Exponential spline for the numerical solutions of linear Fredholm integro-differential equations

Abstract

In this paper, we introduce a new scheme based on the exponential spline function for solving linear second-order Fredholm integro-differential equations. Our approach consists of reducing the problem to a set of linear equations. We prove the convergence analysis of the method applied to the solution of integro-differential equations. The method is described and illustrated with numerical examples. The results reveal that the method is accurate and easy to apply. Moreover, results are compared with the method in (

On the numerical solution of Volterra and Fredholm integral equations using the fractional spline function method

“…where Ẑi , Fi and Fi are the exact solutions of Ẑi , Ẑ( 12 ) i and Ẑ( 32 ) i respectively. By subtracting (13) and (10) we get the following…”

“…Numerical analysis has been used more and more in the fields of applied mathematics since the beginning of the twentieth century, since spline functions are simple to analyze and work with on computer see [2], [3], [10] and [6], they are currently one of the most popular areas of approximation theory. These functions have an important role in mathematics and its technological applications, and also play a significant role in solving integral equations, integro-differential equations, and differential equations.…”

“…Recently, some works have been done on this model type, but with natural orders, see [1], [7], [14], [10], [8], [4], and [15], so that is why the proposed technique will be the beginning of a more detailed work on fractional models and perhaps it can progress the topic. This work is organized as follows: the researchers proposed two new types of fractional Spline method (FSM) to solve Volterra and Fredholm-integral equations by using two systems of equations as shown in Section 2 and Section 3, in Section 4 the convergence of these two fractional Spline models have been investigated, in Section 5 numerical examples are presented to illustrate the applications and effectiveness of the approach.…”

Numerical Method for Solution of Fourth-Order Volterra Integro-Differential Equations by Green’s Function

“…Various algorithms for finding the approximate numerical values are introduced and implemented to find the best results. Some of these are Wavelet-Galerkin method [6], monotone iterative methods [5,20], homotopy perturbation method reproducing kernel [4], Adomian decomposition method [8], Picard-Green's method [7,18], Tau method [11], spectral collocation methods [12], Taylor polynomials [14], Lagrange interpolation [16], exponential spline method [17] and the references therein. Furthermore, higher-order boundary value problems (BVPs) for IDEs have been researched by Agarwal [3] and Morchalo [15].…”

Collocation Approach Based on an Extended Cubic B -Spline for a Second-Order Volterra Partial Integrodifferential Equation

“…Exponential splines were used to find the numerical solutions of linear Fredholm IDEs by Tahernezhad and Jalilian [29]. For approximating linear stochastic IDE of fractional order, Mirzaee and Alipour [30] used a cubic B-spline-based collocation method. For a class of hyperbolic PIDE, Fairweather [31] utilized spline-based collocation technique.…”