Exact solutions of some systems of fractional differential‐difference equations

In this paper, the ‐expansion method is proposed to establish hyperbolic and trigonometric function solutions for fractional differential‐difference equations with the 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. We obtain the hyperbolic and periodic function solutions of the nonlinear time‐fractional Toda lattice equations and relativistic Toda lattice system. The proposed method is more effective and powerful for obtaining exact solutions for nonlinear fractional differential–difference equations and systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method

In this article, our main goal is to render an idea to convert a nonlinear weakly singular Volterra integral equation to a non‐singular one by new fractional‐order Legendre functions. The fractional‐order Legendre functions are generated by change of variable on well‐known shifted Legendre polynomials. We consider a general form of singular Volterra integral equation of the second kind. Then the fractional Legendre–Gauss–Lobatto quadratures formula eliminates the singularity of the kernel of the integral equation. Finally, the Legendre pseudospectral method reduces the solution of this problem to the solution of a system of algebraic equations. This method also can be utilized on fractional differential equations as well. The comparison of results of the presented method and other numerical solutions shows the efficiency and accuracy of this method. Also, the obtained maximum error between the results and exact solutions shows that using the present method leads to accurate results and fast convergence for solving nonlinear weakly singular Volterra integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

New predictor‐corrector scheme for solving nonlinear differential equations with Caputo‐Fabrizio operator

In this paper, we develop a new, simple, and accurate scheme to obtain approximate solution for nonlinear differential equation in the sense of Caputo‐Fabrizio operator. To derive this new predictor‐corrector scheme, which suits on Caputo‐Fabrizio operator, firstly, we obtain the corresponding initial value problem for the differential equation in the Caputo‐Fabrizio sense. Hence, by fractional Euler method and fractional trapeziodal rule, we obtain the predictor formula as well as corrector formula. Error analysis for this new method is derived. To test the validity and simplicity of this method, some illustrative examples for nonlinear differential equations are solved.     

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.     

A numerical algorithm for the solution of nonlinear fractional differential equations via beta‐derivatives

In this paper, the sinc‐collocation method (SCM) is investigated to obtain the solution of the nonlinear fractional order differential equations based on the relatively new defined fractional derivative, beta‐derivative. For this purpose, a theorem is proved for the approximate solution obtained from the SCM. Moreover, convergence analysis of the SCM is presented. To show the efficiency and the simplicity of the proposed method, some examples are solved, and the results are compared with the exact solutions of the considered equations.     

Sine‐cosine wavelet method for fractional oscillator equations

Purpose In this article, a novel computational method is introduced for solving the fractional nonlinear oscillator differential equations on the semi‐infinite domain. The purpose of the proposed method is to get better and more accurate results. Design/methodology/approach The proposed method is the combination of the sine‐cosine wavelets and Picard technique. The operational matrices of fractional‐order integration for sine‐cosine wavelets are derived and constructed. Picard technique is used to convert the fractional nonlinear oscillator equations into a sequence of discrete fractional linear differential equations. Operational matrices of sine‐cosine wavelets are utilized to transformed the obtained sequence of discrete equations into the systems of algebraic equations and the solutions of algebraic systems lead to the solution of fractional nonlinear oscillator equations. Findings The convergence and supporting analysis of the method are investigated. The operational matrices contains many zero entries, which lead to the high efficiency of the method, and reasonable accuracy is achieved even with less number of collocation points. Our results are in good agreement with exact solutions and more accurate as compared with homotopy perturbation method, variational iteration method, and Adomian decomposition method. Originality/value Many engineers can utilize the presented method for solving their nonlinear fractional models.     

Least‐squares mixed Galerkin formulation for variable‐coefficient fractional differential equations with D‐N boundary condition

We propose a least‐squares mixed variational formulation for variable‐coefficient fractional differential equations (FDEs) subject to general Dirichlet‐Neumann boundary condition by splitting the FDE as a system of variable‐coefficient integer‐order equation and constant‐coefficient FDE. The main contributions of this article are to establish a new regularity theory of the solution expressed in terms of the smoothness of the right‐hand side only and to develop a decoupled and optimally convergent finite element procedure for the unknown and intermediate variables. Numerical analysis and experiments are conducted to verify these findings.     

Extremal solutions for p‐Laplacian boundary value problems with the right‐handed Riemann‐Liouville fractional derivative

The paper is concerned with the solvability for several nonlinear boundary value problems of fractional p‐Laplacian differential equation involving the right‐handed Riemann‐Liouville derivative. By applying monotone iterative technique, lower and upper solutions method and the Banach fixed point theorem, sufficient conditions for existence and uniqueness of extremal solutions are obtained and they extend existing results. At last, two examples are provided to illustrate the results.     

On the concept of exact solution for nonlinear differential equations of fractional‐order

In this article, the new exact travelling wave solutions of the nonlinear space‐time fractional Burger's, the nonlinear space‐time fractional Telegraph and the nonlinear space‐time fractional Fisher equations have been found. Based on a nonlinear fractional complex transformation, certain fractional partial differential equations can be turned into ordinary differential equations of integer order in the sense of the Jumarie's modified Riemann–Liouville derivative. The ‐expansion method is effective for constructing solutions to the nonlinear fractional equations, and it appears to be easier and more convenient by means of a symbolic computation system. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and multiplicity of nontrivial solutions for nonlinear fractional differential systems with p‐Laplacian via critical point theory

In this paper, the existence and multiplicity of nontrivial solutions are obtained for nonlinear fractional differential systems with p‐Laplacian by combining the properties of fractional calculus with critical point theory. Firstly, we present a result that a class of p‐Laplacian fractional differential systems exists infinitely many solutions under the famous Ambrosetti‐Rabinowitz condition. Then, a criterion is given to guarantee that the fractional systems exist at least 1 nontrivial solution without satisfying Ambrosetti‐Rabinowitz condition. Our results generalize some existing results in the literature.     

Fractional‐order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro‐differential equations

In this paper, a new two‐dimensional fractional polynomials based on the orthonormal Bernstein polynomials has been introduced to provide an approximate solution of nonlinear fractional partial Volterra integro‐differential equations. For this aim, the fractional‐order orthogonal Bernstein polynomials (FOBPs) are constructed, and its operational matrices of integration, fractional‐order integration, and derivative in the Caputo sense and product operational matrix are derived. These operational matrices are utilized to reduce the under study problem to a nonlinear system of algebraic equations. Using the approximation of FOBPs, the convergence analysis and error estimate associated to the proposed problem have been investigated. Finally, several examples are included to clarify the validity, efficiency, and applicability of the proposed technique via FOBPs approximation.     

Analytical and approximate solutions of fractional‐order susceptible‐infected‐recovered epidemic model of childhood disease

In this work, we make use of the conformable fractional differential transform method (CFDTM) in order to compute an approximate solution of the fractional‐order susceptible‐infected‐recovered (SIR) epidemic model of childhood disease. The method provides the solution in the form of a rapidly convergent series. Two explanatory and illustrative examples are given to represent the efficacy of the obtained results.     

Study and analysis of a two‐dimensional nonconservative fractional order aerosol transport equation

The shifted Legendre collocation method is used to solve the two‐dimensional fractional order aerosol equation with initial and boundary conditions. The solution profile of the equation is presented graphically for different cases. The important feature of the article is graphical exhibitions of the effect of the size of the aerosol particles and also the temporal derivative on the solution profile. The salient feature of the article is the demonstration of lower variation of mass concentration with the change in time level in fractional order systems than that in integer order system. A drive has been taken towards the tabular and pictorial presentations of a comparison of the numerical solution of our proposed method with an analytical solution of an existing problem through error analysis which conforms super‐linearly convergence rate of the proposed method to validate its efficiency and effectiveness.     

Fractional quasi AKNS‐technique for nonlinear space–time fractional evolution equations

This paper aims to formulate the fractional quasi‐inverse scattering method. Also, we give a positive answer to the following question: can the Ablowitz‐Kaup‐Newell‐Segur (AKNS) method be applied to the space–time fractional nonlinear differential equations? Besides, we derive the Bäcklund transformations for the fractional systems under study. Also, we construct the fractional quasi‐conservation laws for the considered fractional equations from the defined fractional quasi AKNS‐like system. The nonlinear fractional differential equations to be studied are the space–time fractional versions of the Kortweg‐de Vries equation, modified Kortweg‐de Vries equation, the sine‐Gordon equation, the sinh‐Gordon equation, the Liouville equation, the cosh‐Gordon equation, the short pulse equation, and the nonlinear Schrödinger equation.     

Global synchronization of fractional‐order and integer‐order N component reaction diffusion systems: Application to biochemical models

This study considers the problem of control and synchronization between fractional‐order and integer‐order, N‐components reaction‐diffusion systems with nonidentical coefficients and different nonlinear parts. The control scheme is designed using the Lyapunov direct method. The results are exemplified by two significant biochemical models, namely, the fractional‐order Lengyel‐Epstein model and the Gray‐Scott model. To illustrate the effectiveness of the proposed scheme, numerical simulations are performed in one and two space dimensions using Homotopy Analysis Method (HAM).     

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     

On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay

A novel collocation method based on Genocchi wavelet is presented for the numerical solution of fractional differential equations and time‐fractional partial differential equations with delay. In this work, to achieve the approximate solution with height accuracy, we employed the operational matrix of integer derivative and the pseudo‐operational matrix of fractional derivative in Caputo sense. Also, based on Genocchi function properties, we presented delay and pantograph operational matrices of Genocchi wavelet functions (GWFs). Due to operational and pseudo‐operational matrices, the equations under this study can be turned into nonlinear algebraic equations with the unknown GWF coefficients. For illustrating the upper bound of error for the proposed method, we estimate the error in the sense of Sobolev space. In addition, to demonstrate the efficacy of the pseudo‐operational matrix of fractional derivative, we investigate the upper bound of error for the mentioned matrix. Finally, the algorithm based on the proposed approach is implemented for some numerical experiments to confirm accuracy and applicability.     

Numerical analysis nonlinear multi‐term time fractional differential equation with collocation method via fractional B‐spline

In this work, we present numerical analysis for nonlinear multi‐term time fractional differential equation which involve Caputo‐type fractional derivatives for . The proposed method is based on utilization of fractional B‐spline basics in collocation method. The scheme can be readily obtained efficient and quite accurate with less computational work numerical result. The proposal approach transform nonlinear multi‐term time fractional differential equation into a suitable linear system of algebraic equations which can be solved by a suitable numerical method. The numerical experiments will be verify to demonstrate the effectiveness of our method for solving one‐ and two‐dimensional multi‐term time fractional differential equation.     

A new numerical approach for single rational soliton solution of Chen‐Lee‐Liu equation with Riesz fractional derivative in optical fibers

In this paper, time‐splitting spectral approximation technique has been proposed for Chen‐Lee‐Liu (CLL) equation involving Riesz fractional derivative. The proposed numerical technique is efficient, unconditionally stable, and of second‐order accuracy in time and of spectral accuracy in space. Moreover, it conserves the total density in the discretized level. In order to examine the results, with the aid of weighted shifted Grünwald‐Letnikov formula for approximating Riesz fractional derivative, Crank‐Nicolson weighted and shifted Grünwald difference (CN‐WSGD) method has been applied for Riesz fractional CLL equation. The comparison of results reveals that the proposed time‐splitting spectral method is very effective and simple for obtaining single soliton numerical solution of Riesz fractional CLL equation.