scholarly journals Numerical Solution of the Fredholm and Volterra Integral Equations by Using Modified Bernstein–Kantorovich Operators

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1193
Author(s):  
Suzan Cival Buranay ◽  
Mehmet Ali Özarslan ◽  
Sara Safarzadeh Falahhesar
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Unknown Function ◽  
Input Data ◽  
Linear Equations ◽  
Numerical Approach ◽  
Volterra Integral Equations ◽  
Test Problems ◽  
The Stability ◽  
Kantorovich Operators

The main aim of this paper is to numerically solve the first kind linear Fredholm and Volterra integral equations by using Modified Bernstein–Kantorovich operators. The unknown function in the first kind integral equation is approximated by using the Modified Bernstein–Kantorovich operators. Hence, by using discretization, the obtained linear equations are transformed into systems of algebraic linear equations. Due to the sensitivity of the solutions on the input data, significant difficulties may be encountered, leading to instabilities in the results during actualization. Consequently, to improve on the stability of the solutions which imply the accuracy of the desired results, regularization features are built into the proposed numerical approach. More stable approximations to the solutions of the Fredholm and Volterra integral equations are obtained especially when high order approximations are used by the Modified Bernstein–Kantorovich operators. Test problems are constructed to show the computational efficiency, applicability and the accuracy of the method. Furthermore, the method is also applied to second kind Volterra integral equations.

Solving the Volterra Integral Equations with Weakly Singular Kernel by Taylor Expansion Methods

2012 ◽  
Vol 220-223 ◽  
pp. 2129-2132
Author(s):  
Li Huang ◽  
Yu Lin Zhao ◽  
Liang Tang
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Unknown Function ◽  
Taylor Expansion ◽  
Closed Form Solution ◽  
Form Solution ◽  
Volterra Integral Equations ◽  
Singular Kernel ◽  
Weakly Singular Kernel ◽  
Weakly Singular

In this paper, we propose a Taylor expansion method for solving (approximately) linear Volterra integral equations with weakly singular kernel. By means of the nth-order Taylor expansion of the unknown function at an arbitrary point, the Volterra integral equation can be converted approximately to a system of equations for the unknown function itself and its n derivatives. This method gives a simple and closed form solution for the integral equation. In addition, some illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method.

Numerical approach for nonlinear system of Fredholm-Volterra integral equations

2021 ◽  
Author(s):  
Zainidin Eshkuvatov ◽  
Husnida Mamatova ◽  
Shahrina Ismail ◽  
Ilyani Abdullah ◽  
Rakhmatillo Aloev
Keyword(s):  
Nonlinear System ◽  
Integral Equations ◽  
Numerical Approach ◽  
Volterra Integral Equations

scholarly journals On the Fredholm integral equation associated with pairs of dual integral equations

1969 ◽  
Vol 16 (3) ◽  
pp. 185-194 ◽  
Author(s):  
V. Hutson
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Bessel Function ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Unknown Function ◽  
Sufficient Conditions ◽  
Neumann Series ◽  
Dual Integral Equations ◽  
Image Position

Consider the Fredholm equation of the second kindwhereand Jv is the Bessel function of the first kind. Here ka(t) and h(x) are given, the unknown function is f(x), and the solution is required for large values of the real parameter a. Under reasonable conditions the solution of (1.1) is given by its Neumann series (a set of sufficient conditions on ka(t) for the convergence of this series is given in Section 4, Lemma 2). However, in many applications the convergence of the series becomes too slow as a→∞ for any useful results to be obtained from it, and it may even happen that f(x)→∞ as a→∞. It is the aim of the present investigation to consider this case, and to show how under fairly general conditions on ka(t) an approximate solution may be obtained for large a, the approximation being valid in the norm of L2(0, 1). The exact conditions on ka(t) and the main result are given in Section 4. Roughly, it is required that 1 -ka(at) should behave like tp(p>0) as t→0. For example, ka(at) might be exp ⌈-(t/ap)⌉.

scholarly journals Resolvent, Natural, and Sumudu Transformations: Solution of Logarithmic Kernel Integral Equations with Natural Transform

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Kevser Köklü
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Logarithmic Kernel ◽  
Sumudu Transform

In this paper, the resolvent of an integral equation was found with natural transform which is a new transformation which converged to Laplace and Sumudu transformations, and the result was confirmed by the Sumudu transform. At the same time, a solution to the first type of logarithmic kernel Volterra integral equations has been produced by the natural transform.

Reduced surface integral equations for Laplacian fields in the presence of layered bodies

2006 ◽  
Vol 84 (12) ◽  
pp. 1049-1061 ◽  
Author(s):  
I R Ciric
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Unknown Function ◽  
Source Region ◽  
Normal Derivative ◽  
Surface Integral ◽  
Field Problem ◽  
Layered Bodies ◽  
High Computational Efficiency ◽  
Reduced Surface

Laplacian potential fields in stratified media are usually analyzed using an integral equation for an unknown function over the union of all the interfaces between regions with different homogeneous materials. In this paper, the field problem is solved using a reduced integral equation involving a single unknown function over only the boundary of the source region. The new integral equation is derived by introducing surface operators to express the potential and its normal derivative on each interface in terms of a single unknown function over the same interface. These operators and the corresponding single functions are obtained recursively, from one interface to the next. Thus, a substantial decrease in the amount of necessary numerical computation and computer memory is achieved especially for systems containing identical layered bodies where the reduction operators are only constructed for one of the bodies. The purpose of this paper is to derive reduced integral equations by directly applying the interface conditions and to show their high computational efficiency for systems of layered bodies.PACS Nos.: 02.30.Rz, 02.70.Pt, 41.20.Cv

scholarly journals A New Numerical Technique for Solving Volterra Integral Equations Using Chebyshev Spectral Method

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Ahmed A. Khidir
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Integral Equations ◽  
Numerical Technique ◽  
Volterra Integral Equations ◽  
New Method ◽  
Algebraic Equations ◽  
Chebyshev Spectral Collocation ◽  
Chebyshev Spectral Method ◽  
Chebyshev Spectral Collocation Method

In this work, we propose a new method for solving Volterra integral equations. The technique is based on the Chebyshev spectral collocation method. The application of the proposed method leads Volterra integral equation to a system of algebraic equations that are easy to solve. Some examples are presented and compared with some methods in the literature to illustrate the ability of this technique. The results demonstrate that the new method is more efficient, convergent, and accurate to the exact solution.

scholarly journals Data dependence of solutions for Fredholm-Volterra integral equations in L2[a, b]

2013 ◽  
Vol 5 (1) ◽  
pp. 5-19
Author(s):  
Szilárd András
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Continuous Dependence ◽  
Main Tool ◽  
Volterra Integral Equations ◽  
Data Dependence ◽  
Solution Operator ◽  
Volterra Type ◽  
Type Integral

Abstract In this paper we study the continuous dependence and the differentiability with respect to the parameter λ ∈ [λ1, λ2] of the solution operator S : [λ1, λ2] → L2[a, b] for a mixed Fredholm-Volterra type integral equation. The main tool is the fiber Picard operators theorem (see [9], [8], [11], [3] and [2]).

Generalizations of the Pollaczek-Khinchin integral equation in the theory of queues

1986 ◽  
Vol 18 (4) ◽  
pp. 952-990 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Waiting Time ◽  
Volterra Integral Equations ◽  
Markov Renewal Process ◽  
Waiting Time Distributions ◽  
Virtual Waiting Time ◽  
Bulk Service ◽  
Constant Coefficients ◽  
Linear Differential

A classical result in queueing theory states that in the stable M/G/1 queue, the stationary distribution W(x) of the waiting time of an arriving customer or of the virtual waiting time satisfies a linear Volterra integral equation of the second kind, of convolution type. For many variants of the M/G/1 queue, there are corresponding integral equations, which in most cases differ from the Pollaczek–Khinchin equation only in the form of the inhomogeneous term. This leads to interesting factorizations of the waiting-time distribution and to substantial algorithmic simplifications. In a number of priority queues, the waiting-time distributions satisfy Volterra integral equations whose kernel is a functional of the busy-period distribution in related M/G/1 queues. In other models, such as the M/G/1 queue with Bernoulli feedback or with limited admissions of customers per service, there is a more basic integral equation of Volterra type, which yields a probability distribution in terms of which the waiting-time distributions are conveniently expressed.For several complex queueing models with an embedded Markov renewal process of M/G/1 type, one obtains matrix Volterra integral equations for the waiting-time distributions or for related vectors of mass functions. Such models include the M/SM/1 and the N/G/1 queues, as well as the M/G/1 queue with some forms of bulk service.When the service-time distributions are of phase type, the numerical computation of waiting-time distributions may commonly be reduced to the solution of systems of linear differential equations with constant coefficients.

scholarly journals Stability and Convergence of Solutions to Volterra Integral Equations on Time Scales

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Eleonora Messina ◽  
Antonia Vecchio
Keyword(s):  
Integral Equations ◽  
Time Scales ◽  
Sufficient Conditions ◽  
Volterra Integral Equations ◽  
Stability And Convergence ◽  
Time Behavior ◽  
Convergence Properties ◽  
Convergence Of Solutions ◽  
Long Time ◽  
The Stability

We consider Volterra integral equations on time scales and present our study about the long time behavior of their solutions. We provide sufficient conditions for the stability and investigate the convergence properties when the kernel of the equations vanishes at infinity.

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