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The method of collocation solves the F-VIE (4) using the approximation (5) through the equations equation(6) rN+1,M+1(si,tj)=gN+1,M+1(si,tj)+��01q(si,y)gN+1,M+1(y,tj)dy+��0tjk(tj,x)gN+1,M+1(si,x)dx?f(si,tj).that for a suitable set of collocation points, we choose Newton-Cotes nodes as (si,tj)=(2i?1/2(N+1),2j?1/2(M+1))(si,tj)=(2i?1/2(N+1),2j?1/2(M+1)) for all i=1,2,��,N+1,j=1,2,��,M+1. [http://www.selleckchem.com/products/MS-275.html MS-275 in vitro] Now, we find Fibonacci coefficients gn,mgn,m introduced in matrix form (5). Using (5) for (si,tj)(si,tj) and (4) we have equation(7) F(si,tj)C+��01q(si,y)F(y,tj)Cdy+��0tjk(tj,x)F(si,x)Cdx=f(si,tj).for i=1,2,��,N+1,j=1,2,��,M+1 Then Eq. (7) can be written in the matrix form as follows equation(8) (F+Q+K)C=A,(F+Q+K)C=A,where F=[F(s1,t1)?F(s1,tM+1)F(s2,t1)?F(s2,tM+1)?F(sN+1,t1)?F(sN+1,tM+1)](N+1)(M+1)��(N+1)(M+1),Q=[��01q(s1,y)F(y,t1)dy?��01q(s1,y)F(y,tM+1)dy��01q(s2,y)F(y,t1)dy?��01q(s2,y)F(y,tM+1)dy?��01q(sN+1,y)F(y,t1)dy?��01q(sN+1,y)F(y,tM+1)dy](N+1)(M+1)��(N+1)(M+1), A=[f(s1,t1)?f(s1,tM+1)f(s2,t1)?f(s2,tM+1)?f(sN+1,t1)?f(sN+1,tM+1)](N+1)(M+1)��1,K=[��0t1k(s1,y)F(y,t1)dy?��0tM+1k(s1,y)F(y,tM+1)dy��0t1k(s2,y)F(y,t1)dy?��0tM+1k(s2,y)F(y,tM+1)dy?��0t1k(sN+1,y)F(y,t1)dy?��0tM+1k(sN+1,y)F(y,tM+1)dy](N+1)(M+1)��(N+1)(M+1). Eq. (7) gives (N+1)(M+1)(N+1)(M+1) algebraic equations with (N+1)(M+1)(N+1)(M+1) unknowns as coefficients of vector C. After solving this linear system, we can approximate [http://www.selleckchem.com/products/GDC-0449.html GDC-0449 mw] the solution of equation (4) with substituting C in (5). Assume that (C[��],��.��)(C[��],��.��) is the Banach space of all [https://en.wikipedia.org/wiki/Quinapyramine Quinapyramine] continuous functions on ���� with norm equation(9) ��g(s,t)��=max(s,t)�ʦ�|g(s,t)|. Furthermore, we denote the error by equation(10) eN+1,M+1(s,t)=|gN+1,M+1(s,t)?g(s,t)|,eN+1,M+1(s,t)=|gN+1,M+1(s,t)?g(s,t)|,where g(s,t),gN+1,M+1(s,t) show the exact and approximate solutions of the two-dimensional Fredholm�CVolterra integral equation, respectively. Theorem 1. Assume that ?f(s,t)f(s,t)in ? Eq (4)is bounded ?, for all ?(s,t)�ʦ�(s,t)�ʦ�, the kernel of Volterra term is bounded such that ?|k(t,x)|��M1,(x,t)�ʦ�|k(t,x)|��M1,(x,t)�ʦ�and the kernel of the Fredholm term is bounded such that ?|q(s,y))|��M2,(s,y)�ʦ�|q(s,y))|��M2,(s,y)�ʦ�.