Fractional-order Legendre functions for solving fractional-order differential equations

In this article, a general formulation for the fractional-order Legendre functions (FLFs) is constructed to obtain the solution of the fractional-order differential equations. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, an efficient and reliable technique for the solution of them is too important. For the concept of fractional derivative we will adopt Caputo’s definition by using Riemann–Liouville fractional integral operator. Our main aim is to generalize the new orthogonal functions based on Legendre polynomials to the fractional calculus. Also a general formulation for FLFs fractional derivatives and product operational matrices is driven. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

Numerical comparison of methods for solving linear 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 linear 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 fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper will present a numerical comparison between the two methods and a conventional method such as the fractional difference method for solving linear differential equations of fractional order. The numerical results demonstrates that the new methods are quite accurate and readily implemented.     

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.     

Analytical approximations for a population growth model with fractional order

In this paper, we apply the homotopy analysis method (HAM) to solve the fractional Volterra’s model for population growth of a species in a closed system. This technique is extended to give solutions for nonlinear fractional integro–differential equations. The whole HAM solution procedure for nonlinear fractional differential equations is established. Further, the accurate analytical approximations are obtained for the first time, which are valid and convergent for all time t. This indicates the validity and great potential of the homotopy analysis method for solving nonlinear fractional integro–differential equations.     

Quadratic spline solution for boundary value problem of fractional order

Fractional differential equations are widely applied in physics, chemistry as well as engineering fields. Therefore, approximating the solution of differential equations of fractional order is necessary. We consider the quadratic polynomial spline function based method to find approximate solution for a class of boundary value problems of fractional order. We derive a consistency relation which can be used for computing approximation to the solution for this class of boundary value problems. Convergence analysis of the method is discussed. Four numerical examples are included to illustrate the practical usefulness of the proposed method.     

On the Solutions of the Generalized Reaction-Diffusion Model for Bacterial Colony

In this paper, the Adomian’s decomposition method has been developed to yield approximate solution of the reaction-diffusion model of fractional order which describe the evolution of the bacterium Bacillus subtilis, which grows on the surface of thin agar plates. The fractional derivatives are described in the Caputo sense. The method introduces a promising tool for solving many linear and nonlinear fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the approach is easy to implement and accurate when applied to partial differential equations of fractional order.     

Efficient analytic method for solving nonlinear fractional differential equations

In this paper, we propose a new approach of the generalized differential transform method (GDTM) for solving nonlinear fractional differential equations. In GDTM, it is a key to derive a recurrence relation of generalized differential transform (GDT) associated with the solution in the given fractional equation. However, the recurrence relations of complex nonlinear functions such as exponential, logarithmic and trigonometry functions have not been derived before in GDTM. We propose new algorithms to construct the recurrence relations of complex nonlinear functions and apply the GDTM with the proposed algorithms to solve nonlinear fractional differential equations. Several illustrative examples are demonstrated to show the effectiveness of the proposed method. It is shown that the proposed technique is robust and accurate for solving fractional differential equations.     

An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations

This paper aims to construct a general formulation for the Jacobi operational matrix of fractional integral operator. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, a reliable and efficient technique for the solution of them is too important. For the concept of fractional derivative we will adopt Caputo’s definition by using Riemann–Liouville fractional integral operator. Our main aim is to generalize the Jacobi integral operational matrix to the fractional calculus. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

Variational iteration method for fractional heat- and wave-like equations

This paper applies the variational iteration method to obtaining analytical solutions of fractional heat- and wave-like equations with variable coefficients. Comparison with the Adomian decomposition method shows that the VIM is a powerful method for the solution of linear and nonlinear fractional differential equations.     

Two‐dimensional shifted Legendre polynomial collocation method for electromagnetic waves in dielectric media via almost operational matrices

In this paper, a numerical solution of fractional partial differential equations (FPDEs) for electromagnetic waves in dielectric media will be discussed. For the solution of FPDEs, we developed a numerical collocation method using an algorithm based on two‐dimensional shifted Legendre polynomials approximation, which is proposed for electromagnetic waves in dielectric media. By implementing the partial Riemann–Liouville fractional derivative operators, two‐dimensional shifted Legendre polynomials approximation and its operational matrix along with collocation method are used to convert FPDEs first into weakly singular fractional partial integro‐differential equations and then converted weakly singular fractional partial integro‐differential equations into system of algebraic equation. Some results concerning the convergence analysis and error analysis are obtained. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2017 John Wiley & Sons, Ltd.     

A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations

In this paper, the pseudo-spectral method is generalized for solving fractional differential equations with initial conditions. For this purpose, an appropriate representation of the solution is presented and the pseudo-spectral differentiation matrix of fractional order is derived. Then, by using pseudo-spectral scheme, the problem is reduced to the solution of a system of algebraic equations. Through several numerical examples, we evaluate the accuracy and performance of our proposed method.     

Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method

Physical processes with memory and hereditary properties can be best described by fractional differential equations due to the memory effect of fractional derivatives. For that reason reliable and efficient techniques for the solution of fractional differential equations are needed. Our aim is to generalize the wavelet collocation method to fractional differential equations using cubic B-spline wavelet. Analytical expressions of fractional derivatives in Caputo sense for cubic B-spline functions are presented. The main characteristic of the approach is that it converts such problems into a system of algebraic equations which is suitable for computer programming. It not only simplifies the problem but also speeds up the computation. Numerical results demonstrate the validity and applicability of the method to solve fractional differential equation.     

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.     

A Short Note on the Quasilinearization Method for Fractional Differential Equations

Recent literature shows that for certain classes of fractional differential equations the monotone iterative technique fails to guarantee the quadratic convergence of the quasilinearization method. The present work proves the quadratic convergence of the quasilinearization method and the existence and uniqueness of the solution of such a class of fractional differential equations. Our analysis depends upon the classical Kantorovich theorem on Newton's method. Various examples are discussed in order to illustrate our approach.     

Novel analytical and numerical techniques for fractional temporal SEIR measles model

In this paper, a fractional temporal SEIR measles model is considered. The model consists of four coupled time fractional ordinary differential equations. The time-fractional derivative is defined in the Caputo sense. Firstly, we solve this model by solving an approximate model that linearizes the four time fractional ordinary differential equations (TFODE) at each time step. Secondly, we derive an analytical solution of the single TFODE. Then, we can obtain analytical solutions of the four coupled TFODE at each time step, respectively. Thirdly, a computationally effective fractional Predictor-Corrector method (FPCM) is proposed for simulating the single TFODE. And the error analysis for the fractional predictor-corrector method is also given. It can be shown that the fractional model provides an interesting technique to describe measles spreading dynamics. We conclude that the analytical and Predictor-Corrector schemes derived are easy to implement and can be extended to other fractional models. Fourthly, for demonstrating the accuracy of analytical solution for fractional decoupled measles model, we applied GMMP Scheme (Gorenflo-Mainardi-Moretti-Paradisi) to the original fractional equations. The comparison of the numerical simulations indicates that the solution of the decoupled and linearized system is close enough to the solution of the original system. And it also indicates that the linearizing technique is correct and effective.     

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.     

The alternative Legendre Tau method for solving nonlinear multi-order fractional differential equations

In this paper, the alternative Legendre polynomials (ALPs) are used to approximate the solution of a class of nonlinear multi-order fractional differential equations (FDEs). First, the operational matrix of fractional integration of an arbitrary order and the product operational matrix are derived for ALPs. These matrices together with the spectral Tau method are then utilized to reduce the solution of the mentioned equations into the one of solving a system of nonlinear algebraic equations with unknown ALP coefficients of the exact solution. The fractional derivatives are considered in the Caputo sense and the fractional integration is described in the Riemann-Liouville sense. Numerical examples illustrate that the present method is very effective for linear and nonlinear multi-order FDEs and high accuracy solutions can be obtained only using a small number of ALPs.     

Numerical Solution for a Variable-Order Fractional Nonlinear Cable Equation via Chebyshev Cardinal Functions

In this paper, a variable-order fractional derivative nonlinear cable equation is considered. It is commonly accepted that fractional differential equations play an important role in the explanation of many physical phenomena. For this reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of class of fractional partial differential equation with variable coefficient of fractional differential equation in various continues functions of spatial and time orders. Our main aim is to generalize the Chebyshev cardinal operational matrix to the fractional calculus. Finally, illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

Numerical technique based on generalized Laguerre and shifted Chebyshev polynomials

In this study, we present a numerical scheme for solving a class of fractional partial differential equations. First, we introduce psi -Laguerre polynomials like psi-shifted Chebyshev polynomials and employ these newly introduced polynomials for the solution of space-time fractional differential equations. In our approach, we project these polynomials to develop operational matrices of fractional integration. The use of these orthogonal polynomials converts the problem under consideration into a system of algebraic equations. The solution of this system provide us the desired results. The convergence of the proposed method is analyzed. Finally, some illustrative examples are included to observe the validity and applicability of the proposed method.     

时不变和时变的Caputo分数阶时滞微分系统的常数变易公式

This paper studies the variation of constant formulae for linear Caputo fractional delay differential systems. We discuss the exponential estimates of the solutions for linear time invariant fractional delay differential systems by using the Gronwall＇s integral inequality. The variation of constant formula for linear time invariant fractional delay differential systems is obtained by using the Laplace transform method. In terms of the superposition principle of linear systems and fundamental solution matrix, we also establish the variation of constant formula for linear time varying fractional delay differential systems. The obtained results generalize the corresponding ones of integer-order delayed differential equations.