Numerical Scheme for Solution of Coupled System of Initial Value Fractional Order Fredholm Integro-Differential Equations with Smooth Solutions

We study shifted legendre polynomials and develop someoperational matrices of integrations. we use these operational matricesand develop new sophisticated technique for numerical solutions to thefollowing coupled system of fredholm integro differential equationsdu(x) = f(x) + 11z 10k11(x, t)u(t)dt + 12z 10k12(x, t)v (t)dt,dv (x) = g(x) + 21z 10k21(x, t)u(t)dt + 22z 10k22(x, t)v (t)dt,u(0) = c1, v (0) = c2,where d is fractional derivative of order  with respect to x, 0 <  6 1,11, 12, 21, 22 are real constants, f, g 2 c([0, 1]) and k11, k12, k21,k22 2 c([0, 1]×[0, 1]). we develop simple procedure to reduce the coupledsystem of equations to a system of algebraic equations without discretizingthe system. we provide examples and numerical simulations toshow the applicability and simplicity of our results and to demonstratethat the results obtained using the new technique matches well with theexact solutions of the problem. we also provide error analysis.