A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation

Maleknejad, K.; Hashemizadeh, E.; Ezzati, Reza

doi:10.1016/j.cnsns.2010.05.006

Cited by 107 publications

(62 citation statements)

References 20 publications

(24 reference statements)

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“…In this section, we explain both first and second kind Volterra integral equations, with regular and weakly singular kernels, which are available in the existing literature [1][2][3]5] to verify the accuracy of our formulation presented in the previous section. The convergence of each linear Volterra integral equations is calculated by The exact solution is…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Now the unknown parameters i a are determined by solving the system of equations (6) and substituting these values of parameters in (3), we get the approximate solution ) ( x j of the integral equation (2). Now, we consider the Volterra integral equation (VIE) of the second kind [1] given by…”

Section: Formulation Of Integral Equation In Matrix Formmentioning

confidence: 99%

“…They can be differentiated and integrated without difficulty. Bernstein s approximation were used in [2] by Maleknejad et al to find out the Numerical solution of Volterra integral equations. Bhattacharya and Mandal in [3] and Shirin and Islam in [4] studied on Volterra and Fredholm Integral Equations Using Bernstein Polynomials to find out their numerical solutions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical Solutions Of Volterra Integral Equations Using Legendre Polynomials

Rahman

Islam

2013

GANIT: J. Bangladesh Math. Soc.

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In this paper, Legendre piecewise polynomials are used to approximate the solutions of linear Volterra integral equations. Both second and first kind integral equations with regular as well as weakly singular kernels are considered. A matrix formulation is given for linear Volterra integral equations by the technique of Galerkin method. Numerical examples are considered to verify the accuracy of the proposed derivations, and the numerical solutions in this paper are also compared with the existing methods in the published literature.

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Section: Numerical Examplesmentioning

confidence: 99%

Section: Formulation Of Integral Equation In Matrix Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Numerical Solutions Of Volterra Integral Equations Using Legendre Polynomials

Rahman

Islam

2013

GANIT: J. Bangladesh Math. Soc.

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“…These problems have application in mathematics, physics, and engineering. Recently, using polynomials have been common to solve these equations, see [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Bernstein Multiscaling polynomials and application by solving Volterra integral equations

2016

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In this paper, we present a direct computational method to solve Volterra integral equations. The proposed method is a direct method based on approximate functions with the Bernstein Multiscaling polynomials. In this method, using operational matrices, the integral equation turns into a system of equations. Our approach can solve nonlinear integral equations of the first kind and the second kind with piecewise solution. The computed operational matrices in this article are exact and new. The comparison of obtained solutions with the exact solutions shows that this method is acceptable. We also compared our approach with two direct and expansion-iterative methods based on the block-pulse functions. Our method produces a system, which is more economical, and the solutions are more accurate. Moreover, the stability of the proposed method is studied and analyzed by examining the noise effect on the data function. The appropriateness of noisy solutions with the amount of noise approves that the method is stable.

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“…The Linear integro-differential equations play a major role in many physical processes such as Nano hydrodynamic [1], drop wise condensation [2], biologic [3] and others. The various numerical methods exist for solving LFIDEs for example variation iteration method [4], Adomian decomposition method [5], Chebyshev Polynomials [6], Bernstein's approximation [7]. PIM was applied successfully for solving boundary value problems [8].…”

Section: Introductionmentioning

confidence: 99%