A Fredholm integral equation of the second kind is an equation of the form

Where a and b and the functions f and K are given. To approximate the function u on the interval [a, b], a partition x0 = a

Are solved for u(x0), u(x1), · · · , u(xm). The integrals are approximated using quadrature formulas based on the nodes x0, · ·, xm. In our problem, a = 0, b = 1, f (x) = x2, and K(x, t) = e|xˆ’t|.
a. Show that the linear system
u(0) = f (0) + 1/2 [K(0, 0)u(0) + K(0, 1)u(1)],
u(1) = f (1) + 1/2 [K(1, 0)u(0) + K(1, 1)u(1)]
Must be solved when the Trapezoidal rule is used
b. Set up and solve the linear system that results when the Composite Trapezoidal rule is used with n = 4.
c. Repeat part (b) using the Composite Simpson's rule.

Transcribed Image Text:

a(x) = f(x) + 1 K (x, t)u(t) dt .b u(xfKau) dt, for eachi 0,m,