Abstract
This paper is concerned with a numerical method based on the improved block-pulse basis functions (IBPFs). It is done mainly to solve linear and nonlinear Volterra and Fredholm integral equations of the second kind. These equations can be simplified into a linear system of algebraic equations by using IBPFs and their operational matrix of integration. After that, the system can be programmed and solved using Mathematica. The changes made to the method obviously improved - as it will be shown in the numerical examples - the time taken by the program to solve the system of algebraic equations. Also, it is reflected in the accuracy of the solution. This modification works perfectly and improved the accuracy over the regular block–pulse basis functions (BPF). A slight change in the intervals of the BPF changes the whole technique to a new easier and more accurate technique. This change has worked well while solving different types of integral equations. The accompanied theorems of the IBPF technique and error estimation are stated and proved. The paper also dealt with the uniqueness and convergence theorems of the solution. Numerical examples are presented to illustrate the efficiency and accuracy of the method. The tables and required graphs are also shown to prove and demonstrate the efficiency.
Highlights
In recent years, there has been a growing interest in the formulation of many engineering and physical problems in terms of integral equations
To crystallize the presentation of the current paper, the rest of it is organized as follows: In Section 2, we describe improved block-pulse basis functions (IBPFs) and their properties, how to apply them to linear and nonlinear different types of integral equations
Notice that the collocation points are taken as the midpoints of the subintervals of the IBPF
Summary
There has been a growing interest in the formulation of many engineering and physical problems in terms of integral equations. The completeness of IBPFs guarantees an arbitrary small mean square error that can be obtained from a real bounded function This has only a finite number of discontinuous points in the interval x [0, 1) by increasing the number of terms in the improved block pulse series. To crystallize the presentation of the current paper, the rest of it is organized as follows: In Section 2, we describe IBPFs and their properties, how to apply them to linear and nonlinear different types of integral equations. We can deduce that the integration of the vector < ( ) defined in Eq (11) can be represented approximately as American Journal of Mathematical and Computer Modelling 2021; 6(2): 19-34 This operational matrix may be modified and rewritten to get better results than that used in the work of F.
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