A general framework for the numerical analysis of high-order finite difference solvers for nonlinear multi-term time-space fractional partial differential equations with time delay

Hendy, Ahmed S.; Zaky, Mahmoud A.; Staelen, Rob De

doi:10.1016/j.apnum.2021.06.010

Cited by 26 publications

(6 citation statements)

References 26 publications

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“…3,[13][14][15][16] For that matter, credible and competent techniques for solving FPDEs are demanded. Researchers' assays have resulted in presenting efficient schemes for numerical solutions of different categories of FPDEs, such as weighted finite difference methods, [17][18][19][20][21] Laplace-Sumudu transform, 22 modified double Laplace decomposition method, 23 Bernoulli wavelet method, 24 Tau method, 25 Galerkin method, 26 Petrov-Galerkin method, 27 Legendre wavelet method, 28 Crank-Nickolson method, 29 Haar Wavelet Picard method, 30 collocation method, 31,32 operational Chebyshev wavelets method, 33 and Taylor wavelets method. 34 Classical and fractional KdV-Burgers-Kuramoto equations are nonlinear partial differential equations that explain instability systems and unstable drift waves in plasma.…”

Section: Introductionmentioning

confidence: 99%

A pseudo‐operational collocation method for variable‐order time‐space fractional KdV–Burgers–Kuramoto equation

Sadri¹,

Hosseini²,

Hinçal³

et al. 2023

Math Methods in App Sciences

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The idea of this work is to provide a pseudo-operational collocation scheme to deal with the solution of the variable-order time-space fractional KdV-Burgers-Kuramoto equation (VOSTFKBKE). Such the fractional partial differential equation (FPDE) has three characteristics of dissipation, dispersion, and instability, which make this equation is used to model many phenomena in diverse fields of physics. Numerical solutions are sought in a linear combination of two-dimensional Jacobi polynomials as basis functions. In order to approximate unknown functions in terms of the basis vector, pseudo-operational matrices are constructed to avoid integration. An error bound of the residual function is estimated in a Jacobi-weighted space in the L 2 norms. Numerical results are compared with exact ones and those reported by other researchers to demonstrate the effectiveness of the recommended method.

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Section: Introductionmentioning

confidence: 99%

A pseudo‐operational collocation method for variable‐order time‐space fractional KdV–Burgers–Kuramoto equation

Sadri¹,

Hosseini²,

Hinçal³

et al. 2023

Math Methods in App Sciences

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“…Due to the tremendous difficulty of finding exact solutions for many types of FDEs, many researchers have been interested in obtaining analytical and numerical solutions for such problems. Some of them are the finite difference (Hendy et al (2021)), predictor-corrector (Kumar and Gejji (2019)), wavelet operational matrix (Yi and Huang ( 2014)), Galerkin-Legendre spectral (Zaky et al (2020)), fractional-order Boubaker wavelets (Rabiei and Razzaghi (2021)), fractional Jacobi collocation (Abdelkawy et al (2020)) and artificial neural network (Pakdaman et al 2017) methods. Because of the globality of fractional calculus, it is preferable to use global numerical techniques to solve FDEs of various types.…”

Section: Introductionmentioning

confidence: 99%

Robust spectral treatment for time-fractional delay partial differential equations

2023

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Fractional delay differential equations (FDDEs) and time-fractional delay partial differential equations (TFDPDEs) are the focus of the present research. The FDDEs is converted into a system of algebraic equations utilizing a novel numerical approach based on the spectral Galerkin (SG) technique. The suggested numerical technique is likewise utilized for TFDPDEs. In terms of shifted Jacobi polynomials, suitable trial functions are developed to fulfill the initial-boundary conditions of the main problems. According to the authors, this is the first time utilizing the SG technique to solve TFDPDEs. The approximate solution of five numerical examples is provided and compared with those of other approaches and with the analytic solutions to test the superiority of the proposed method.

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“…In [30], the Adomian decomposition method was applied to solve impulsive nonclassical type differential equations with the Caputo fractional operator. Very recently, a number of studies in the direction of efficiency and performance of various computational methods have been given by researchers [31][32][33][34][35][36][37][38][39]. In [40], authors have defined a novel fractional-order Lagrangian to study the motion of a beam on a nanowire.…”

Section: Introductionmentioning

confidence: 99%

An Implementation of the Generalized Differential Transform Scheme for Simulating Impulsive Fractional Differential Equations

Odibat

Ertürk

Kumar³

et al. 2022

Mathematical Problems in Engineering

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In this research study, the generalized differential transform scheme has been applied to simulate impulsive differential equations with the noninteger order. One specific tool of the implemented scheme is that it converts the problems into a recurrence equation that finally leads easily to the solution of the considered problem. The validity and reliability of this method have successfully been accomplished by applying it to simulate the solution of some equations. It is shown that the considered method is very suitable and efficient for solving classes of fractional-order initial value problems for impulsive differential equations and might find wide applications.

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A general framework for the numerical analysis of high-order finite difference solvers for nonlinear multi-term time-space fractional partial differential equations with time delay

Cited by 26 publications

References 26 publications

A pseudo‐operational collocation method for variable‐order time‐space fractional KdV–Burgers–Kuramoto equation

A pseudo‐operational collocation method for variable‐order time‐space fractional KdV–Burgers–Kuramoto equation

Robust spectral treatment for time-fractional delay partial differential equations

An Implementation of the Generalized Differential Transform Scheme for Simulating Impulsive Fractional Differential Equations

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