Advances in Mathematical Physics

Volume 2019, Article ID 1705651, 10 pages

https://doi.org/10.1155/2019/1705651

## Solution of Nonlinear Volterra Integral Equations with Weakly Singular Kernel by Using the HOBW Method

^{1}Department of Mathematics, Benha Faculty of Engineering, Benha University, Benha, Egypt^{2}Department of Mathematics, College of Sciences and Human Studies at Howtat Sudair, Majmaah University, Al–Majmaah 11952, Saudi Arabia^{3}Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA^{4}Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia^{5}College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China^{6}International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa

Correspondence should be addressed to Mohamed R. Ali; ge.ude.ub.tihb@ader.demahom and Mohamed M. Mousa; moc.liamg@mmmm.gne.rd

Received 2 December 2018; Revised 12 December 2018; Accepted 16 December 2018; Published 3 February 2019

Academic Editor: Soheil Salahshour

Copyright © 2019 Mohamed R. Ali et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We present a new numerical technique to discover a new solution of Singular Nonlinear Volterra Integral Equations (SNVIE). The considered technique utilizes the Hybrid Orthonormal Bernstein and Block-Pulse functions wavelet method (HOBW) to solve the weakly SNVIE including Abel’s equations. We acquire the HOBW implementation matrix of the integration to derive the procedure of solving these kind integral equations. The explained technique is delineated with two numerical cases to demonstrate the benefit of the technique used by us. At last, the exchange uncovers the way that the strategy utilized here is basic in usage.

#### 1. Introduction

In the current literature, there are many different applications of SNVIE in various areas, such as mathematical physics, electrochemistry, scattering theory, heat conduction, semiconductors, population dynamics, and fluid flow [1, 2]. Numerical strategies for the SNVIE are spline collocation methods [3], Newton–Cotes methods [4], extrapolation algorithm [5], and Hermite-collocation method [6]. The most popular methods for talking about the such equations are introduced, such as homotopy asymptotic method [7], Nyström interpolant method [8], Mesh method [9], Tau method [10], Laplace transform [11], orthonormal Bernstein, and block-pulse functions [12–17].

Wavelet theory is a moderately new and considered as a rising territory in the field of applied science and engineering. Wavelets allow the accurate representation of a lot of functions. The wavelet technique is a new numerical technique utilized for dissolving the fractional equations. SNVIE has numerous applications in different zones, for example, semiconductors’ mathematical chemistry, chemical reactions, physics, scattering theory, electrochemistry, seismology, metallurgy, fluid flow, and population dynamics [2, 18–20].

In 1823, Niels Henrik Abel derived the equationwhere is an unknown function and is a given function. This equation is an example of a nonhomogeneous Volterra equation of first kind with weak singularity. Abel obtained this equation while studying the motion of a particle on a smooth curve lying on a vertical plane. The physical depiction of this condition is given in [21] as pursues. Abel thought about the issue in traditional mechanics, which is that of deciding the time a molecule brings to slide openly down a smooth settled bend in a vertical* xy*-plane (in Figure 1), from any settled point on the bend to its absolute bottom (the starting point 0). If means the mass of the molecule and signifies the condition of the smooth bend where is a differentiable function of , at that point we acquire the vitality protection condition aswhere is the speed of the molecule at the position at time , if the molecule tumbles from rest at time from the point , and represents acceleration due to gravity. The connection (2) can be expressed asby utilizing the arc-length , estimated from the starting point to the point, where a less sign has been utilized in the square root since diminishes with time amid the fall of the molecule. Using the formulawe can composeBy integrating both sides of (5), we obtain where is the total time of fall of the particle, from the point to the origin Therefore, we havewhere . In this way, we can find that the time of descent of the particle,* T*, can be resolved totally by utilizing the recipe (7), if the state of the curve , and consequently the function is known. On the off chance that we consider, on the other hand, the issue of assurance of the state of the bend, when the time of fall is known, which is the historic Abel’s problem, then the relation (7) is an integral equation for the unknown function , which is known as Abel’s integral equation.