Two dimensional Volterra integral equation with singular kernels in contact problems and its numerical computations

2021 ◽  
Vol 15 (7) ◽  
pp. 303-319
Author(s):  
A. M. Al-Bugami
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Contact Problems ◽  
Two Dimensional ◽  
Numerical Computations ◽  
Singular Kernels

scholarly journals Runge-Kutta and Block by Block Methods to Solve Linear Two-Dimensional Volterra Integral Equation with Continuous Kernel

2016 ◽  
Vol 11 (10) ◽  
pp. 5705-5714
Author(s):  
Abeer Majed AL-Bugami
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solution ◽  
Two Dimensional ◽  
Block Method ◽  
Numerical Examples ◽  
Block Methods ◽  
Runge Kutta ◽  
Method R

In this paper, the existence and uniqueness of solution of the linear two dimensional Volterra integral equation of the second kind with Continuous Kernel are discussed and proved.RungeKutta method(R. KM)and Block by block method (B by BM) are used to solve this type of two dimensional Volterra integral equation of the second kind. Numerical examples are considered to illustrate the effectiveness of the proposed methods and the error is estimated.

scholarly journals Fractional Order Operational Matrix Method for Solving Two-Dimensional Nonlinear Fractional Volterra Integro-Differential Equations

2021 ◽  
Vol 45 (4) ◽  
pp. 571-585
Author(s):  
AMIRAHMAD KHAJEHNASIRI ◽  
◽  
M. AFSHAR KERMANI ◽  
REZZA EZZATI ◽  
◽  
...  
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Fractional Order ◽  
Volterra Integral Equation ◽  
Matrix Method ◽  
Operational Matrix ◽  
Basis Functions ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Operational Matrix Method

This article presents a numerical method for solving nonlinear two-dimensional fractional Volterra integral equation. We derive the Hat basis functions operational matrix of the fractional order integration and use it to solve the two-dimensional fractional Volterra integro-differential equations. The method is described and illustrated with numerical examples. Also, we give the error analysis.

scholarly journals Efficient Numerical Algorithm for the Solution of Nonlinear Two-Dimensional Volterra Integral Equation Arising from Torsion Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
A. M. Al-Bugami
Keyword(s):  
Integral Equation ◽  
Cross Section ◽  
Volterra Integral Equation ◽  
Kernel Method ◽  
Degenerate Kernel ◽  
Two Dimensional ◽  
Torsion Problem ◽  
Block Method ◽  
Elliptical Cross ◽  
Nonlinear Viscoelastic

In this article, an effective method is given to solve nonlinear two-dimensional Volterra integral equations of the second kind, which is arising from torsion problem for a long bar that consists of the nonlinear viscoelastic material type with a fixed elliptical cross section. First, the existence of a unique solution of this problem is discussed, and then, we find the solution of a nonlinear two-dimensional Volterra integral equation (NT-DVIE) using block-by-block method (B-by-BM) and degenerate kernel method (DKM). Numerical examples are presented, and their results are compared with the analytical solution to demonstrate the validity and applicability of the method.

scholarly journals A Mechanical Quadrature Method for Solving Delay Volterra Integral Equation with Weakly Singular Kernels

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Li Zhang ◽  
Jin Huang ◽  
Yubin Pan ◽  
Xiaoxia Wen
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Richardson Extrapolation ◽  
Quadrature Method ◽  
Mechanical Quadrature ◽  
Proportional Delays ◽  
Weakly Singular ◽  
Uniqueness Of The Solution ◽  
Singular Kernels ◽  
Mechanical Quadrature Method

In this work, a mechanical quadrature method based on modified trapezoid formula is used for solving weakly singular Volterra integral equation with proportional delays. An improved Gronwall inequality is testified and adopted to prove the existence and uniqueness of the solution of the original equation. Then, we study the convergence and the error estimation of the mechanical quadrature method. Moreover, Richardson extrapolation based on the asymptotic expansion of error not only possesses a high accuracy but also has the posterior error estimate which can be used to design self-adaptive algorithm. Numerical experiments demonstrate the efficiency and applicability of the proposed method.

scholarly journals Solving two-dimensional nonlinear fuzzy Volterra integral equations by homotopy analysis method

2021 ◽  
Vol 54 (1) ◽  
pp. 11-24
Author(s):  
Atanaska Georgieva
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Homotopy Analysis Method ◽  
Volterra Integral Equations ◽  
Two Dimensional ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Fuzzy Solution

Abstract The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

Sign in / Sign up

Export Citation Format

Share Document