scholarly journals On the Analysis of Numerical Methods for Nonstandard Volterra Integral Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
H. S. Mamba ◽  
M. Khumalo
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Numerical Solutions ◽  
Trapezoidal Rule ◽  
Collocation Methods ◽  
Point Collocation ◽  
The One ◽  
Theoretical Results

We consider the numerical solutions of a class of nonlinear (nonstandard) Volterra integral equation. We prove the existence and uniqueness of the one point collocation solutions and the solution by the repeated trapezoidal rule for the nonlinear Volterra integral equation. We analyze the convergence of the collocation methods and the repeated trapezoidal rule. Numerical experiments are used to illustrate theoretical results.

scholarly journals On the numerical solutions for nonlinear Volterra-Fredholm integral equations

2022 ◽  
Vol 40 ◽  
pp. 1-11
Author(s):  
Parviz Darania ◽  
Saeed Pishbin
Keyword(s):  
Nonlinear System ◽  
Integral Equations ◽  
Numerical Experiments ◽  
Numerical Solutions ◽  
Coefficient Matrix ◽  
Collocation Methods ◽  
Fredholm Integral Equations ◽  
Stability Estimates ◽  
Lower Triangular ◽  
Theoretical Results

In this note, we study a class of multistep collocation methods for the numerical integration of nonlinear Volterra-Fredholm Integral Equations (V-FIEs). The derived method is characterized by a lower triangular or diagonal coefficient matrix of the nonlinear system for the computation of the stages which, as it is known, can beexploited to get an efficient implementation. Convergence analysis and linear stability estimates are investigated. Finally numerical experiments are given, which confirm our theoretical results.

SOLVING TWO-POINT BOUNDARY VALUE PROBLEMS OF FRACTIONAL DIFFERENTIAL EQUATIONS VIA SPLINE COLLOCATION METHODS

2010 ◽  
Vol 01 (01) ◽  
pp. 117-132 ◽  
Author(s):  
NINGMING NIE ◽  
YANMIN ZHAO ◽  
MIN LI ◽  
XIANGTAO LIU ◽  
SALVADOR JIMÉNEZ ◽  
...  
Keyword(s):  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Boundary Value ◽  
Collocation Methods ◽  
Existence And Uniqueness Theorem ◽  
Spline Collocation ◽  
Fractional Differential ◽  
Theoretical Results ◽  
Fractional Boundary Value Problems

In this article, we introduce spline collocation methods to solve fractional boundary value problems (BVPs). The existence and uniqueness theorem of collocation solutions is studied, and the error estimate of collocation solutions is discussed. Numerical experiments are presented to demonstrate the theoretical results.

scholarly journals Convergence analysis of piecewise continuous collocation methods for higher index integral algebraic equations of the Hessenberg type

2013 ◽  
Vol 23 (2) ◽  
pp. 341-355 ◽  
Author(s):  
Babak Shiri ◽  
Sedaghat Shahmorad ◽  
Gholamreza Hojjati
Keyword(s):  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Convergence Properties ◽  
Piecewise Continuous ◽  
The Given ◽  
Uniqueness Of A Solution ◽  
Theoretical Results

In this paper, we deal with a system of integral algebraic equations of the Hessenberg type. Using a new index definition, the existence and uniqueness of a solution to this system are studied. The well-known piecewise continuous collocation methods are used to solve this system numerically, and the convergence properties of the perturbed piecewise continuous collocation methods are investigated to obtain the order of convergence for the given numerical methods. Finally, some numerical experiments are provided to support the theoretical results.

On the numerical solutions of Fredholm–Volterra integral equation

2003 ◽  
Vol 146 (2-3) ◽  
pp. 713-728 ◽  
Author(s):  
M.A. Abdou ◽  
Khamis I. Mohamed ◽  
A.S. Ismail
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Numerical Solutions

scholarly journals NUMERICAL SOLUTION OF HIGHER INDEX NONLINEAR INTEGRAL ALGEBRAIC EQUATIONS OF HESSENBERG TYPE USING DISCONTINUOUS COLLOCATION METHODS

2014 ◽  
Vol 19 (1) ◽  
pp. 99-117 ◽  
Author(s):  
Babak Shiri
Keyword(s):  
Numerical Solution ◽  
Convergence Analysis ◽  
Large Class ◽  
Numerical Experiments ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Linear And Nonlinear ◽  
Theoretical Results

In this paper, we deal with a system of linear and nonlinear integral algebraic equations (IAEs) of Hessenberg type. Convergence analysis of the discontinuous collocation methods is investigated for the large class of IAEs based on the new definitions. Finally, some numerical experiments are provided to support the theoretical results.

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 On a nonlinear inverse problem in viscoelasticity

2009 ◽  
Vol 31 (3-4) ◽  
Author(s):  
H. D. Bui ◽  
S. Chaillat
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Volterra Integral Equation ◽  
Boundary Data ◽  
Viscoelastic Body ◽  
Low Frequency ◽  
Cyclic Loads ◽  
Nonlinear Inverse Problem ◽  
Method Of Solution ◽  
The One

We consider an inverse problem for determining an inhomogeneity in a viscoelastic body of the Zener type, using Cauchy boundary data, under cyclic loads at low frequency. We show that the inverse problem reduces to the one for the Helmholtz equation and to the same nonlinear Calderon equation given for the harmonic case. A method of solution is proposed which consists in two steps: solution of a source inverse problem, then solution of a linear Volterra integral equation.

scholarly journals Nonlocal problem with the integral condition for a loaded heate equation

2021 ◽  
pp. 47-56
Author(s):  
М.М. Сагдуллаева
Keyword(s):  
Integral Equation ◽  
Heat Equation ◽  
Regular Solution ◽  
Volterra Integral Equation ◽  
Existence And Uniqueness ◽  
Nonlocal Problem ◽  
Integral Condition ◽  
Local Problem ◽  
Non Local ◽  
Loaded Term

В работе рассмотрена нелокальная задача с интегральным условием для нагруженного уравнения теплопроводности, где нагруженное слагаемое представляет собой производную второго порядка от неизвестной функции в начале координат. Доказано существование и единственность регулярного решения. С помощью функции Грина и тепловых потенциалов доказанао существование регулярного решения исследуемой задачи. Доказательство основано на редукции поставленной задачи к интегральному уравнению Вольтерра второго рода со слабой особенностью. Из разрешимости полученных интегральных уравнений Вольтерра следует существование единственного решения поставленной задачи. In this paper, we consider a non-local problem with the integral condition for the loaded heat equation, where the loaded term is a derivative of the second order from an unknown function at the origin. The existence and uniqueness of a regular solution is proven. Using the Green’s functions and thermal potentials, the existence of a regular solution to this problem is proved. The proof is based on the reduction of the formulated problem to the second kind Volterra integral equation with a weak singularity. The solvability of the obtained Volterra integral equations implies the existence of a unique solution to the problem.

Modified Quadrature Method for Solving BIEs of Steady-State Anisotropic Heat Conduction Problems

2014 ◽  
Vol 687-691 ◽  
pp. 1354-1358
Author(s):  
Xin Luo ◽  
Jin Huang
Keyword(s):  
Integral Equation ◽  
Heat Conduction ◽  
Steady State ◽  
Heat Conduction Equation ◽  
Numerical Solutions ◽  
Trapezoidal Rule ◽  
Conduction Equation ◽  
Quadrature Method ◽  
Singular Kernels ◽  
Anisotropic Heat Conduction

In this paper, steady-state anisotropic heat conduction equation can be converted into the first kind integral equation, then modified quadrature formula based on trapezoidal rule is used to deal the integrals with singular kernels. In addition, Sidi transformation is applied to remove the singularities at concave points in concave polygons. This technique improves the accuracy of numerical solutions of the heat conduction equation. Numerical results show the convergence rate of the proposed method is the order three.

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