Quantum mechanics

Exact solutions in general relativity

Philosophy

Differential equation

Applied mathematics

Convergence (economics)

Perturbation (astronomy)

Economics

A priori and a posteriori

Physics

Volterra integral equation

Mathematics

Epistemology

Fredholm integral equation

Mathematical analysis

Uniqueness

Economic growth

Finance

Fractional calculus

Integral equation

Order (exchange)

International Journal of Computer Mathematics

The present work considers the approximation of solutions of a type of fractional-order Volterra-Fredholm integro-differential equations, where the fractional derivative is introduced in Caputo sense.In addition, we also present several applications of the fractional-order differential equations and integral equations.Here, we provide a sufficient condition for existence and uniqueness of the solution and also obtain an a priori bound of the solution of the present problem.Then, we discuss about the higher-order model equation which can be written as a system of equations whose orders are less than or equal to one.Next, we present an approximation of the solution of this problem by means of a perturbation approach based on homotopy analysis.Also, we discuss the convergence analysis of the method.It is observed through different examples that the adopted strategy is a very effective one for good approximation of the solution, even for higherorder problems.It is shown that the approximate solutions converge to the exact solution, even for higher-order fractional differential equations.In addition, we show that the present method is highly effective compared to the existed method and produces less error.

A perturbation-based approach for solving fractional-order Volterra–Fredholm integro differential equations and its convergence analysis