Analytical solutions of some integral fractional differential–difference equations

2019 ◽  
Vol 34 (01) ◽  
pp. 2050009 ◽  
Author(s):  
Jian-Gen Liu ◽  
Xiao-Jun Yang ◽  
Yi-Ying Feng
Keyword(s):  
Invariant Subspace ◽  
Difference Equations ◽  
Analytical Solutions ◽  
Volterra Equation ◽  
Burgers Equation ◽  
Space Time ◽  
Subspace Method ◽  
Lattice Equation ◽  
Fractional Differential

The invariant subspace method (ISM) is a powerful tool for investigating analytical solutions to fractional differential–difference equations (FDDEs). Based on previous work by other people, we apply the ISM to the space-time fractional differential and difference equations, including the cases of the scalar space-time FDDEs and the multi-coupled space-time FDDEs. As a result, we obtain some new analytical solutions to the well-known scalar space-time Lotka–Volterra equation, the space-time fractional generalized Hybrid lattice equation and the space-time fractional Burgers equation as well as two couple space-time FDDEs. Furthermore, some properties of the analytical solutions are illustrated by graphs.

Invariant Subspace Method and Exact Solutions of Certain Nonlinear Time Fractional Partial Differential Equations

2015 ◽  
Vol 18 (1) ◽  
Author(s):  
Ramajayam Sahadevan ◽  
Thangarasu Bakkyaraj
Keyword(s):  
Exact Solutions ◽  
Invariant Subspace ◽  
Toda Lattice ◽  
Burgers Equation ◽  
Kdv Equation ◽  
Subspace Method ◽  
Fractional Partial Differential Equations ◽  
Fractional Partial Differential Equation ◽  
Partial Differential ◽  
Fractional Differential

AbstractWe show, using invariant subspace method, how to derive exact solutions to the time fractional Korteweg-de Vries (KdV) equation, potential KdV equation with absorption term, KdV-Burgers equation and a time fractional partial differential equation with quadratic nonlinearity. Also we extend the invariant subspace method to nonlinear time fractional differential-difference equations and derive exact solutions of the time fractional discrete KdV and Toda lattice equations.

scholarly journals A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives

2020 ◽  
Vol 5 (2) ◽  
pp. 35-48 ◽  
Author(s):  
Kamal Ait Touchent ◽  
Zakia Hammouch ◽  
Toufik Mekkaoui
Keyword(s):  
Partial Differential Equations ◽  
Numerical Methods ◽  
Differential Equations ◽  
Exact Solutions ◽  
Invariant Subspace ◽  
Fractional Derivatives ◽  
Subspace Method ◽  
Partial Differential ◽  
Singular Kernels ◽  
Fractional Differential

AbstractIn this work, the well known invariant subspace method has been modified and extended to solve some partial differential equations involving Caputo-Fabrizio (CF) or Atangana-Baleanu (AB) fractional derivatives. The exact solutions are obtained by solving the reduced systems of constructed fractional differential equations. The results show that this method is very simple and effective for constructing explicit exact solutions for partial differential equations involving new fractional derivatives with nonlocal and non-singular kernels, such solutions are very useful to validate new numerical methods constructed for solving partial differential equations with CF and AB fractional derivatives.

Numerical computation of fractional Lotka-Volterra equation arising in biological systems

2015 ◽  
Vol 4 (2) ◽  
Author(s):  
Ramswroop ◽  
Jagdev Singh ◽  
Devendra Kumar
Keyword(s):  
Exact Solution ◽  
Difference Equations ◽  
Volterra Equation ◽  
Auxiliary Function ◽  
Transform Method ◽  
Homotopy Analysis ◽  
Effective Role ◽  
Fractional Differential ◽  
And Control

AbstractIn this paper, we present the homotopy analysis transform method (HATM) to solve fractional Lotka- Volterra equation, which describes the long term servival of species. The HATM solutions, denotes less error compare with their respective exact solution for alpha = 1. In addition to non-proposed techniques, HATMis valid for both small and large parameters, it also provides us with a simpleway to adjust and control the parameter hbar and auxiliary function H(t), which play effective role for convergence solutions of fractional differential-difference equations (FDDEs).

scholarly journals Optimal Perturbation Iteration Method for Solving Fractional Model of Damped Burgers’ Equation

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 958 ◽  
Author(s):  
Sinan Deniz ◽  
Ali Konuralp ◽  
Mnauel De la Sen
Keyword(s):  
Laplace Transform ◽  
Analytical Solutions ◽  
Differential Form ◽  
Fractional Derivatives ◽  
Burgers Equation ◽  
Optimal Perturbation ◽  
Fractional Differential ◽  
Fractional Type ◽  
Iteration Technique ◽  
Fractional Differential Form

The newly constructed optimal perturbation iteration procedure with Laplace transform is applied to obtain the new approximate semi-analytical solutions of the fractional type of damped Burgers’ equation. The classical damped Burgers’ equation is remodeled to fractional differential form via the Atangana–Baleanu fractional derivatives described with the help of the Mittag–Leffler function. To display the efficiency of the proposed optimal perturbation iteration technique, an extended example is deeply analyzed.

Extension of the Homotopy Perturbation Method for Solving Nonlinear Differential-Difference Equations

2010 ◽  
Vol 65 (12) ◽  
pp. 1060-1064 ◽  
Author(s):  
Mohamed Medhat Mousa ◽  
Aidarkan Kaltayev ◽  
Hasan Bulut
Keyword(s):  
Perturbation Method ◽  
Exact Solutions ◽  
Difference Equations ◽  
Analytical Solutions ◽  
Homotopy Perturbation Method ◽  
Lattice Equation ◽  
Convenient Tool ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions ◽  
De Vries

In this paper, we have extended the homotopy perturbation method (HPM) to find approximate analytical solutions for some nonlinear differential-difference equations (NDDEs). The discretized modified Korteweg-de Vries (mKdV) lattice equation and the discretized nonlinear Schr¨odinger equation are taken as examples to demonstrate the validity and the great potential of the HPM in solving such NDDEs. Comparisons are made between the results of the presented method and exact solutions. The obtained results reveal that the HPM is a very effective and convenient tool for solving such kind of equations.

scholarly journals Symmetry Classification and Exact Solutions of a Variable Coefficient Space-Time Fractional Potential Burgers’ Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Manoj Gaur ◽  
K. Singh
Keyword(s):  
Exact Solutions ◽  
Variable Coefficient ◽  
Burgers Equation ◽  
Space Time ◽  
Subspace Method ◽  
Group Invariant ◽  
Infinitesimal Generators ◽  
Symmetry Properties ◽  
Coefficient Space ◽  
Group Invariant Solutions

We investigate the symmetry properties of a variable coefficient space-time fractional potential Burgers’ equation. Fractional Lie symmetries and corresponding infinitesimal generators are obtained. With the help of the infinitesimal generators, some group invariant solutions are deduced. Further, some exact solutions of fractional potential Burgers’ equation are generated by the invariant subspace method.

On exact solutions of fractional differential-difference equations with Ψ-Riemann–Liouville derivative

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Rajagopalan Ramaswamy ◽  
Mohamed S. Abdel Latif ◽  
Amr Elsonbaty ◽  
Abas H. Abdel Kader
Keyword(s):  
Exact Solutions ◽  
Difference Equations ◽  
Three Dimensional ◽  
Generalized Solutions ◽  
Lattice Equation ◽  
Closed Form Solutions ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
First Time ◽  
Rl Fractional Derivative

Abstract The aim of this work is to modify the invariant subspace method (ISM) in order to obtain closed form solutions of fractional differential-difference equations with Ψ-Riemann–Liouville (Ψ-RL) fractional derivative for first time. We have investigated the cases of two-dimensional and the three-dimensional invariant subspaces (ISs) in the suggested scheme. Using the modified ISM, new exact generalized solutions for the general fractional mKdV Lattice equation and the fractional Volterra lattice system are obtained. Compared with similar solution techniques in literature, the presented solution scheme is highly efficient and is capable to find new general exact solutions which cannot be attained by other methods.

Homotopy Perturbation Method for Solving Nonlinear Differential- Difference Equations

2010 ◽  
Vol 65 (6-7) ◽  
pp. 511-517 ◽  
Author(s):  
Mohamed M. Mousa ◽  
Aidarkhan Kaltayev
Keyword(s):  
Schrödinger Equation ◽  
Perturbation Method ◽  
Exact Solutions ◽  
Difference Equations ◽  
Analytical Solutions ◽  
Schrodinger Equation ◽  
Homotopy Perturbation Method ◽  
Nonlinear Schrodinger Equation ◽  
Lattice Equation ◽  
Homotopy Perturbation

In this paper, the homotopy perturbation method (HPM) is extended to obtain analytical solutions for some nonlinear differential-difference equations (NDDEs). The discretized modified Kortewegde Vries (mKdV) lattice equation and the discretized nonlinear Schrodinger equation are taken as examples to illustrate the validity and the great potential of the HPM in solving such NDDEs. Comparisons between the results of the presented method and exact solutions are made. The results reveal that the HPM is very effective and convenient for solving such kind of equations.

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