Taylor Collocation Matlab Code for Solving Volterra–Fredholm Integral Equations

This code implements Taylor Collocation method to compute numerical solutions of Volterra–Fredholm integral equations as explained down. The code gives the following results:

• Chart of the numerical solution compared with the exact solution
• Error chart of this method
• Elapsed Time of the calculations
• Upper Bound Error of Taylor Collocation method
• The coefficients of Taylor Polynomial

Required Inputs:

• The interval of the solution function [a b]
• The constant of the series c
• The power of the supposed series N
• The constant  Lambda 1
• The constant  Lambda 2
• A(x) Value
• B(x) Value
• The following functions h(x), h(t), K1(x,t), K2(x,t), f(x) and Yex(x)

Where: h(x): Function is the upper limit of the first integral. h(t): the same as h(x) function, but with substituting t instead of x. f(x): The functions out of the Integrals. K1(x,t): The function inside the first integral. K2(x,t): The function inside the second integral. Yex(x): The Exact solution (it exist for comparison, if not just disable the lines 110,111,114 and 121 and you keep getting the numerical results). The full form of Voltera Fredholm equation is as the following:

Implemented in the code six examples you can uncomment any of them and run it, the following values is for the sake of example:

a = 0; b = 1; N = 8;
L1= 1; L2= 1;
c=0;
% Functional Inputs
A = 1; B = 0;
h = x; ht = t;
f = cos(x)*(0.5*x – 0.25) + 0.25*cos(2-x);
k1=sin(x-t); k2=cos(x-t);
Yex = sin(x); % Exact Solution

Taylor Collocation Method:

Taylor polynomial is of degree N at x=c, where y(c)^(m), m=0,1,2,…,N, are the coefficients to be determined. In addition, to compute the Taylor coefficients we use the collocation points defined by Xn. Where the values Xn are spread out over the interval [a,b].

References:

Keyan Wang, Qisheng Wang, Taylor collocation method and convergence analysis for the Volterra–Fredholm integral equations,  at School of Mathematics and Computational Science, Wuyi University, Jiangmen, Guangdong 529020, PR China.