Numerical methods for nonlinear partial differential equations of fractional order

In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when applied to partial differential equations of fractional order.     

Solitary wave solutions for a time-fraction generalized Hirota–Satsuma coupled KdV equation by an analytical technique

In this work, we implement a relatively analytical technique, the homotopy perturbation method (HPM), for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo derivatives. This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations which applied in engineering mathematics. The corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations. He’s homotopy perturbation method (HPM) which does not need small parameter is implemented for solving the differential equations. It is predicted that HPM can be found widely applicable in engineering.     

Approximate analytical solutions of fractional gas dynamic equations

In this article, differential transform method (DTM) has been successfully applied to obtain the approximate analytical solutions of the nonlinear homogeneous and non-homogeneous gas dynamic equations, shock wave equation and shallow water equations with fractional order time derivatives. The true beauty of the article is manifested in its emphatic application of Caputo fractional order time derivative on the classical equations with the achievement of the highly accurate solutions by the known series solutions and even for more complicated nonlinear fractional partial differential equations (PDEs). The method is really capable of reducing the size of the computational work besides being effective and convenient for solving fractional nonlinear equations. Numerical results for different particular cases of the equations are depicted through graphs.     

Solving systems of fractional differential equations using differential transform method

This paper presents approximate analytical solutions for systems of fractional differential equations using the differential transform method. The fractional derivatives are described in the Caputo sense. The application of differential transform method, developed for differential equations of integer order, is extended to derive approximate analytical solutions of systems of fractional differential equations. The solutions of our model equations are calculated in the form of convergent series with easily computable components. Some examples are solved as illustrations, using symbolic computation. The numerical results show that the approach is easy to implement and accurate when applied to systems of fractional differential equations. The method introduces a promising tool for solving many linear and nonlinear fractional differential equations.     

The homotopy perturbation method applied to the nonlinear fractional Kolmogorov–Petrovskii–Piskunov equations

The fractional derivatives in the sense of Caputo, and the homotopy perturbation method are used to construct approximate solutions for nonlinear Kolmogorov–Petrovskii–Piskunov (KPP) equations with respect to time and space fractional derivatives. Also, we apply complex transformation to convert a time and space fractional nonlinear KPP equation to an ordinary differential equation and use the homotopy perturbation method to calculate the approximate solution. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.     

Solving nonlinear fractional partial differential equations using the homotopy analysis method

In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional KdV, K(2,2), Burgers, BBM‐Burgers, cubic Boussinesq, coupled KdV, and Boussinesq‐like B(m,n) equations with initial conditions, which are introduced by replacing some integer‐order time derivatives by fractional derivatives. The homotopy analysis method for partial differential equations of integer‐order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions of the studied models are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A novel method for travelling wave solutions of fractional Whitham–Broer–Kaup,fractional modified Boussinesq and fractional approximate long wave equations in shallow water

In this paper, the analytical approximate traveling wave solutions of Whitham–Broer–Kaup (WBK) equations, which contain blow‐up solutions and periodic solutions, have been obtained by using the coupled fractional reduced differential transform method. By using this method, the solutions were calculated in the form of a generalized Taylor series with easily computable components. The convergence of the method as applied to the WBK equations is illustrated numerically as well as analytically. By using the present method, we can solve many linear and nonlinear coupled fractional differential equations. The results justify that the proposed method is also very efficient, effective and simple for obtaining approximate solutions of fractional coupled modified Boussinesq and fractional approximate long wave equations. Numerical solutions are presented graphically to show the reliability and efficiency of the method. Moreover, the results are compared with those obtained by the Adomian decomposition method (ADM) and variational iteration method (VIM), revealing that the present method is superior to others. Copyright © 2014 John Wiley & Sons, Ltd.     

Reduced differential transform method for nonlinear integral member of Kadomtsev–Petviashvili hierarchy differential equations

The main objective of this paper is to use the reduced differential to transform method (RDTM) for finding the analytical approximate solutions of two integral members of nonlinear Kadomtsev–Petviashvili (KP) hierarchy equations. Comparing the approximate solutions which obtained by RDTM with the exact solutions to show that the RDTM is quite accurate, reliable and can be applied for many other nonlinear partial differential equations. The RDTM produces a solution with few and easy computation. This method is a simple and efficient method for solving the nonlinear partial differential equations. The analysis shows that our analytical approximate solutions converge very rapidly to the exact solutions.     

Exact solutions of nonlinear time fractional partial differential equations by sub‐equation method

In this article, the sub‐equation method is presented for finding the exact solutions of a nonlinear fractional partial differential equations. For this, the fractional complex transformation method has been used to convert fractional‐order partial differential equation to ordinary differential equation. The fractional derivatives are described in Jumarie's the modified Riemann–Liouville sense. We apply to this method for the nonlinear time fractional differential equations. With the aid of symbolic computation, a variety of exact solutions for them are obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical approach to differential equations of fractional order

In this paper, the variational iteration method and the Adomian decomposition method are implemented to give approximate solutions for linear and nonlinear systems of differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper presents a numerical comparison between the two methods for solving systems of fractional differential equations. Numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order.     

Application of homotopy analysis method for fractional Swift Hohenberg equation – Revisited

In this article, the homotopy analysis method is used to obtain the approximate analytical solutions of the non-linear Swift Hohenberg equation with fractional time derivative. The fractional derivative is described in Caputo sense. Numerical results reveal that the method is easy to implement, reliable and accurate when applied to time fractional nonlinear partial differential equations. Effects of parameters of physical importance on the probability density function and the convergence of the approximate series solution using residual error formula with the proper choices of auxiliary parameter for various fractional Brownian motions and standard motion are depicted through graphs and tables for different particular cases.     

Infinite Sequence Solutions for Space-Time Fractional Symmetric Regularized Long Wave Equation

In this paper, we investigate the space-time fractional symmetric regularized long wave equation. By using the Bäcklund transformations and nonlinear superposition formulas of solutions to Riccati equation, we present infinite sequence solutions for space-time fractional symmetric regularized long wave equation. This method can be extended to solve other nonlinear fractional partial differential equations.     

Numerical solutions and solitary wave solutions of fractional KDV equations using modified fractional reduced differential transform method

In this paper, the modified fractional reduced differential transform method (MFRDTM) has been proposed and it is implemented for solving fractional KdV (Korteweg-de Vries) equations. The fractional derivatives are described in the Caputo sense. In this paper, the reduced differential transform method is modified to be easily employed to solve wide kinds of nonlinear fractional differential equations. In this new approach, the nonlinear term is replaced by its Adomian polynomials. Thus the nonlinear initial-value problem can be easily solved with less computational effort. In order to show the power and effectiveness of the present modified method and to illustrate the pertinent features of the solutions, several fractional KdV equations with different types of nonlinearities are considered. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of fractional KdV equations.     

Analysis of nonlinear fractional partial differential equations with the homotopy analysis method

In this paper, the time fractional partial differential equations are investigated by means of the homotopy analysis method. This technique is extended to study the partial differential equations of fractal order for the first time. The accurate series solutions are obtained. This indicates the validity and great potential of the homotopy analysis method for solving nonlinear fractional partial differential equations.     

Solving a system of nonlinear fractional partial differential equations using homotopy analysis method

In this article, the homotopy analysis method (HAM) has been employed to obtain solutions of a System of nonlinear fractional partial differential equations. This indicates the validity and great potential of the homotopy analysis method for solving system of fractional partial differential equations. The fractional derivative is described in the Caputo sense.     

The first integral method for some time fractional differential equations

In this paper, the fractional derivatives in the sense of modified Riemann–Liouville derivative and the first integral method are employed for constructing the exact solutions of nonlinear time-fractional partial differential equations. The power of this manageable method is presented by applying it to several examples. This approach can also be applied to other nonlinear fractional differential equations.     

Exact solutions for nonlinear partial differential equations via Exp‐function method

In this article, the Exp‐function method is applied to nonlinear Burgers equation and special fifth‐order partial differential equation. Using this method, we obtain exact solutions for these equations. The method is straightforward and concise, and its applications are promising. This method can be used as an alternative to obtain analytical and approximate solutions of different types of nonlinear differential equations. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A New Approach for Solving a Class of Delay Fractional Partial Differential Equations

In this article, a new numerical approach has been proposed for solving a class of delay time-fractional partial differential equations. The approximate solutions of these equations are considered as linear combinations of Müntz–Legendre polynomials with unknown coefficients. Operational matrix of fractional differentiation is provided to accelerate computations of the proposed method. Using Padé approximation and two-sided Laplace transformations, the mentioned delay fractional partial differential equations will be transformed to a sequence of fractional partial differential equations without delay. The localization process is based on the space-time collocation in some appropriate points to reduce the fractional partial differential equations into the associated system of algebraic equations which can be solved by some robust iterative solvers. Some numerical examples are also given to confirm the accuracy of the presented numerical scheme. Our results approved decisive preference of the Müntz–Legendre polynomials with respect to the Legendre polynomials.     

A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor's formula

In this article, a novel numerical method is proposed for nonlinear partial differential equations with space- and time-fractional derivatives. This method is based on the two-dimensional differential transform method (DTM) and generalized Taylor's formula. The fractional derivatives are considered in the Caputo sense. Several illustrative examples are given to demonstrate the effectiveness of the present method. Results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented elsewhere. Results also show that the numerical scheme is very effective and convenient for solving nonlinear partial differential equations of fractional order.     

Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

By the rapid growth of available data, providing data-driven solutions for nonlinear (fractional) dynamical systems becomes more important than before. In this paper, a new fractional neural network model that uses fractional order of Jacobi functions as its activation functions for one of the hidden layers is proposed to approximate the solution of fractional differential equations and fractional partial differential equations arising from mathematical modeling of cognitive-decision-making processes and several other scientific subjects. This neural network uses roots of Jacobi polynomials as the training dataset, and the Levenberg-Marquardt algorithm is chosen as the optimizer. The linear and nonlinear fractional dynamics are considered as test examples showing the effectiveness and applicability of the proposed neural network. The numerical results are compared with the obtained results of some other networks and numerical approaches such as meshless methods. Numerical experiments are presented confirming that the proposed model is accurate, fast, and feasible.