In the last decade, theoretical and applied studies were done in order to provide a suitable definition of fractional derivative, which meets all the requirement of a derivative in its primary sense. It was concluded by some eminent researchers that the Riemann-Liouville version was the most suitable. However, many numerical approximation of fractional derivative were done with Caputo version. This paper addresses the numerical approximation of fractional differentiation based on the Riemann-Liouville definition, from power-law kernel to generalized Mittag-Leffler-law via exponential-decay-law.

Numerical approximation of Riemann-Liouville definition of fractional derivative: From Riemann-Liouville to Atangana-Baleanu

Lie symmetry analysis to the time fractional generalized fifth-order KdV equation

The fractional partial differential equations stand for natural phenomena all over the world from science to engineering. When it comes to obtaining the solutions of these equations, there are many various techniques in the literature. Some of these give to us approximate solutions; others give to us analytical solutions. In this paper, we applied the modified trial equation method (MTEM) to the one-dimensional nonlinear fractional wave equation (FWE) and time fractional generalized Burgers equation. Then, we submitted 3D graphics for different value of .

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The Modified Trial Equation Method for Fractional Wave Equation and Time Fractional Generalized Burgers Equation

Some physical problems found in nature can follow the power law; others can follow the Mittag-Leffler law and others the exponential decay law. On the other hand, one can observe in nature a physical problem that combines the three laws, it is therefore important to provide a new fractional operator that could possibly be used to model such physical problem. In this paper, we suggest a fractional operator power-law-exponential-Mittag-Leffler kernel with three fractional orders. Some very useful properties are obtained. Numerical solutions were obtained for three examples proposed. The results show that the new fractional operators are powerful mathematical tools to model complex problems.

New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications

In this paper, group analysis of the time fractional Harry-Dym equation with Riemann–Liouville derivative is performed. Its maximal symmetry group in Lie’s sense and the corresponding optimal system of subgroups are determined. Similarity reductions of the equation under study are performed. As a result, the reduced fractional ordinary differential equations are deduced, and some group invariant solutions in explicit form are obtained as well.

Symmetries and exact solutions of the time fractional Harry-Dym equation with Riemann-Liouville derivative

Invariant analysis of time fractional generalized Burgers and Korteweg–de Vries equations

Abstract We 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.

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

A systematic method is given to derive Lie point symmetries of nonlinear fractional ordinary differential equations and illustrate its applicability through the fractional Riccati equation and nonlinear fractional ordinary differential equation of Lienard type with Riemann–Liouville fractional derivative. Using the obtained Lie point symmetries, we construct their exact solutions wherever possible.

Invariant analysis of nonlinear fractional ordinary differential equations with Riemann-Liouville fractional derivative

We consider the well-known nonlinear Hirota equation (NLH) with fractional time derivative and derive its periodic wave solution and approximate analytic solitary wave solution using the homotopy analysis method (HAM). We also apply HAM to two coupled time fractional NLHs and construct their periodic wave solution and approximate solitary wave solution. We observe that the obtained periodic wave solution in both cases can be written in terms of the Mittag–Leffler function when the convergence control parameter $${c}_0$$
 equals $$-1$$
 . Convergence of the obtained solution is discussed. The derived approximate analytic solution and the effect of time-fractional order $$\alpha $$
 are shown graphically.

On solutions of two coupled fractional time derivative Hirota equations

In this article, we consider the well known nonlinear Schrodinger equation (NLS) with fractional time derivative and derive its approximate analytical solution using the homotopy analysis method (HAM) We also applied HAM to two coupled time fractional NLS and constructed its approximate solution The question of convergence of the obtained solution is discussed The obtained approximate analytical periodic wave solution, solitary wave solution and the effect of time fractional order $$\alpha $$
 are shown graphically

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T. Bakkyaraj

Papers

Invariant analysis of time fractional generalized Burgers and Korteweg–de Vries equations

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

Invariant analysis of nonlinear fractional ordinary differential equations with Riemann-Liouville fractional derivative

On solutions of two coupled fractional time derivative Hirota equations

Approximate Analytical Solution of Two Coupled Time Fractional Nonlinear Schrödinger Equations