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Unformatted text preview: is an equation of the form u ( x ) = f ( x ) + Z b a K ( x, t ) u ( t ) dt, where a and b and the functions f and K are given. To approximate the function u on the interval [ a, b ], a partition a = x < x 1 < . . . < x m = b is selected and the equations u ( x i ) = f ( x i ) + Z b a K ( x i , t ) u ( t ) dt, i = 0 , 1 , . . . , m are solved for u ( x ) , u ( x 1 ) , . . . , u ( x m ). For this problem, consider a = 0, b = 1, f ( x ) = 3 x 2 + 4-2 e x-e 1-x , and K ( x, t ) = e | x-t | . 1 2 1. Show that u ( x ) = x 2 is the solution to the integral equation. 2. Set up and solve the linear system that results when the Compos-ite Trapezoidal rule is used with m = 10 , 20 , 40 , 80 , 160 to obtain a vector ( g , . . . , g m ). For each value of m , compute the error e m = max ≤ i ≤ m | u ( x i )-g i | . Do a loglog plot of the error as a function of h = 1 n , and determine the order of convergence of the overall method....
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