Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel

Işık, Osman Raşit; Sezer, Mehmet; Güney, Zekeriya

doi:10.1016/j.amc.2011.01.114

Cited by 31 publications

(24 citation statements)

References 14 publications

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“…An mth order linear Volterra integro-differential equation with weakly singular kernel (WSVIDE) having variable coefficients can be expressed as [1,2]…”

Section: Volterra Integro-differential Equations With Weakly Singularmentioning

confidence: 99%

“…(3,7), its numerical solutions have become an important subject for researchers. Numerical solution algorithms have been given by collocation methods based on Bernstein polynomials [1] and Bessel polynomials [2] for ξ = 0.5. Tang [13] presented a numerical solution for the first order WSVIDEs by using spline collocation method.…”

Section: Volterra Integro-differential Equations With Weakly Singularmentioning

confidence: 99%

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Jacobi polynomial solutions of Volterra integro-differential equations with weakly singular kernel

Bahşı¹,

Çevik²,

Sezer³

2018

ntmsci

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An advanced matrix method is formulated for the solution of Volterra integro-differential equations with weakly singular kernel by using orthogonal Jacobi polynomials. Employing this practical method, it is possible to solve various types of these equations routinely in a systematic fashion. An error estimation procedure is prescribed to estimate the error of the basic Jacobi method and then the error correction term is added to the basic method to obtain more accurate results. Four test experiments are provided to confirm the validity and systematic approach of the advanced method. These experiments also certified that this advanced method surpasses the basic Jacobi method, as well as several alternative approaches.

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“…An mth order linear Volterra integro-differential equation with weakly singular kernel (WSVIDE) having variable coefficients can be expressed as [1,2]…”

Section: Volterra Integro-differential Equations With Weakly Singularmentioning

confidence: 99%

Section: Volterra Integro-differential Equations With Weakly Singularmentioning

confidence: 99%

Jacobi polynomial solutions of Volterra integro-differential equations with weakly singular kernel

Bahşı¹,

Çevik²,

Sezer³

2018

ntmsci

Self Cite

View full text Add to dashboard Cite

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“…Therefore, it can be used for finding computational errors on the nodes. The following theorem is proved in [41] for the Bernstein series solutions p N .t/. We modify the theorem and its proof for the Bessel series as follows.…”

Section: Error Bound and Stability Of The Solutionmentioning

confidence: 99%

A numerical approach for solving a class of the nonlinear Lane-Emden type equations arising in astrophysics

Yüzbaşı

2011

Math. Meth. Appl. Sci.

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In this paper, a collocation method based on the Bessel polynomials is presented for the approximate solution of a class of the nonlinear Lane-Emden type equations, which have many applications in mathematical physics. The exact solution can be obtained if the exact solution is polynomial. In other cases, such as an increasing number of nodes, a good approximation can be obtained with applicable errors. In addition, the method is presented with error and stability analysis. The numerical results show the effectiveness of the method for this type of equations. Comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate.

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“…The Adomian decomposition method and the variational iteration method have been used to provide the approximate solution of the Volterra integral equation with a weakly singular kernel in the reproducing kernel space . Collocation method based on the Bernstein polynomials has been introduced for the approximate solution of a class of linear Volterra integro‐differential equations with weakly singular kernel in Işik et al…”

Section: Introductionmentioning

confidence: 99%

“…8 The Adomian decomposition method and the variational iteration method have been used to provide the approximate solution of the Volterra integral equation with a weakly singular kernel in the reproducing kernel space. 9 Collocation method based on the Bernstein polynomials has been introduced for the approximate solution of a class of linear Volterra integro-differential equations with weakly singular kernel in Işik et al 10 Also, there are few numerical methods for solving partial integro-differential equations. For instance, an efficient matrix method based on two-dimensional orthonormal Bernstein polynomials have been presented to solve linear and nonlinear weakly singular partial integro-differential equations in Mirzaee et al 11 Numerical solution of a Volterra integro-differential equation of parabolic type with memory term subject to initial boundary value conditions has been presented in Fahim et al 12 Spline collocation methods have been proposed for the spatial discretization of a class of hyperbolic partial integro-differential equations in Fairweather.…”

Section: Introductionmentioning

confidence: 99%

Numerical scheme for solving singular fractional partial integro‐differential equation via orthonormal Bernoulli polynomials

Samadyar

Mirzaee

2019

Int J Numerical Modelling

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In this paper, an efficient matrix method based on 2D orthonormal Bernoulli polynomials are developed to obtain numerical solution of weakly singular fractional partial integro‐differential equations (FPIDEs). Operational matrix of integration, almost operational matrix of integration, and product operational matrix are employed to transform the solution of Volterra singular FPIDEs to the system of linear or nonlinear algebraic equations. The obtained algebraic system can be easily solved by using an appropriate numerical method. Also, some useful theorems concerning the convergence and error analysis associated to the mentioned approach are proven in the current paper. Finally, some numerical examples are included to illustrate the accuracy, efficiency, and applicability of the proposed approach.

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Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel

Cited by 31 publications

References 14 publications

Jacobi polynomial solutions of Volterra integro-differential equations with weakly singular kernel

Jacobi polynomial solutions of Volterra integro-differential equations with weakly singular kernel

A numerical approach for solving a class of the nonlinear Lane-Emden type equations arising in astrophysics

Numerical scheme for solving singular fractional partial integro‐differential equation via orthonormal Bernoulli polynomials

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