Bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations

Bazm, Sohrab

doi:10.1016/j.cam.2014.07.018

Cited by 49 publications

(25 citation statements)

References 20 publications

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“…2n (0), and c (1) 2n−1 (0) can be obtained, we return to Eq. (9), and we seek the solution of this equation in the form:…”

Section: Separation Of Variables Methodsmentioning

confidence: 99%

“…It is well known that the integral equations govern many mathematical models of various phenomena in physics, economy, biology, engineering, and even in mathematics and other fields of science. The illustrative examples of such models can be found in the literature (see, e.g., [1][2][3][4][5][6][7][8]). Many problems of mathematical physics, applied mathematics, and engineering are reduced to Volterra-Fredholm integral equations, see [9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

“…A numerical method is used to translate the Volterra-Fredholm integral Eq. (1) to Volterra integral equations of the second kind with the continuous kernel, the ideas are interesting, and this area caught the attention of many researchers, having so many applications.…”

Section: Introductionmentioning

confidence: 99%

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On a discussion of Volterra–Fredholm integral equation with discontinuous kernel

2020

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The purpose of this paper is to establish the general solution of a Volterra-Fredholm integral equation with discontinuous kernel in a Banach space. Banach's fixed point theorem is used to prove the existence and uniqueness of the solution. By using separation of variables method, the problem is reduced to Volterra integral equations of the second kind with continuous kernel. Normality and continuity of the integral operator are also discussed.

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“…2n (0), and c (1) 2n−1 (0) can be obtained, we return to Eq. (9), and we seek the solution of this equation in the form:…”

Section: Separation Of Variables Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On a discussion of Volterra–Fredholm integral equation with discontinuous kernel

2020

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“…Among these methods, we refer to wavelet method, homotopy perturbation method, collocation method, and meshless method . A novel algorithm to get approximate solution of these equations is to express the solution as linear combination of orthogonal or nonorthogonal basis functions and polynomials such as block‐pulse functions, hat functions, Bernoulli polynomials, Legendre polynomials, Bessel polynomials, Chebyshev polynomials, Fibonacci polynomials, and orthonormal Bernstein polynomials …”

Section: Introductionmentioning

confidence: 99%

On the numerical solution of stochastic quadratic integral equations via operational matrix method

Mirzaee

Samadyar

2018

Math Methods in App Sciences

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In this paper, stochastic operational matrix of integration based on delta functions is applied to obtain the numerical solution of linear and nonlinear stochastic quadratic integral equations (SQIEs) that appear in modelling of many real problems. An important advantage of this method is that it dose not need any integration to compute the constant coefficients. Also, this method can be utilized to solve both linear and nonlinear problems. By using stochastic operational matrix of integration together collocation points, solving linear and nonlinear SQIEs converts to solve a nonlinear system of algebraic equations, which can be solved by using Newton's numerical method. Moreover, the error analysis is established by using some theorems. Also, it is proved that the rate of convergence of the suggested method is O(h 2 ). Finally, this method is applied to solve some illustrative examples including linear and nonlinear SQIEs. Numerical experiments confirm the good accuracy and efficiency of the proposed method. KEYWORDSBrownian motion process, error analysis, operational matrix, quadratic integral equations, stochastic integral equations | INTRODUCTIONThe various kinds of integral equations are important mathematical tools for describing knowledge models that appear in different areas of applied science. Because of extensive application of integral equations and not having the exact solutions in many cases, numerical solution of integral equations has attracted researcher's attention to develop numerical method for approximating solution of these equations. Among these methods, we refer to wavelet method, 1-4 homotopy perturbation method, 5-8 collocation method, 9,10 and meshless method. 11-13 A novel algorithm to get approximate solution of these equations is to express the solution as linear combination of orthogonal or nonorthogonal basis functions and polynomials such as block-pulse functions, 14,15 hat functions, 16,17 Bernoulli polynomials, 18 Legendre polynomials, 19 Bessel polynomials, 20 Chebyshev polynomials, 21 Fibonacci polynomials, 22 and orthonormal Bernstein polynomials. 23 Mathematical modeling of various phenomena in real world is leaded to a special kind of integral equations named quadratic integral equations. Quadratic integral equations always arise in many problems of mathematical physics and chemical such as theory of radiative transfer, the kinetic theory of gases, the theory of neutron transport, the queuing theory, and the traffic theory and many other applications. Existence solution and numerical method to solve these type of integral equations have been studied in previous papers. [24][25][26][27][28] Recently, with increasing computational power, it becomes feasible to utilize more accurate equations to model some problems that are often dependent on a noise source. So modeling these problems naturally require the use of

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“…Such as mathematical economics and optimal control theory [1,2], boundary value problems of mathematics physics in [3]. There is a fact that many integral equations are usually difficult to solve analytically.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of two-dimensional Fredholm integral equations via modification of barycentric rational interpolation

Pan¹,

Huang²

2017

Proceedings of the 2017 2nd International Conference on Automation, Mechanical Control and Computational Engineering (AMCCE 201

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Abstract. We have presented a modified barycentric rational interpolation method for solving two-dimensional integral equations. The present method can accurately approximate the exact solution. We also compare with the Lagrange interpolation method and Nystöm method. The proposed method can achieve higher numerical accuracy than other two methods. At last, we give some examples to illustrate the validity of the presented method.

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Bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations

Cited by 49 publications

References 20 publications

On a discussion of Volterra–Fredholm integral equation with discontinuous kernel

On a discussion of Volterra–Fredholm integral equation with discontinuous kernel

On the numerical solution of stochastic quadratic integral equations via operational matrix method

Numerical solution of two-dimensional Fredholm integral equations via modification of barycentric rational interpolation

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