Numerical analysis of an operational Jacobi Tau method for fractional weakly singular integro-differential equations

Mokhtary, P.

doi:10.1016/j.apnum.2017.06.010

Cited by 32 publications

(13 citation statements)

References 17 publications

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“…which is a system of linear algebraic equations in terms of the unknown elements of the vector C. After solving system (27), we obtain the numerical solution of problem (1) from (24).…”

Section: Methodsmentioning

confidence: 99%

“…The operational matrices of P ( ) = P (1) and P (1− ) = P (0.5) are given in Section 3. We obtain matrices of D, K, and F as following: By using these matrices in (27) and solving system of algebraic equations, we obtain c 1,0 = 0, c 1,1 = 0.408248, c 1,2 = 0, c 2,0 = 0.707107, c 2,1 = 0.408248, c 2,2 = 0.…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…This kind of equations appears in the field of elasticity and fracture mechanics, 20 radiative equilibrium, 21 heat conduction problem, 22 and so on. During the recent years, a few methods have been used to solve the fractional integro-differential equations with weakly singular kernels numerically, such as the second kind Chebyshev polynomials, 23 spline collocation, 24 second kind Chebyshev wavelets (SCW) method, 25 Legendre wavelets (LW), 26 an operational Jacobi Tau method, 27 hat functions, 28 and the shifted Jacobi polynomials. 29 The various applications of fractional integro-differential equations with weakly singular kernels is our main motivation from the present work.…”

Section: Introductionmentioning

confidence: 99%

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A fast numerical algorithm based on the Taylor wavelets for solving the fractional integro‐differential equations with weakly singular kernels

Keshavarz

Ordokhani

2019

Math Methods in App Sciences

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In this paper, a fast numerical algorithm based on the Taylor wavelets is proposed for finding the numerical solutions of the fractional integro-differential equations with weakly singular kernels. The properties of Taylor wavelets are given, and the operational matrix of fractional integration is constructed. These wavelets are utilized to reduce the solution of the given fractional integro-differential equation to the solution of a linear system of algebraic equations. Also, convergence of the proposed method is studied. Illustrative examples are included to demonstrate the validity and applicability of the technique. KEYWORDS convergence, numerical solution, operational matrix of fractional integration, Taylor wavelets, weakly singular integro-differential equations Math Meth Appl Sci. 2019;42:4427-4443.wileyonlinelibrary.com/journal/mma

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“…which is a system of linear algebraic equations in terms of the unknown elements of the vector C. After solving system (27), we obtain the numerical solution of problem (1) from (24).…”

Section: Methodsmentioning

confidence: 99%

Section: Illustrative Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A fast numerical algorithm based on the Taylor wavelets for solving the fractional integro‐differential equations with weakly singular kernels

Keshavarz

Ordokhani

2019

Math Methods in App Sciences

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“…Fractional integro-differential equations with a weakly singular kernel are used to model a lot of different physical problems, such as heat conduction problem 1 and elasticity and fracture mechanics. 2 There are several numerical methods for the fractional integro-differential equations such as hybrid collocation method, 3 the Jacobi spectral-collocation method, 4 operational Jacobi Tau method, 5 and discontinuous Galerkin method. 6 However, the fractional integro-differential equations with a weakly singular kernel are solved by only a few methods such as Legendre wavelets method 7 and second Chebyshev wavelets methods.…”

Section: Introductionmentioning

confidence: 99%

Exact solution of a class of fractional integro‐differential equations with the weakly singular kernel based on a new fractional reproducing kernel space

Chen

Yang

2018

Math Methods in App Sciences

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In this paper, we construct a new fractional weighted reproducing kernel space, which is the minimum space containing the exact solution. The closed form of the reproducing kernel is obtained. Using this fractional reproducing kernel space, a class of fractional integro-differential equations with a weakly singular kernel is solved. The error estimation is given. The final numerical experiments demonstrate the correctness of the theory and the effectiveness of the method.KEYWORDS fractional integro-differential equation, fractional reproducing kernel space, weakly singular kernel 1 0 k(x, )u( )d + g(x), m − 1 < ≤ m, u (k) (0) = 0, k = 0, 1, · · · , m − 1,Math Meth Appl Sci. 2018;41:3841-3855.wileyonlinelibrary.com/journal/mma

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“…Ren et al solved the time-fractional convection dispersion equations in [21]. However, the methods for obtaining the analytic solutions of those FDEs are still lacking due to the complexity of FCT (see [22,23]). …”

Section: Introductionmentioning

confidence: 99%

An anomalous diffusion model based on a new general fractional operator with the Mittag-Leffler function of Wiman type

et al. 2018

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In this article, an anomalous diffusion model via a new Liouville-Caputo general fractional operator with the Mittag-Leffler function of Wiman type is investigated for the first time. The convergence of the series solution for the problem is discussed with the aid of the Laplace transform. The anomalous diffusion processes are compared to the characteristics of the conventional diffusion graphically. The results show that the new Liouville-Caputo general fractional operator is effective in characterizing and solving the problems of the anomalous diffusion.

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Numerical analysis of an operational Jacobi Tau method for fractional weakly singular integro-differential equations

Cited by 32 publications

References 17 publications

A fast numerical algorithm based on the Taylor wavelets for solving the fractional integro‐differential equations with weakly singular kernels

A fast numerical algorithm based on the Taylor wavelets for solving the fractional integro‐differential equations with weakly singular kernels

Exact solution of a class of fractional integro‐differential equations with the weakly singular kernel based on a new fractional reproducing kernel space

An anomalous diffusion model based on a new general fractional operator with the Mittag-Leffler function of Wiman type

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