An approach based on fractional-order Lagrange polynomials for the numerical approximation of fractional order non-linear Volterra-Fredholm integro-differential equations

This article discussed and analyzed a numerical technique based on fractional-order Lagrange polynomials to solve a class of fractional-order non-linear Volterra-Fredholm integro-differential equations. The fractional derivative has been considered of Caputo type. The existence and uniqueness of the continuous solution have been discussed for the given problem. In this approach, first using the Laplace transform, fractional-order Lagrange polynomials operational matrices of fractional integration have been derived. Then using these operational matrices, the continuous problem has been reduced into a system of algebraic equations. The error analysis also has been carried out and an upper error bound estimate for the approximate solution has been given in L-2-norm. It is also shown that as the number of fractional-order Lagrange polynomials increases, the approximation error approaches to zero rapidly. Further, some numerical examples are discussed to verify the accuracy and efficiency of the proposed numerical technique and to validate our theoretical findings.