Collocation Approach for the Computational Solution Of Fredholm-Volterra Fractional Order of Integro-Differential Equations

Abstract

In this work, a collocation technique is used to determine the computational solution to fractional order Fredholm-Volterra integro-differential equations with boundary conditions using Caputo sense. We obtained the linear integral form of the problem and transformed it into a system of linear algebraic equations using standard collocation points. The matrix inversion approach is adopted to solve the algebraic equation and substituted it into the approximate solution. We established the uniqueness and convergence of the method and some modelled numerical examples are provided to demonstrate the method’s correctness and efficiency. It is observed that the results obtained by our new method are accurate and performed better than the results obtained in the literature. The study will be useful to engineers and scientists. It is advantageous because it addresses the difficulty in tackling fractional order Fredholm-Volterra integro-differential problems using a simple collocation strategy. The approach has the advantage of being more accurate and reducing computer running time.

Collocation Method for the Numerical Solution of Multi-Order Fractional Differential Equations

“…Their decision was to use an operational matrix for fractional integration based on boubakar polynomi-als. Collocation approach for the computational solution of fredholm-volterra fractional order of integro-differential equations was presented by [18]. They solved the problem by first obtaining the linear integral form of it and then transforming it into a system of linear algebraic equations by making use of conventional collocation points.…”

A Polynomial Approximation for the Numerical Solution of First Order Volterra Integro-Differential Equations

“…Some methods for determining the numerical solution of integro-differential equations include: Bernstein Method [14], Adomian decompositions method [2,3], Finite difference-Simpson method [17], Collocation method by [4,5,6,7,21,22], Hybrid linear multistep method [8,9], Chebyshev-Galerkin method [10], Bernoulli matrix method [11], Differential transform method [12], Lagrange Interpolation [13], Differential Transformation [15], Block pulse functions operational matrices [19] Chebyshev polynomials [16], Optimal Auxiliary Function Method (OAFM) [18] and Spectral Homotopy Analysis Method [20]. We consider first order Volterra integro-differential equation of the form 𝑦 ′ (𝑥) = 𝑔(𝑥) + ∫ 𝑘(𝑥, 𝑡)𝑦(𝑡)𝑑𝑡 𝑥 0 (1) with the initial condition 𝑦(0) = 𝑞…”

A Numerical Method using Collocation Approach for the Solution of Volterra-Fredholm Integro-Differential Equations

“…A trial solution function of unknown constants that conform with the differential equations together with the initial conditions were assumed and substituted into the equations under consideration. Collocation approach for the computational solution of Fredholm-Volterra Fractional order of integro-differential equations was presented by [22]. They obtained the linear integral form of the problem and transformed it into a system of linear algebraic equations using standard collocation points.…”