A numerical method based on finite difference for solving fractional delay differential equations

Moghaddam, Behrouz Parsa; Mostaghim, Z. S.

doi:10.1016/j.jtusci.2013.07.002

Cited by 60 publications

(27 citation statements)

References 10 publications

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“…From Table 1, we observe that the absolute errors obtained using Bernstein's polynomials appear comparatively better than the Legendre's polynomials. Test example 4.2 Here, we consider the fractional Mackey-Glass equation [11] defined using Caputo derivatives as, Figure 4 shows the different solution plot for fixed delay τ = 0.5, and Fig. 5 elaborates the approximated solution y(t) for varying time delay τ to 0.5, 0.6, 0.7 and 0.8.…”

Section: Resultsmentioning

confidence: 99%

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An Approximate Method for Solving Fractional Delay Differential Equations

Pandey

Kumar

Mohaptra

2016

Int. J. Appl. Comput. Math

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This paper presents an approximate method for solving a kind of fractional delay differential equations defined in terms of Caputo fractional derivatives. The approximate method is based on the application of the Bernstein's operational matrix of fractional differentiation. First, Bernstein operational matrix of fractional differentiation is presented generalizing the idea of Bernstein's operational matrix of derivative for integer orders, and then applied to solve the nonlinear fractional delay differential equations. The operational matrix method combined with the typical tau method reduces the fractional delay differential equation into system of nonlinear equations. Solving these nonlinear equations the desired solution is achieved. Two different cases of the fractional delay differential equations are illustrated and solved using the presented method. Numerical results and discussions demonstrate the applicability of the proposed method.

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Section: Resultsmentioning

confidence: 99%

“…FDDE's are considered by the researchers and methods for solving the FDDE's are discussed. In [11], finite difference scheme is applied to solve the FDDE's. Wang [12] used the Adams-Bashforth-Moulton method together with the linear interpolation method to approximate the solution of the delayed fractional order differential equations.…”

Section: Introductionmentioning

confidence: 99%

An Approximate Method for Solving Fractional Delay Differential Equations

Pandey

Kumar

Mohaptra

2016

Int. J. Appl. Comput. Math

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“…In recent decades, FDEs have been the focus of attention as a probable representation for description of anomalous diffusion and relaxation phenomena which are seen in a wide range of science and engineering fields [39][40][41][42][43][44], with applications in transport of fluid in porous media, diffusion of plasma, diffusion at liquid surfaces, surface growth and two-dimensional rotating flow. However, many recent researches [45,46] showed that fractional diffusion equations cannot totally represent some more complicated diffusion processes, whose diffusion behaviours depend on the spatial variation or time evolution.…”

Section: Introductionmentioning

confidence: 99%

Application of a Legendre collocation method to the space–time variable fractional-order advection–dispersion equation

Mallawi

Alzaidy

Hafez

2019

Journal of Taibah University for Science

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In this paper, an efficient spectral collocation method is presented for solving the space-time variable-order fractional advection-dispersion equations (ST-VO-FADE). The proposed method is based on the Legendre collocation spectral procedure together with the Legendre operational matrices for fractional derivatives, described in the sense of Riemann-Liouville and Caputo. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations, which greatly simplifies the solution process. The validity of the method is demonstrated by solving two numerical examples. Finally, comparisons between the algorithm derived in this paper and the existing algorithms are given, which show that our numerical schemes exhibit better performances than the existing ones. ARTICLE HISTORY

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“…Fractional calculus [2][3][4][5][6][7][8] has gained a considerable importance over the past few decades, due to the fact that it represents real-time physical models better and more accurately as compared with integer-order derivatives and integrals. Fractional delay differential equation is the generalization of delay differential equation.…”

Section: Introductionmentioning

confidence: 99%

Modified wavelets–based algorithm for nonlinear delay differential equations of fractional order

Iqbal

Shakeel

Mohyud‐Din

et al. 2017

Advances in Mechanical Engineering

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Most of the physical phenomena located around us are nonlinear in nature and their solutions are of great significance for scientists and engineers. In order to have a better representation of these physical phenomena, fractional calculus is developed. Some of these nonlinear physical models can be represented in the form of delay differential equations of fractional order. In this article, a new method named Gegenbauer Wavelets Steps Method is proposed using Gegenbauer polynomials and method of steps for solving nonlinear fractional delay differential equations. Method of steps is used to convert the fractional nonlinear fractional delay differential equation into a fractional nonlinear differential equation and then Gegenbauer wavelet method is applied at each iteration of fractional differential equation to find the solution. To check the accuracy and efficiency of the proposed method, the proposed method is implemented on different nonlinear fractional delay differential equations including singular-type problems also.

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A numerical method based on finite difference for solving fractional delay differential equations

Cited by 60 publications

References 10 publications

An Approximate Method for Solving Fractional Delay Differential Equations

An Approximate Method for Solving Fractional Delay Differential Equations

Application of a Legendre collocation method to the space–time variable fractional-order advection–dispersion equation

Modified wavelets–based algorithm for nonlinear delay differential equations of fractional order

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