. A common problem of mathematical physics is that of solving the Fredhol integral equation f(x)-ф(x) +1 K(x,t)ф(t)dt, where the functions f (x) and K(x, t) are given and the problem is to obtain ф(x). (a) Describe a numerical method for solving this equation. (b) Solve the following equation דר o(x) x2+/ 3(0.5 sin 3x-ty?)o(t) dt. Compare to the exact solution φ(x) = sin 3x

Answer 1

1)� This type of equation can be solved using degenerate method f(x) = phi (x) + lambda integral^b_sa K(x, t) phi (t) dt let K(x, t) = alpha_1 (x) beta (t) + d_2 (x) beta (t) + d_3 (x) beta_3 (t) {we can take in general n also} is lambda integral^b_a alpha (x) beta (t) phi (t) dt = lambda alpha (x) c_1 slly for other is f(x) = phi (x) + lambda C_1 alpha_1 (x) + lambda C_2 d_2 (x) + lambda c_3 alpha_3 (x) rightarrow 1 integral^b_a 1 times beta (x) dx implies integral^b_a beta (x) y(x) dx = integral^b_a f(x) beta_1 (x) dx + lambda c_1 integral^b_a alpha_1 (x) beta_1 (x) dx + lambda c_2 integral^b_a alpha_2 (x) beta_1 (x) dx + lambda c_3 integral^b_a alpha_3 (x) beta (x) dx as C_1 = lambda_1 + lambda c_1 alpha_11 + lambda c_2 alpha_12 \r\n