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 some operational matrices of integrations. We use these operational matrices and develop new sophisticated technique for numerical solutions to the following coupled system of fredholm integro differential equations DU(x) = f(x) + 11 Z 1 0 K11(x, t)U(t)dt + 12 Z 1 0 K12(x, t)V (t)dt, DV (x) = g(x) + 21 Z 1 0 K21(x, t)U(t)dt + 22 Z 1 0 K22(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 coupled system of equations to a system of algebraic equations without discretizing the system. We provide examples and numerical simulations to show the applicability and simplicity of our results and to demonstrate that the results obtained using the new technique matches well with the exact solutions of the problem. We also provide error analysis.