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

Mohamed R Ali,Mohamed M Mousa,Wen-Xiu Ma

doi:10.1155/2019/1705651

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.

Highlights

In the current literature, there are many different applications of Singular Nonlinear Volterra Integral Equations (SNVIE) in various areas, such as mathematical physics, electrochemistry, scattering theory, heat conduction, semiconductors, population dynamics, and fluid flow [1, 2]
We present a new numerical technique to discover a new solution of Singular Nonlinear Volterra Integral Equations (SNVIE)
(x ut − t)0.5 dt where u(t) is an unknown function and f(x) is a given function. This equation is an example of a nonhomogeneous Volterra equation of first kind with weak singularity

Summary

Introduction

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]. 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, 18,19,20]. Where u(t) is an unknown function and f(x) is a given function This equation is an example of a nonhomogeneous Volterra equation of first kind with weak singularity. 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 T is known, which is the historic Abel’s problem, the relation (7) is an integral equation for the unknown function u(y), which is known as Abel’s integral equation. We summarize the process of dissolving weakly singular-Volterra integral equations based on the HOBW implementation matrix method.

The HOBW Method and Operational Matrix of the Integration

HOBW Operational Matrix

Solution of Nonlinear Volterra Integral Equations via the HOBW Method

Numerical Examples

Conclusion