Question

Mathews and Fink show how Simpson's rule can be used to approximate the solution of an integral equation (Problem 7, page 377). The procedure is outlined as follows Let the integral equation be given as u (x) = x2 + 0.1?(x2 +t)u(t) dt. This could. for example, be the expression for the velocity of some object at position x. Note here that t is just a dummy variable used for integration purposes. To solve this integral equation via Simpson's rule with h- 0.5, we let to 0, t0.5 and ti-1. We then have 0.5 3 0 Substituting Eq (1) into the integral equation we then have 6 Substituting xo -0, 21 -0.5, ^2 -1 into Eq (2) gives us a system of linear equations 60 v-0.25 (0.25v0 +3vi +1.2502) 60 60 which can be solved to give vo 0.0273. ul = 0.2866, u2 = 1.0646. Substituting these values back into Eq (2) and simplifying the algebra gives us the solution to the integral equation v(x)1.0373052 0.027297 One can check the validity of the solution by substituting it back into the right hand side of the integral equation, integrating and simplifying the right hand side. This should compare well with the approximate solution given by Eq (3). This is a technique known as a self-consistency check and is common throughout applied mathematics, science and engineering (a) 15 marks] Using the ideas presented above, use the Simpson rule with h 0.5 to approximate the solution of the integral equation given by (x)0.25(+t) (t) dt

Answer 1

Given that P (x) = C_0 + C_1 x + C_2 x^2 Therefore integral^h_-h P (x) dx = integral^h_-h (C_0 + C_1 x + C_2 x^2) dx = [C_0 x + C_1 x^2/2 + C_2 x^3/3]^h_-h = C_0 (h - (-h)) + C_1/2 [h^2 - (-h)^2] + C_2/3 [h^2 - (-h)^3] = C_0 (h + h) + C_1/2 (h^2 - h^2) + C_2/3 (h^3 + h^3) = 2 C_0 h + 2 C_2 h^3/3 Therefore integral^h_-h P (x) dx = 2 C_0 h + 2 C_2 h^3/3