An efficient analytical approach for fractional Lakshmanan-Porsezian-Daniel model

In this paper, the q -homotopy analysis transform method (q -HATM) is applied to find the solution for the fractional Lakshmanan-Porsezian-Daniel (LPD) model. The LPD model is the generalization of the non-linear Schrödinger (NLS) equation. The proposed method is graceful fusions of Laplace transform technique with q -homotopy analysis scheme, and the derivative is considered in Caputo sense. In order to validate and illustrate the efficiency of the proposed method, we analysed the projected model in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured for the three different cases in terms of 3D and contour plots for diverse values of the fractional order. The obtained results confirm that the future method is easy to implement, highly methodical, and very effective to analyse the behaviour of complex non-linear fractional differential equations exist in the connected areas of science and engineering.     

Analysis of fractional Swift-Hohenberg equation using a novel computational technique

In this paper, the approximated analytical solution for fractional Swift-Hohenberg (S-H) equation is found with the aid of novel technique called q -homotopy analysis transform method (q -HATM) . To ensure the applicability and efficiency of the proposed algorithm, we consider non-linear arbitrary-order S-H equation in presence and absence of dispersive term. The convergence analysis for the projected problem is presented, and the numerical simulations have been conducted to verify the future scheme is reliable and accurate. Further, the effect of bifurcation and dispersive parameters with physical importance on the probability density function for distinct fractional Brownian and standard motions are presented through plots. The obtained results elucidate that the proposed technique is easy to implement and very effective to analyse the complex problems that arose in science and technology.     

A new analytical modelling for fractional telegraph equation via Laplace transform

The main aim of the present work is to propose a new and simple algorithm for space-fractional telegraph equation, namely new fractional homotopy analysis transform method (HATM). The fractional homotopy analysis transform method is an innovative adjustment in Laplace transform algorithm (LTA) and makes the calculation much simpler. The proposed technique solves the nonlinear problems without using Adomian polynomials and He’s polynomials which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation transform method (HPTM). The beauty of the paper is error analysis which shows that our solution obtained by proposed method converges very rapidly to the known exact solution. The numerical solutions obtained by proposed method indicate that the approach is easy to implement and computationally very attractive. Finally, several numerical examples are given to illustrate the accuracy and stability of this method.     

Analytical approximate solutions of Riesz fractional diffusion equation and Riesz fractional advection–dispersion equation involving nonlocal space fractional derivatives

In this paper, we consider the analytical solutions of fractional partial differential equations (PDEs) with Riesz space fractional derivatives on a finite domain. Here we considered two types of fractional PDEs with Riesz space fractional derivatives such as Riesz fractional diffusion equation (RFDE) and Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second‐order space derivative with the Riesz fractional derivative of order α∈(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first‐order and second‐order space derivatives with the Riesz fractional derivatives of order β∈(0,1] and of order α∈(1,2] respectively. Here the analytic solutions of both the RFDE and RFADE are derived by using modified homotopy analysis method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. Here the space fractional derivatives are defined as Riesz fractional derivatives. Copyright © 2015 John Wiley & Sons, Ltd.     

The comparison of two reliable methods for accurate solution of time‐fractional Kaup–Kupershmidt equation arising in capillary gravity waves

In this paper, we compared two different methods, one numerical technique, viz Legendre multiwavelet method, and the other analytical technique, viz optimal homotopy asymptotic method (OHAM), for solving fractional‐order Kaup–Kupershmidt (KK) equation. Two‐dimensional Legendre multiwavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is used to compute the numerical solution of nonlinear time‐fractional KK equation. The approximate solutions of time fractional Kaup–Kupershmidt equation thus obtained by Legendre multiwavelet method are compared with the exact solutions as well as with OHAM. The present numerical scheme is quite simple, effective, and expedient for obtaining numerical solution of fractional KK equation in comparison to analytical approach of OHAM. Copyright © 2016 John Wiley & Sons, Ltd.     

Approximate analytical solution of two-dimensional space-time fractional diffusion equation

This work presents an iterative scheme for the numerical solution of the space-time fractional two-dimensional advection–reaction–diffusion equation applying homotopy perturbation with Laplace transform using Caputo fractional-order derivatives. The solution obtained is beneficial and significant to analyze the modeling of superdiffusive systems and subdiffusive system, anomalous diffusion, transport process in porous media. This iterative technique presents the combination of homotopy perturbation technique, and Laplace transforms with He's polynomials, which can further be applied to numerous linear/nonlinear two-dimensional fractional models to computes the approximate analytical solution. In the present method, the nonlinearity can be tackle by He's polynomials. The salient features of the present scientific work are the pictorial presentations of the approximate numerical solution of the two-dimensional fractional advection–reaction–diffusion equation for different particular cases of fractional order and showcasing of the damping effect of reaction terms on the nature of probability density function of the considered two-dimensional nonlinear mathematical models for various situations.     

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order α(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order β(0,1) and of order α(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.     

A novel attempt for finding comparatively accurate solution for sine‐Gordon equation comprising Riesz space fractional derivative

In this paper, a numerical procedure involving Chebyshev wavelet method has been implemented for computing the approximate solution of Riesz space fractional sine‐Gordon equation (SGE). Two‐dimensional Chebyshev wavelet method is implemented to calculate the numerical solution of space fractional SGE. The fractional SGE is considered as an interpolation between the classical SGE (corresponding to α = 2) and nonlocal SGE (corresponding to α = 1). As a consequence, the approximate solutions of fractional SGE obtained by using Chebyshev wavelet approach were compared with those derived by using modified homotopy analysis method with Fourier transform. Copyright © 2015 John Wiley & Sons, Ltd.     

Modified iteration method for solving fractional gas dynamics equation

This paper is devoted to the time‐fractional gas dynamics equation with Caputo derivative. Fractional operators are very natural tools to model memory‐dependent phenomena. Modified iteration method is proposed to obtain the approximate and analytical solution of the fractional gas dynamics equation. This method is a combined form of the new iteration method and Laplace transform. Modified iteration method really is powerful and simple method compared with other methods. Existence and uniqueness of solution are proven. Numerical results for different cases of the equation are obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

A new Rabotnov fractional-exponential function-based fractional derivative for diffusion equation under external force

This work suggested a new generalized fractional derivative which is producing different kinds of singular and nonsingular fractional derivatives based on different types of kernels. Two new fractional derivatives, namely Yang-Gao-Tenreiro Machado-Baleanu and Yang-Abdel-Aty-Cattani based on the nonsingular kernels of normalized sinc function and Rabotnov fractional-exponential function are discussed. Further, we presented some interesting and new properties of both proposed fractional derivatives with some integral transform. The coupling of homotopy perturbation and Laplace transform method is implemented to find the analytical solution of the new Yang-Abdel-Aty-Cattani fractional diffusion equation which converges to the exact solution in term of Prabhaker function. The obtained results in this work are more accurate and proposed that the new Yang-Abdel-Aty-Cattani fractional derivative is an efficient tool for finding the solutions of other nonlinear problems arising in science and engineering.     

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L_∞ ‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.     

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.     

A certain family of fractional wavelet transformations

In the present paper, a fractional wavelet transform of real order α is introduced, and various useful properties and results are derived for it. These include (for example) Perseval's formula and inversion formula for the fractional wavelet transform. Multiresolution analysis and orthonormal fractional wavelets associated with the fractional wavelet transform are studied systematically. Fractional Fourier transforms of the Mexican hat wavelet for different values of the order α are compared with the classical Fourier transform graphically, and various remarkable observations are presented. A comparative study of the various results, which we have presented in this paper, is also represented graphically.     

On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law

The present article deals with a fractional extension of the vibration equation for very large membranes with distinct special cases. The fractional derivative is considered in Atangana-Baleanu sense. A numerical algorithm based on homotopic technique is employed to examine the fractional vibration equation. The stability analysis is conducted for the suggested scheme. The maple software package is utilized for numerical simulation. In order to illustrate the effects of space, time, and order of Atangana-Baleanu derivative on the displacement, the outcomes of this study are demonstrated graphically. The results revel that the Atangana-Baleanu fractional derivative is very efficient in describing vibrations in large membranes.     

Finite difference spectral approximation for the time‐space fractional telegraph equation and its parameter estimation

The object of this paper is to present the numerical solution of the time‐space fractional telegraph equation. The proposed method is based on the finite difference scheme in temporal direction and Fourier spectral method in spatial direction. The fast Fourier transform (FFT) technique is applied to practical computation. The stability and convergence analysis are strictly proven, which shows that this method is stable and convergent with (2?α) order accuracy in time and spectral accuracy in space. Moreover, the Levenberg‐Marquardt (L‐M) iterative method is employed for the parameter estimation. Finally, some numerical examples are given to confirm the theoretical analysis.     

New predictor-corrector approach for nonlinear fractional differential equations: error analysis and stability

In this paper, the predictor-corrector approach is used to propose two algorithms for the numerical solution of linear and non-linear fractional differential equations (FDE). The fractional order derivative is taken to be in the sense of Caputo and its properties are used to transform FDE into a Volterra-type integral equation. Simpson''s 3/8 rule is used to develop new numerical schemes to obtain the approximate solution of the integral equation associated with the given FDE. The error and stability analysis for the two methods are presented. The proposed methods are compared with the ones available in the literature. Numerical simulation is performed to demonstrate the validity and applicability of both the proposed techniques. As an application, the problem of dynamics of the new fractional order non-linear chaotic system introduced by Bhalekar and Daftardar-Gejji is investigated by means of the obtained numerical algorithms.     

A local hybrid kernel meshless method for numerical solutions of two-dimensional fractional cable equation in neuronal dynamics

This study deals with obtaining numerical solutions of two-dimensional (2D) fractional cable equation in neuronal dynamics by using a recently introduced meshless method. In solution process at first stage, time derivatives that are appeared in the considered problem are discretized by using finite difference method. Then a meshless method based on hybridization of Gaussian and cubic kernels is developed in local fashion. The problem is solved both on regular and irregular domians. L_∞ and RMS error norms are calculated and compared with other numerical methods in literature as well as exact solutions. Also, obtained condition numbers are monitored. Numerical simulations show that local hybrid kernel meshless method is a thriving method for solving 2D fractional cable equation on regular and irregular domians.     

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.     

Numerical simulation of a class of fractional subdiffusion equations via the alternating direction implicit method

In this article, a new numerical technique is proposed for solving the two‐dimensional time fractional subdiffusion equation with nonhomogeneous terms. After a transformation of the original problem, standard central difference approximation is used for the spatial discretization. For the time step, a new fractional alternating direction implicit (FADI) scheme based on the L₁ approximation is considered. This FADI scheme is constructed by adding a small term, so it is different from standard FADI methods. The solvability, unconditional stability and H¹ norm convergence are proved. Finally, numerical examples show the effectiveness and accuracy of our proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 531–547, 2016     

Numerical solution of the multiterm time‐fractional diffusion equation based on reproducing kernel theory

In this study, we present an efficient computational method for finding approximate solution of the multi term time‐fractional diffusion equation. The approximate solution is presented in the form of a finite series in a reproducing kernel Hilbert space. The convergence of proposed method is studied under some hypothesis which provides the theoretical basis of proposed method for solving the considered equation. Finally, some numerical experiments are considered to examine the efficiency of proposed method in the sense of accuracy and CPU time.