Non-polynomial spline functions and Quasi-linearization to approximate nonlinear Volterra integral equation

Abstract

In this work, we want to use the Non-polynomial spline basis and Quasi-linearization method to solve the nonlinear Volterra integral equation. When the iterations of the Quasilinear technique employed in nonlinear integral equation we obtain a linear integral equation then by using the Non-polynomial spline functions and collocation method the solution of the integral equation can be approximated. Analysis of convergence is investigated. At the end, some numerical examples are presented to show the effectiveness of the method.

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

“…In our knowledge, so far the exponential spline functions have not been yet applied for approximating the second-order integro-differential equations. In this study, according to the exponential method in [31], a suitable method is presented to approximate secondorder integro-differential equations. The proposed algorithm is novel for second-order integro-differential equations.…”

“…In this paper, based on the non-polynomial spline basis and quasilinearization method to solve the nonlinear Volterra integral equation [31], we want to use the non-polynomial spline functions to develop a numerical method for the solution of the Fredholm integrodifferential equation…”

Non-polynomial fractional spline method for solving Fredholm integral equations

“…proposed Numerical methods for solving Fredholm integral equations of the second kind. And for other works see [4] and [25] Non-polynomial spline functions are used to find approximate solutions to a variety of problems, including integral equations [10], [26], [23], and [13], and differential equations [3], [16], [30], [11] and [12],…”

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

“…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.…”