Numerical solution of system of Volterra integral equations with weakly singular kernels and its convergence analysis

Maleknejad, K.; Ostadi, A.

doi:10.1016/j.apnum.2016.12.005

Cited by 11 publications

(9 citation statements)

References 16 publications

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“…[25] which the obtained results in Ref. [25] are shown in Tables 4 and 5. Comparing the results, shows that the method applied in this paper gives more accurate solutions than SE-Sinc method.…”

Section: Exact Solutionmentioning

confidence: 67%

“…The problem (28) has been solved by SE-Sinc method in Ref. [25] which the obtained results in Ref. [25] are shown in Tables 4 and 5.…”

Section: Exact Solutionmentioning

confidence: 99%

“…Thus a variety of numerical and approximate methods have been developed to solve these, such as operational matrices, [19] homotopy perturbation method (HPM), [20] homotopy analysis method (HAM), [21] fractional differential transform method, [22] extrapolation method, [23] Legendre wavelets, [24] and Sinc approximation with the single exponential (SE). [25] Li [26] proposed a novel application of Taylor expansion method for approximate solution of linear ordinary differential equations with variable coefficients. The method expanded by Li and his co-authors to solve Abel's integral equation, [18,27] Riccati equation, [28] an integral equation with fixed singularity for a cruciform crack, [29] a class of linear integro-differential equations, [30] and fractional integro-differential equations.…”

Section: Introductionmentioning

confidence: 99%

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An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

Didgar¹,

Vahidi²

2018

Commun. Theor. Phys.

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In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singularVolterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution, which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.

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Section: Exact Solutionmentioning

confidence: 67%

“…The problem (28) has been solved by SE-Sinc method in Ref. [25] which the obtained results in Ref. [25] are shown in Tables 4 and 5.…”

Section: Exact Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

Didgar¹,

Vahidi²

2018

Commun. Theor. Phys.

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“…In [9], Mirzaee and Rafei solved the nonlinear two-dimensional Volterra integral equations. In the references [10][11][12][13][14][15], the authors solved linear and nonlinear system of Volterra integral equations numerically. This paper describes the numerical solutions of (TDVIDE).…”

Section: Introductionmentioning

confidence: 99%

Two Dimensional Volterra Integrodifferential Equation of Heat Conduction and Diffusion Problems and Its Numerical Solutions

Al‐Bugami

2022

Preprint

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This paper deals with new equation in the form Two-dimensional Volterra integrodifferential equation (TDVIDE), which is related of the heat conduction and diffusion problems. Runge–Kutta method (RKM) and Bolck-by-Block method (BBM) have been applied to the problem. Numerical examples are presented, and their results are compared with the analytical solution to demonstrate the validity and applicability of the methods.

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“…Sinc methods have been proposed and studied by Stenger [21]. The sinc method has been increasingly used for solving ordinary differential equations [22][23][24][25], partial differential equations [26][27][28][29], and integral equations [30][31][32][33]; it not only has exponential convergence rate, but also can deal with singular problems well. In recent years, the sinc method has been also frequently employed for the numerical solution of fractional partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

Sinc-Galerkin method for solving the time fractional convection–diffusion equation with variable coefficients

Chen

2020

Adv Differ Equ

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In this paper, a new numerical algorithm for solving the time fractional convection–diffusion equation with variable coefficients is proposed. The time fractional derivative is estimated using the $L_{1}$ L 1 formula, and the spatial derivative is discretized by the sinc-Galerkin method. The convergence analysis of this method is investigated in detail. The numerical solution is $2-\alpha$ 2 − α order accuracy in time and exponential rate of convergence in space. Finally, some numerical examples are given to show the effectiveness of the numerical scheme.

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Numerical solution of system of Volterra integral equations with weakly singular kernels and its convergence analysis

Cited by 11 publications

References 16 publications

An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

Two Dimensional Volterra Integrodifferential Equation of Heat Conduction and Diffusion Problems and Its Numerical Solutions

Sinc-Galerkin method for solving the time fractional convection–diffusion equation with variable coefficients

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