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.

Numerical solution of nonlinear stochastic Itô – Volterra integral equation driven by fractional Brownian motion

2020 ◽  
Vol 37 (9) ◽  
pp. 3243-3268
Author(s):  
S. Saha Ray ◽  
S. Singh
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Stochastic Integral ◽  
Bernstein Polynomials ◽  
Numerical Results ◽  
Content Type ◽  
Stochastic Integral Equations

Purpose This paper aims to study fractional Brownian motion and its applications to nonlinear stochastic integral equations. Bernstein polynomials have been applied to obtain the numerical results of the nonlinear fractional stochastic integral equations. Design/methodology/approach Bernstein polynomials have been used to obtain the numerical solutions of nonlinear fractional stochastic integral equations. The fractional stochastic operational matrix based on Bernstein polynomial has been used to discretize the nonlinear fractional stochastic integral equation. Convergence and error analysis of the proposed method have been discussed. Findings Two illustrated examples have been presented to justify the efficiency and applicability of the proposed method. The corresponding obtained numerical results have been compared with the exact solutions to establish the accuracy and efficiency of the proposed method. Originality/value To the best of the authors’ knowledge, nonlinear stochastic Itô–Volterra integral equation driven by fractional Brownian motion has been for the first time solved by using Bernstein polynomials. The obtained numerical results well establish the accuracy and efficiency of the proposed method.

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 The Existence and Uniqueness of the Solution of a Nonlinear Fredholm–Volterra Integral Equation with Modified Argument via Geraghty Contractions

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 29
Author(s):  
Maria Dobriţoiu
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Existence And Uniqueness ◽  
Metric Spaces ◽  
Nonlinear Integral Equations ◽  
Uniqueness Of The Solution

Using some of the extended fixed point results for Geraghty contractions in b-metric spaces given by Faraji, Savić and Radenović and their idea to apply these results to nonlinear integral equations, in this paper we present some existence and uniqueness conditions for the solution of a nonlinear Fredholm–Volterra integral equation with a modified argument.

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 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 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 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.

Rods falling near a vertical wall

1977 ◽  
Vol 83 (2) ◽  
pp. 273-287 ◽  
Author(s):  
W. B. Russel ◽  
E. J. Hinch ◽  
L. G. Leal ◽  
G. Tieffenbruck
Keyword(s):  
Integral Equation ◽  
Viscous Fluid ◽  
Numerical Solution ◽  
Vertical Wall ◽  
Numerical Results ◽  
Slender Body ◽  
Asymptotic Results ◽  
Slender Body Theory ◽  
Friction Coefficients ◽  
Body Theory

As an inclined rod sediments in an unbounded viscous fluid it will drift horizontally but will not rotate. When it approaches a vertical wall, the rod rotates and so turns away from the wall. Illustrative experiments and a slender-body theory of this phenomenon are presented. In an incidental study the friction coefficients for an isolated rod are found by numerical solution of the slender-body integral equation. These friction coefficients are compared with the asymptotic results of Batchelor (1970) and the numerical results of Youngren ' Acrivos (1975), who did not make a slender-body approximation.

scholarly journals Integral equations of the first kind of Sonine type

2003 ◽  
Vol 2003 (57) ◽  
pp. 3609-3632 ◽  
Author(s):  
Stefan G. Samko ◽  
Rogério P. Cardoso
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Lebesgue Space ◽  
Inverse Operator ◽  
Orlicz Spaces ◽  
Hypersingular Integral ◽  
The Real ◽  
Integrable Kernel

A Volterra integral equation of the first kindKφ(x):≡∫−∞xk(x−t)φ(t)dt=f(x)with a locally integrable kernelk(x)∈L1loc(ℝ+1)is called Sonine equation if there exists another locally integrable kernelℓ(x)such that∫0xk(x−t)ℓ(t)dt≡1(locally integrable divisors of the unit, with respect to the operation of convolution). The formal inversionφ(x)=(d/dx)∫0xℓ(x−t)f(t)dtis well known, but it does not work, for example, on solutions in the spacesX=Lp(ℝ1)and is not defined on the whole rangeK(X). We develop many properties of Sonine kernels which allow us—in a very general case—to construct the real inverse operator, within the framework of the spacesLp(ℝ1), in Marchaud form:K−1f(x)=ℓ(∞)f(x)+∫0∞ℓ′(t)[f(x−t)−f(x)]dtwith the interpretation of the convergence of this “hypersingular” integral inLp-norm. The description of the rangeK(X)is given; it already requires the language of Orlicz spaces even in the case whenXis the Lebesgue spaceLp(ℝ1).

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