Newton's Method for Solving Hilfer Fractional Volterra-Fredholm Integro Differential Equations

In this paper, we apply Newton's method to solve a class of integro-differential equations of the Volterra-Fredholm type with nonlocal characteristics, involving almost sectorial operators and Hilfer fractional derivatives. Since these equations play a key role in mathematics, engineering, biology, physics, chemistry, control theory and economy, finding an appropriate solution is important. By Newton's method, we linearize a nonlinear Volterra-Fredholm integro-differential equation which is equivalent to a category of Volterra-Fredholm integro-differential equations with nonlocal properties, incorporating almost sectorial operators and Hilfer fractional derivatives and nearly sectorial operators. This technique has been shown to be an effective tool for the numerical solution of initial value problems in nonlinear integro-differential equations. The convergence analysis is also investigated.