Analytical and numerical results of fractional differential-difference equations

Abstract In this paper, we examine the fractional differential-difference equation (FDDE) by employing the proposed sensitivity approach (SA) and Adomian transformation method (ADTM). In SA the nonlinear differential-difference equation is converted to infinite linear equations which have a wide criterion to solve for the analytical solution. By ADTM, the FDDE is converted into ordinary differential-difference equation that can be solved. We test both the techniques through some test problems which are arising in nonlinear dynamical systems and found that ADTM is equivalently appropriate and simpler method to handle than SA.

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General Fractional Dynamics

Mathematics ◽

10.3390/math9131464 ◽

2021 ◽

Vol 9 (13) ◽

pp. 1464

Author(s):

Vasily E. Tarasov

Keyword(s):

Fractional Calculus ◽

Exact Solutions ◽

Arbitrary Order ◽

Nonlinear Dynamical Systems ◽

Fractional Integrals ◽

Fractional Dynamics ◽

Fractional Systems ◽

Nonlinear Dynamical ◽

Fractional Differential ◽

Solutions Of Equations

General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time. GFDynamics implies research and obtaining results concerning the general form of nonlocality, which can be described by general-form operator kernels and not by its particular implementations and representations. In this paper, the concept of “general nonlocal mappings” is proposed; these are the exact solutions of equations with GFI and GFD at discrete points. In these mappings, the nonlocality is determined by the operator kernels that belong to the Sonin and Luchko sets of kernel pairs. These types of kernels are used in general fractional integrals and derivatives for the initial equations. Using general fractional calculus, we considered fractional systems with general nonlocality in time, which are described by equations with general fractional operators and periodic kicks. Equations with GFI and GFD of arbitrary order were also used to derive general nonlocal mappings. The exact solutions for these general fractional differential and integral equations with kicks were obtained. These exact solutions with discrete timepoints were used to derive general nonlocal mappings without approximations. Some examples of nonlocality in time are described.

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Exact Solutions for Fractional Differential-Difference Equations by (G'/G)-Expansion Method with Modified Riemann-Liouville Derivative

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.2014.m798 ◽

2016 ◽

Vol 8 (2) ◽

pp. 293-305 ◽

Cited By ~ 10

Author(s):

Ahmet Bekir ◽

Ozkan Guner ◽

Burcu Ayhan ◽

Adem C. Cevikel

Keyword(s):

Difference Equation ◽

Exact Solutions ◽

Difference Equations ◽

Applied Mathematics ◽

Expansion Method ◽

New Exact Solutions ◽

Liouville Derivative ◽

Nonlinear Lattice ◽

Fractional Differential ◽

Differential Difference Equation

AbstractIn this paper, the (G'/G)-expansion method is suggested to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential difference equation into its differential difference equation of integer order. With the aid of symbolic computation, we choose nonlinear lattice equations to illustrate the validity and advantages of the algorithm. It is shown that the proposed algorithm is effective and can be used for many other nonlinear lattice equations in mathematical physics and applied mathematics.

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A highly accurate numerical method for solving nonlinear time‐fractional differential difference equation

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.5883 ◽

2019 ◽

Author(s):

Muhammad Khalid ◽

Fareeha Sami Khan ◽

Mariam Sultana

Keyword(s):

Difference Equation ◽

Numerical Method ◽

Fractional Differential ◽

Differential Difference Equation

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Extension of Khan’s Homotopy Transformation Method via Optimal Parameter for Differential Difference Equations

Journal of Applied Mathematics ◽

10.1155/2014/813474 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8

Author(s):

Mohamed S. Mohamed ◽

Khaled A. Gepreel ◽

Faisal A. Al-Malki ◽

Nouf Altalhi

Keyword(s):

Difference Equation ◽

Difference Equations ◽

Numerical Solutions ◽

Optimal Parameter ◽

Transformation Method ◽

Series Solutions ◽

Transform Method ◽

Homotopy Perturbation ◽

Physical Problems ◽

Differential Difference Equation

A new scheme, deduced from Khan’s homotopy perturbation transform method (HPTM) (Khan, 2014; Khan and Wu, 2011) via optimal parameter, is presented for solving nonlinear differential difference equations. Simple but typical examples are given to illustrate the validity and great potential of Khan’s homotopy perturbation transform method (HPTM) via optimal parameter in solving nonlinear differential difference equation. The numerical solutions show that the proposed method is very efficient and computationally attractive. It provides more realistic series solutions that converge very rapidly for nonlinear real physical problems. The results reveal that the method is very effective and simple. This method gives more reliable results as compared to other existing methods available in the literature. The numerical results demonstrate the validity and applicability of the method.

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