An analytical approach for systems of fractional differential equations by means of the innovative homotopy perturbation method

We have applied the new approach of homotopy perturbation method (NAHPM) for partial differential system equations featuring time-fractional derivative. The Caputo-type of fractional derivative is considered in this paper. A combination of NAHPM and multiple fractional power series form has been used the first time to present analytical solution. In order to illustrate the simplicity and ability of the suggested approach, some specific and clear examples have been given. All numerical calculations in this manuscript have been carried out with Mathematica

Approximate solution for solving fractional Riccati differential equations via trigonometric basic functions

In this paper, a method has been proposed to finding a numerical function for the Riccati differential equations of non integer order (FRDEs), in which trigonometric basic functions are used. First, by defining trigonometric basic functions, we define the values of the transformation function in relation to trigonometric basis functions (TBFs). Following that, the numerical function is defined as a linear combination of trigonometric base functions and values of transform function which is named trigonometric transform method (TTM), and the convergence of the method is also presented. To get a numerical solution function with discrete derivatives of the solution function, we have determined the numerical solution function which satisfies the FRDEs. In the end, the algorithm of the method is elaborated with several examples. Numerical results obtained show that the proposed algorithm gives very good numerical solutions. In one example, we have presented an absolute error comparison of some numerical methods. Keywords: Trigonometric transform, Riccati differential equations, Basic functions, Caputo derivativ

Solving fractional Bratu’s equations using a semi-analytical technique

Finding the solution of the fractional Bratu’s differential equations(FBDEs) in this paper is based on a semi-analytical iterative approach.Temimi and Ansari introduced this method and called it TAM.Three examples, with their approximate solutions, are presented in thisway to show its suitability, convenience, simplicity and efficiency. Theresults demonstrate that the advantage of this method to other methodsis that there are no limiting conditions for nonlinear fractional differentialequations with initial conditions or boundary conditions. Regardingthe help of the softwareMathematica, all the results have been obtainedand the calculations have been done

Analysis of solution for system of nonlinear fractional Burger differential equations based on multiple fractional power series

We have applied the new approach of homotopic perturbation method (NHPM) for Burger differential system equations featuring time-fractional derivative. A combination of NHPM and multiple fractional power series form has been used the first time to present analytical solution. In order to illustrate the simplicity and ability of the suggested approach, some specific and clear examples have been given. All numerical calculations in this manuscript have been carried out with Mathematica

Approximate method for solving strongly fractional nonlinear problems using fuzzy transform

In this research work, we have shown that it is possible to use fuzzy transform method (FTM) for approximate solution of strongly fractional nonlinear problems. In numerical methods, in order to approximate a function on a particular interval, only a restricted number of points are employed. However, what makes the F-transform preferable to other methods is that it makes use of all points in this interval. The comparison of the time used in minutes is given for two derivatives Caputo derivative and Caputo-Fabrizio derivative

An optimal method for approximating the delay differential equations of noninteger order

Abstract The main purpose of this paper is to use a method with a free parameter, named the optimal asymptotic homotopy method (OHAM), in order to obtain the solution of delay differential equations, delay partial differential equations, and a system of coupled delay equations featuring fractional derivative. This method is preferable to others since it has faster convergence toward homotopy perturbation method, and the convergence rate can be set as a controlled area. Various examples are given to better understand the use of this method. The approximate solutions are compared with exact solutions as well

Variational Iteration Method and He’s Polynomials for Time-Fractional Partial Differential Equations

In this work, we have applied the variational iteration method and He’s polynomials to solve partial differential equation (PDEs) with time-fractional derivative. The variational homotopy perturbation iteration method (VHPIM) is presented in two steps. Some illustrative examples are given in order to show the ability and simplicity of the approach. All numerical calculations in this manuscript were performed on a PC applying some programs written in Maple18

A reliable mixed method for singular integro-differential equations of non-integer order

It is our goal in this article to apply a method which is based on the assumption that combines two methods of conjugating collocation and multiple shooting method. The proposed method can be used to find the numerical solution of singular fractional integro-differential boundary value problems (SFIBVPs)         Dϑ y(t) + η ∫0t  (t−s)ς−1 y(s) ds = g(t),    1 < ϑ ≤ 2,    0 < ς< 1,    η ∈ ℝ, where Dϑ denotes the Caputo derivative of order ϑ. Meanwhile, in a separate section the existence and uniqueness of this method is also discussed. Two examples are presented to illustrate the application and further understanding of the methods