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

2015 ◽  
Vol 11 (5) ◽  
pp. 5220-5229
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 Block-by-Block Method for Solving Nonlinear Volterra-Fredholm Integral Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-8
Author(s):  
Abdallah A. Badr
Keyword(s):  
Integral Equation ◽  
Fredholm Integral Equation ◽  
Existence And Uniqueness ◽  
Time Dependent ◽  
Volterra Kernel ◽  
Block Method ◽  
Numerical Examples ◽  
Uniqueness Of The Solution ◽  
Fredholm Kernel

We consider a nonlinear Volterra-Fredholm integral equation (NVFIE) of the second kind. The Volterra kernel is time dependent, and the Fredholm kernel is position dependent. Existence and uniqueness of the solution to this equation, under certain conditions, are discussed. The block-by-block method is introduced to solve such equations numerically. Some numerical examples are given to illustrate our results.

NON-LIPSCHITZ BACKWARD STOCHASTIC VOLTERRA TYPE EQUATIONS WITH JUMPS

2007 ◽  
Vol 07 (04) ◽  
pp. 479-496 ◽  
Author(s):  
ZHIDONG WANG ◽  
XICHENG ZHANG
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Volterra Integral Equation ◽  
Existence And Uniqueness ◽  
Jump Process ◽  
Uniqueness Of Solution ◽  
Volterra Type

In this paper, we prove the existence and uniqueness of solution for the backward stochastic Volterra integral equation with non-Lipschitz coefficients and driven by Brownian motion and jump process. Moreover, when the equation is driven only by Brownian motion, we also study the continuity of the solution with respect to the time.

Fuzzy double controlled metric spaces and related results

2021 ◽  
Vol 40 (5) ◽  
pp. 9977-9985
Author(s):  
Naeem Saleem ◽  
Hüseyin Işık ◽  
Salman Furqan ◽  
Choonkil Park
Keyword(s):  
Integral Equation ◽  
Metric Space ◽  
Existence And Uniqueness ◽  
Metric Spaces ◽  
Banach Contraction Principle ◽  
Uniqueness Of Solution ◽  
Contraction Principle ◽  
Triangular Inequality ◽  
Banach Contraction ◽  
The Banach Contraction Principle

In this paper, we introduce the concept of fuzzy double controlled metric space that can be regarded as the generalization of fuzzy b-metric space, extended fuzzy b-metric space and controlled fuzzy metric space. We use two non-comparable functions α and β in the triangular inequality as: M q ( x , z , t α ( x , y ) + s β ( y , z ) ) ≥ M q ( x , y , t ) ∗ M q ( y , z , s ) . We prove Banach contraction principle in fuzzy double controlled metric space and generalize the Banach contraction principle in aforementioned spaces. We give some examples to support our main results. An application to existence and uniqueness of solution for an integral equation is also presented in this work.

scholarly journals On the existence and uniqueness of solution to Volterra equation on a time scale

2019 ◽  
Vol 27 (3) ◽  
pp. 177-194
Author(s):  
Bartłomiej Kluczyński
Keyword(s):  
Integral Equation ◽  
Sobolev Space ◽  
Time Scale ◽  
Existence And Uniqueness ◽  
Nonlinear Term ◽  
Volterra Equation ◽  
Uniqueness Of Solution ◽  
Inversion Theorem ◽  
T Tau

AbstractUsing a global inversion theorem we investigate properties of the following operator\matrix{\matrix{ V(x)( \cdot ): = {x^\Delta }( \cdot ) + \int_0^ \cdot {v\left( { \cdot ,\tau ,x,\left( \tau \right)} \right)} \Delta \tau , \hfill \cr \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x(0) = 0, \hfill \cr}\cr {} \cr }in a time scale setting. Under some assumptions on the nonlinear term v we then show that there exists exactly one solution {x_y} \in W_{\Delta ,0}^{1,p}\left( {{{[0,1]}_\mathbb{T}},{\mathbb{R}^N}} \right) to the associated integral equation\left\{ {\matrix{{{x^\Delta }(t) + \int_0^t {v\left( {t,\tau ,x\left( \tau \right)} \right)} \Delta \tau = y(t)\,\,\,for\,\Delta - a.e.\,\,\,t \in {{[0.1]}_\mathbb{T}},} \cr {x(0) = 0,} \cr } } \right.which is considered on a suitable Sobolev space.

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