scholarly journals Modified Block Pulse Functions for Numerical Solution of Stochastic Volterra Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
K. Maleknejad ◽  
M. Khodabin ◽  
F. Hosseini Shekarabi
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Rate Of Convergence ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Volterra Integral Equations ◽  
Numerical Example ◽  
Block Pulse Functions ◽  
A New Technique ◽  
Modified Block

We present a new technique for solving numerically stochastic Volterra integral equation based on modified block pulse functions. It declares that the rate of convergence of the presented method is faster than the method based on block pulse functions. Efficiency of this method and good degree of accuracy are confirmed by a numerical example.

scholarly journals Numerical Solution of System of Three Nonlinear Volterra Integral Equations Using Implicit Trapezoidal

2017 ◽  
Vol 10 (1) ◽  
pp. 44
Author(s):  
Dalal Adnan Maturi ◽  
Honaida Mohammed Malaikah
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Numerical Solution ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Volterra Integral Equations

In this project, we will be find numerical solution of Volterra Integral Equation of Second kind through using Implicit trapezoidal and that by using Maple 17 program, then we found that numerical solution was highly accurate when it was compared with exact solution.

scholarly journals THE METHOD OF NUMERICAL SOLUTION OF NONLINEAR VOLTERRA INTEGRAL EQUATIONS OF THE FIRST KIND

2018 ◽  
pp. 10-18
Author(s):  
Karakeev T.T. ◽  
Mustafayeva N.T.
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Numerical Methods ◽  
Numerical Solution ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Initial Point ◽  
Nonlinear Problems ◽  
Volterra Integral Equations ◽  
Algebraic Equations

When considering systems of differential equations with very general boundary conditions, exact solution methods encounter great difficulties, which become insurmountable in the study of nonlinear problems. In this case it is necessary to apply to certain numerical methods. It is important to note that the use of numerical methods often allows you to abandon the simplified interpretation of the mathematical model of the process. The problems of numerical solution of nonlinear Volterra integral equations of the first kind with a differentiable kernel, which degenerates at the initial point of the diagonal, are studied in the paper. This equation is reduced to the Volterra integral equation of the third kind and a numerical method is developed on the basis of that regularized equation. The convergence of the numerical solution to the exact solution of the Volterra integral equation of the first kind is proved, an estimate of the permissible error and a recursive formula of the computational process are obtained. Keywords: nonlinear integral equation, system of nonlinear algebraic equations, error vectors, the Volterra equation, small parameter, numerical methods.

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.

scholarly journals Orthonormal Bernstein polynomials for Volterra integral equations of the second kind

2020 ◽  
Vol 9 (1) ◽  
pp. 9
Author(s):  
Jafar Biazar ◽  
Hamed Ebrahimi
Keyword(s):  
Mathematical Modeling ◽  
Integral Equation ◽  
Numerical Method ◽  
Integral Equations ◽  
Method Of Moments ◽  
Volterra Integral Equation ◽  
Bernstein Polynomials ◽  
Volterra Integral Equations ◽  
Linear Volterra Integral Equations ◽  
Relative Errors

The purpose of this research is to provide an effective numerical method for solving linear Volterra integral equations of the second kind. The mathematical modeling of many phenomena in various branches of sciences lead into an integral equation. The proposed approach is based on the method of moments (Galerkin- Ritz) using orthonormal Bernstein polynomials. To solve a Volterra integral equation, the ap-proximation for a solution is considered as an expansion in terms of Bernstein orthonormal polynomials. Ultimately, the usefulness and extraordinary accuracy of the proposed approach will be verified by a few examples where the results are plotted in diagrams, Also the re-sults and relative errors are presented in some Tables.  

scholarly journals Splines of the Fourth Order Approximation and the Volterra Integral Equations

2021 ◽  
Vol 20 ◽  
pp. 475-488
Author(s):  
I.G. Burova ◽  
A.G. Doronina ◽  
D.E. Zhilin
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Integral Equations ◽  
Fourth Order ◽  
Volterra Integral Equation ◽  
Numerical Experiments ◽  
Order Approximation ◽  
Order Of Approximation ◽  
Polynomial Splines ◽  
Local Splines

This paper is a continuation of a series of papers devoted to the numerical solution of integral equations using local interpolation splines. The main focus is given to the use of splines of the fourth order of approximation. The features of the application of the polynomial and non-polynomial splines of the fourth order of approximation to the solution of Volterra integral equation of the second kind are discussed. In addition to local splines of the Lagrangian type, integro-differential splines are also used to construct computational schemes. The comparison of the solutions obtained by different methods is carried out. The results of the numerical experiments are presented.

scholarly journals Existence and Numerical Solution of the Volterra Fractional Integral Equations of the Second Kind

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Abdon Atangana ◽  
Necdet Bildik
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Fractional Integral ◽  
Numerical Results ◽  
Uniqueness Of The Solution ◽  
Picard Method ◽  
Rule Method

This work presents the possible generalization of the Volterra integral equation second kind to the concept of fractional integral. Using the Picard method, we present the existence and the uniqueness of the solution of the generalized integral equation. The numerical solution is obtained via the Simpson 3/8 rule method. The convergence of this scheme is presented together with numerical results.

scholarly journals NUMERICAL SOLUTION OF MIXED LINEAR VOLTERRA-FREDHOLM INTEGRAL EQUATIONS BY MODIFIED BLOCK PULSE FUNCTIONS

2021 ◽  
Vol 5 (1) ◽  
pp. 1
Author(s):  
Ayyubi Ahmad
Keyword(s):  
Linear System ◽  
Error Analysis ◽  
Numerical Solution ◽  
Integral Equations ◽  
Operational Matrix ◽  
Fredholm Integral Equations ◽  
Algebraic Equations ◽  
Block Pulse Functions ◽  
Operational Matrix Of Integration ◽  
Modified Block

A numerical method based on modified block pulse functions is proposed for solving the mixed linear Volterra-Fredholm integral equations. We obtain an integration operational matrix of modified block pulse functions on interval [0,T). A modified block pulse functions and their operational matrix of integration, the mixed linear Volterra-Fredholm integral equations can be reduced to a linear system of algebraic equations. The rate of convergence is O(h) and error analysis of the proposed method are discussed. Some examples are provided to show that the proposed method have a good degree of accuracy.

scholarly journals COLLOCATION METHOD FOR FUZZY VOLTERRA INTEGRAL EQUATIONS OF THE SECOND KIND

2020 ◽  
Vol 25 (1) ◽  
pp. 146-166 ◽  
Author(s):  
Zahra Alijani ◽  
Urve Kangro
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Integral Equations ◽  
Collocation Method ◽  
Volterra Integral Equation ◽  
Basis Function ◽  
Convergence Rates ◽  
Numerical Examples ◽  
Convergence Estimates ◽  
Upper And Lower Functions

In this paper we consider fuzzy Volterra integral equation of the second kind whose kernel may change sign. We give conditions for smoothness of the upper and lower functions of the solution. For numerical solution we propose the collocation method with two different basis function sets: triangular and rectangular basis. The smoothness results allow us to obtain the convergence rates of the methods. The proposed methods are illustrated by numerical examples, which confirm the theoretical convergence estimates.

scholarly journals EXISTENCE OF RESOLVENT FOR VOLTERRA INTEGRAL EQUATIONS ON TIME SCALES

2010 ◽  
Vol 82 (1) ◽  
pp. 139-155 ◽  
Author(s):  
MURAT ADıVAR ◽  
YOUSSEF N. RAFFOUL
Keyword(s):  
Integral Equation ◽  
Qualitative Analysis ◽  
Integral Equations ◽  
Time Scales ◽  
Volterra Integral Equation ◽  
Sufficient Conditions ◽  
Volterra Integral Equations ◽  
Future Research ◽  
Shift Operators ◽  
Image Position

AbstractWe introduce the concept of ‘shift operators’ in order to establish sufficient conditions for the existence of the resolvent for the Volterra integral equation on time scales. The paper will serve as the foundation for future research on the qualitative analysis of solutions of Volterra integral equations on time scales, using the notion of the resolvent.

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