Search

We couple the L1 discretization for the Caputo derivative in time with the spectral Galerkin method in space to devise a scheme that solves quasilinear subdiffusion equations. Both the diffusivity and the source are allowed to be nonlinear functions of the solution. We prove the stability and convergence of the method with spectral accuracy in space. The temporal order depends on the regularity of the solution in time. Furthermore, we support our results with numerical simulations that utilize parallelism for spatial discretization. Moreover, as a side result, we find exact asymptotic values of the error constants along with their remainders for discretizations of the Caputo derivative and fractional integrals. These constants are the smallest possible, which improves previously established results from the literature. [ABSTRACT FROM AUTHOR]

The fractional generalized Bagley–Torvik equation (FGB-TE) is a mathematical description of the motion of an immersed plate in a Newtonian fluid. The analytical study of the FGB-TE with uncertain initial conditions and two independent fractional orders is usually complex and difficult. Therefore, it is necessary to develop a proper and effective technique to solve the fuzzy fractional generalized Bagley–Torvik equation (FFGB-TE) analytically. This paper presents the analytical fuzzy solution of the FFGB-TEs based on the concept of the fuzzy Caputo generalized Hukuhara differentiability (g H -differentiability) using the fuzzy Laplace transform (FLT) technique. The closed-form solution of FFGB-TEs is presented for both the homogeneous and non-homogeneous cases in terms of the Mittag-Leffler function (MLF) involving double series. Several significant results are introduced and proven with true reasoning. We illustrate our proposed analytical approach with the help of several demonstrative examples. To enhance the novelty of the proposed work, we solved the FFGB-TE as an application of the motion of an immersed plate and visualized their graphs to support the theoretical results. • Analytical solutions of fuzzy fractional Bagley-Torvik equation are investigated. • The potential solutions are extracted using fuzzy Caputo g H -differentiability. • Closed-form solutions of proposed scheme are presented using Mittag-Leffler function. • Several significant results are presented using the proposed analytical approach. • The real-world problem is solved as an application of the proposed study. [ABSTRACT FROM AUTHOR]

In this paper we study the Controllability and the generalized Ulam-Hyers stability of Riemann-Liouville fractional order linear differential equations with variable coefficients in Banach spaces. We obtained the desired results by means of fixed point principle and fractional calculus. Furthermore, we used the Gronwall's inequality to investigate above type of stability for Riemann-Liouville fractional order linear differential equation with time dependent coefficients under some specific assumptions. An example is also provided at the end to illustrate the theoretical work. [ABSTRACT FROM AUTHOR]

In this work, the Optimal Homotopy Asymptotic Method (OHAM) is prolifically implemented to find the optimal solutions of fractional order of fuzzy differential equations. We inspect the competence of the method by examining fractional order time-dependent first order fuzzy initial value problems in the Caputo derivative sense. The concepts of fuzzy sets theory and fractional calculus were used to present a new general fuzzification formulation of the OHAM method followed by fuzzy analysis for the proposed fractional problems are presented in detail. OHAM is recognized as a feasible and convergent method for solving linear and nonlinear models, and it provides a convenient technique for controlling the convergence of approximate solutions optimally. The method's capability is demonstrated by obtaining approximate solutions to linear and nonlinear fuzzy initial value problems in various orders of fractional derivative. The numerical results acknowledge that OHAM is a feasible and transformation mechanism for solving linear and nonlinear fuzzy fractional initial value problems. Also, the results ensure that the method used was succinct, efficient, and simple to use, to deal with a more comprehensive fuzzy fractional differential equation. [ABSTRACT FROM AUTHOR]

We consider a class of boundary value problems for linear fractional weakly singular integro-differential equations with Caputo fractional derivatives and integral boundary conditions. Using an integral equation reformulation of the boundary value problem, we first study the regularity of the exact solution and its Caputo derivative. Based on the obtained regularity properties and by using suitable smoothing transformations along with spline collocation techniques, the numerical solution of the problem is discussed. Optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given. A numerical illustration is also presented. [ABSTRACT FROM AUTHOR]

In this paper, we propose a semi numerical-analytical method, called Fractional Reduced Differential Transform Method (FRDTM), for finding exact and approximate solutions of fractional Helmholtz equation with appropriate initial conditions. The fractional derivatives are demonstrated in the Caputo sense. The solutions are given in the form of series with easily computable terms, then with the help of Mittag-Leffler function, we find the exact solutions of the fractional Helmholtz equations. Three examples are given to demonstrate the applicability of FRDTM. [ABSTRACT FROM AUTHOR]

The time-fractional heat conduction equation with the Caputo derivative is solved for an infinite plane with an external half-infinite crack which surfaces are exposed to the heat flux loading. The solution is obtained using the integral transform technique and is expressed in terms of the Mittag-Leffler function. The stress intensity factor is calculated for different values of the order of fractional derivative. Numerical results are illustrated graphically. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we construct a discontinuous Galerkin finite element scheme for the multi-order fractional ordinary differential equation. The analysis of the stability shows the scheme is L 2 stable. The existence and uniqueness of the numerical solution are discussed in detail. The convergence study gives the approximation orders under L 2 norm and L ∞ norm. Numerical examples demonstrate the effectiveness of the theoretical results. The oscillation phenomena are also found during numerical tracing a non-linear multi-order fractional initial problem. [ABSTRACT FROM AUTHOR]

Abstract This work deals with the generalization of the Lorentz model. It is made using the Caputo fractional derivative of order 0 < γ ≤ 1 in the frequency domain. The fractional complex dielectric function, index of refraction, reflectance, electric susceptibility were obtained. It is shown that all of them depend not only on the frequency of applied field, but also on the fractional order of the differential equation 0 < γ ≤ 1. This opens the possibility to describe more accurately the experimental results in complex systems. The ordinary results are obtained when γ = 1. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a new fractional derivative of the Caputo type is proposed and some basic properties are studied. The form of the definition shows that the new derivative is the natural extension of the Caputo one, and that it yields the Caputo derivative with designated memory length. By adaptively changing the memory length, the new definition is capable of capturing local memory effect in a distinct way, which is critical in modelling complex systems where the short memory properties has to be considered. Another attractive property of the new derivative is that it is naturally associated with the Riemann–Liouville definition and as a result, the well established Grünwald–Letnikov approach for numerically solving the fractional differential equation can be readily embedded to approximate the solution of differential equation that involves the new derivatives. Numerical simulations demonstrate the changeable memory effect of the new definition. Highlights • The paper presents a new fractional derivative of the Caputo type. • Some important properties and integral transform of the new derivative is derived. • The new derivative offers particular value for capturing the local memory effect. [ABSTRACT FROM AUTHOR]

This paper is devoted to the application of the method of lines to solve one-dimensional diffusion equation where the classical (integer) second derivative is replaced by a fractional derivative of the Caputo type of order α less than 2 as the space derivative. A system of initial value problems approximates the solution of the fractional diffusion equation with spline approximation of the Caputo derivative. The result is a numerical approach of order O ( Δ x 2 + Δ t m ) , where Δx and Δt denote spatial and temporal step-sizes, and 1 ≤  m  ≤ 5 is an integer which is set by an ODE integrator that we used. The convergence and numerical stability of the method are considered, and numerical tests to investigate the efficiency and feasibility of the scheme are provided. [ABSTRACT FROM AUTHOR]

In this paper, the approximate analytic solution of any order space-time fractional differential equations is constructed by means of semi-analytical method, named as residual power series method (RPSM). The first step is to reduce space-time fractional differential equation to either a space fractional differential equations or a time fractional differential equations before applying RSPM. The main step is to obtain fractional power series solutions by RSPM. At the final step, it is shown that RPSM is very efficacious, plain and powerful for obtaining the solution of any-order space-time fractional differential equations in the form of fractional power series by illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, a discrete form is proposed for solving time fractional convection–diffusion equation. Firstly, we obtain a time discrete scheme based on finite difference method. Secondly, we prove that the time discrete scheme is unconditionally stable, and the numerical solution converges to the exact one with order O ( τ 2 − α ) , where τ is the time step size. Finally, two numerical examples are proposed respectively, to verify the order of convergence. [ABSTRACT FROM AUTHOR]

The main aim of the current paper is to propose an efficient numerical technique for solving two-dimensional space-multi-time fractional Bloch–Torrey equations. The current research work is a generalization of [6] . The temporal direction is based on the Caputo fractional derivative with multi-order fractional derivative and the spatial directions are based on the Riemann–Liouville fractional derivative. Thus, to achieve a numerical technique, the time variable is discretized using a finite difference scheme with convergence order O ( τ 2 − α ) . Also, the space variable is discretized using the finite element method. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, four test problems have been illustrated to verify the efficiency and simplicity of the proposed technique on irregular computational domains. [ABSTRACT FROM AUTHOR]

This paper investigates robust finite-time stability problem of fractional-order systems (FOSs) with time-varying delay and nonlinear perturbation. Based on a generalized Gronwall inequality, a delay-dependent sufficient condition for robust finite-time stability of such systems is established in terms of the Mittag-Leffler function. Compared with the finite-time stability conditions in the literature, our condition is simple and convenient for testing and application. The obtained results are applied to finite-time stability of linear uncertain FOSs and linear non-autonomous FOSs with time-varying delays. A numerical example is given to show the effectiveness of the obtained result. [ABSTRACT FROM AUTHOR]

In this paper, we present a hybridizable discontinuous Galerkin (HDG) method for solving a class of fractional boundary value problems involving Caputo derivatives. The HDG methods have the computational advantage of eliminating all internal degrees of freedom and the only globally coupled unknowns are those at the element interfaces. Furthermore, the global stiffness matrix is tridiagonal, symmetric, and positive definite. Internal degrees of freedom are recovered at an element-by-element postprocessing step. We carry out a series of numerical experiments to ascertain the performance of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a spatial fractional-order thermal transport equation with the Caputo derivative is proposed to describe convective heat transfer of nanofluids within disordered porous media in boundary layer flow. This equation arises naturally when the effect of anomalous migration of nanoparticles on heat transfer is considered. The numerical results show that local Nusselt numbers of four different kinds of nanofluids are all inversely proportional to the fractional derivative exponent β . Based on this finding, it is concluded that the anomalous diffusion of nanoparticles improves the convective heat transfer of nanofluids and the space fractional thermal transport equation may serve as a candidate model for studying nanofluids. Additionally, the effects of other involved physical parameters on temperature distribution and Nusselt number are presented and analyzed. [ABSTRACT FROM AUTHOR]

A simple and inexpensive preprocessing of an initial–boundary value problem with a Caputo time derivative is shown theoretically and numerically to yield an enhanced convergence rate for the L1 scheme. The same preprocessing can also be used with other methods for time-dependent problems. [ABSTRACT FROM AUTHOR]

In this paper, the semi-analytic numerical technique for the solution of time-space fractional telegraph equation is applied. This numerical technique is based on coupling of the homotopy analysis method and sumudu transform. It shows the clear advantage with mess methods like finite difference method and also with polynomial methods similar to perturbation and Adomian decomposition methods. It is easily transform the complex fractional order derivatives in simple time domain and interpret the results in same meaning. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a new numerical scheme to solve the time fractional nonlinear Klein–Gordon equation. The fractional derivative is described in the Caputo sense. The method consists of expanding the required approximate solution as the elements of Sinc functions along the space direction and shifted Chebyshev polynomials of the second kind for the time variable. The proposed scheme reduces the solution of the main problem to the solution of a system of nonlinear algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces accurate results. [ABSTRACT FROM AUTHOR]

Fractional calculus has been found to be a promising area of research for information processing and modeling of some physical systems. In this paper, we propose a fractional gradient descent method for the backpropagation (BP) training of neural networks. In particular, the Caputo derivative is employed to evaluate the fractional-order gradient of the error defined as the traditional quadratic energy function. The monotonicity and weak (strong) convergence of the proposed approach are proved in detail. Two simulations have been implemented to illustrate the performance of presented fractional-order BP algorithm on three small datasets and one large dataset. The numerical simulations effectively verify the theoretical observations of this paper as well. [ABSTRACT FROM AUTHOR]

In this paper, we derive a high order numerical scheme for variable order fractional ordinary differential equation by establishing a second order numerical approximation to variable order Riemann–Liouville fractional derivative. The scheme is strictly proved to be stable and convergent with second order accuracy, which is higher than some recently derived schemes. Finally, some numerical examples are presented to demonstrate the theoretical analysis and verify the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The paper presents a new numerical method for solving eigenvalue problems for fractional differential equations. It combines two techniques: the method of external excitation (MEE) and the backward substitution method (BSM). The first one is a mathematical model of physical measurements when a mechanical, electrical or acoustic system is excited by some source and resonant frequencies can be determined by using the growth of the amplitude of oscillations near these frequencies. The BSM consists of replacing the original equation by an approximate equation which has an exact analytic solution with a set of free parameters. These free parameters are determined by the use of the collocation procedure. Some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results. The numerical results show that the proposed method is of a high accuracy and is efficient for solving of a wide class of eigenvalue problems. [ABSTRACT FROM AUTHOR]

In this paper, based on the magnetohydrodynamics approach, the blood flow along with magnetic particles through a circular cylinder is studied. The fluid is acted by an oscillating pressure gradient and an external magnetic field. The study is based on a mathematical model with Caputo fractional derivatives. The model of ordinary fluid, corresponding to time-derivatives of integer order, is obtained as a particular case. Closed forms of the fluid velocity and magnetic particles velocity are obtained by means of the Laplace and finite Hankel transforms. Effects of the order of Caputo's time-fractional derivatives and of the external magnetic field on flow parameters of both blood and magnetic particles are studied. Numerical simulations and graphical illustrations are used in order to study the influence of the fractional parameter α , Reynolds number and Hartmann number on the fluid and particles velocity. The results highlights that, models with fractional derivatives bring significant differences compared to the ordinary model. This fact can be an important advantage for some practical problems. It also results that the blood velocity, as well as that of magnetic particles, is reduced under influence of the exterior magnetic field. [ABSTRACT FROM AUTHOR]

In this paper, homotopy perturbation method (HPM) is applied to solve fractional partial differential equations (PDEs) with proportional delay in t and shrinking in x . The method do not require linearization or small perturbation. The fractional derivatives are taken in the Caputo sense. The present method performs extremely well in terms of efficiency and simplicity. Numerical results for different particular cases of α are presented graphically. [ABSTRACT FROM AUTHOR]

In the present paper, a wavelet method based on the Hermite wavelet expansion along with operational matrices of fractional derivative and integration is proposed for finding the numerical solution to a coupled system of nonlinear time-fractional Jaulent–Miodek (JM) equations. Consequently, the approximate solutions of fractional Jaulent–Miodek equations acquired by using Hermite wavelet technique were compared with those derived by using optimal homotopy asymptotic method (OHAM) and exact solutions. The present proposed numerical technique is easy, expedient and powerful in computing the numerical solution of coupled system of nonlinear fractional differential equations like Jaulent–Miodek equations. [ABSTRACT FROM AUTHOR]

In this paper, a new wavelet method based on the Hermite wavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is proposed to solve time-fractional modified Fornberg–Whitham (mFW) equation. The approximate solutions of time fractional modified Fornberg–Whitham equation which are obtained by Hermite wavelet method are compared with the exact solutions as well as the solutions obtained by optimal homotopy asymptotic method (OHAM). The present numerical scheme is quite simple, effective and expedient for obtaining numerical solution of fractional modified Fornberg–Whitham equation. [ABSTRACT FROM AUTHOR]

This paper presents a multi-domain Legendre–Gauss pseudospectral method for approximate solutions of the fractional Volterra’s model for population growth of a species in a closed system. Fractional Volterra’s model is a fractional Volterra integro-differential equation. The fractional derivative is considered in the Caputo sense. In this method, the fractional integro-differential equation is first replaced with a singular Volterra integro-differential equation (SVIDE). Then, by choosing a step-size, the replaced problem is converted into a sequence of SVIDEs in subintervals. The difficulty in SVIDEs due to singularity is overcome here by utilizing integration by parts. The obtained problems in subintervals are then step by step reduced to systems of algebraic equations using collocation. We give some numerical applications to show validity and high accuracy of the proposed technique. [ABSTRACT FROM AUTHOR]

This paper concerns the abstract Cauchy problem of fractional evolution equations with almost sectorial operators. The suitable mild solutions of evolution equations with Riemann–Liouville derivative and Caputo derivative are introduced respectively. The existence theorems of mild solutions of Cauchy problems are established. The results obtained here improve and generalize some known results. Finally, an example is given for demonstration. [ABSTRACT FROM AUTHOR]

In this paper we propose a nonpolynomial collocation method for solving a class of terminal (or boundary) value problems for differential equations of fractional order α , 0 < α < 1 . The approach used is based on the equivalence between a problem of this type and a Fredholm integral equation of a particular form. Taking into account the asymptotic behaviour of the solution of this problem, we propose a nonpolynomial collocation method on a uniform mesh. We study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained. In order to illustrate the theoretical results and the performance of the method we present several numerical examples. [ABSTRACT FROM AUTHOR]

We consider a class of boundary value problems for nonlinear fractional differential equations involving Caputo-type fractional derivatives. Using an integral equation reformulation of the boundary value problem, some regularity properties of the exact solution are derived. Based on these properties and spline collocation techniques, the numerical solution of boundary value problems by suitable non-polynomial approximations is discussed. Optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given. Theoretical results are tested by two numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a comparison for the solutions of nonlinear Swift–Hohenberg equation with time-space fractional derivatives has been analyzed. The two most promising techniques, fractional variational iteration method (FVIM) and the homotopy analysis method have been chosen for the comparison. The two different definitions of fractional calculus are considered to solve time-fractional derivative separately for the considered approaches. Also, the space fractional derivative is described in the Reisz sense. Analytical and numerical solutions for various combinations of the parameters are obtained. Numerical comparisons have been made for different values of parameters and depicted. [Copyright &y& Elsevier]

Abstract: In this paper, we establish a local stable manifold theorem near a hyperbolic equilibrium point for planar fractional differential equations. The construction of this stable manifold is based on the associated Lyapunov–Perron operator. An example is provided to illustrate the result. [Copyright &y& Elsevier]

Abstract: In this paper, we applied relatively new analytical techniques, the homotopy analysis method (HAM) and the Adomian’s decomposition method (ADM) for solving time-fractional Fornberg–Whitham equation. The homotopy analysis method contains the auxiliary parameter, which provides us with a simple way to adjust and control the convergence region of solution series. The fractional derivatives are described in the Caputo sense. A comparison is made the between HAM and ADM results. The present methods performs extremely well in terms of efficiency and simplicity. Numerical results for different particular cases of the problem are presented. [Copyright &y& Elsevier]

• PV power intra-hour forecasting is dealt with. • A novel method is proposed based on fractional calculus, more specifically the Caputo derivative has been used. • The use of Caputo derivative allows considering information on the trend of the forecasting variable. • The prediction tool allows to provide a closed form for the predictor which is particularly suitable for online applications. • A sensitivity analysis shows that the inclusion of information on the trend of the forecasting variable in recent past could lead to accurate results. Intra-hour photovoltaic power forecasting provides essential information for real time optimal control of microgrids. At this purpose, a critical issue is the selection of the forecasting method. The choice of a forecasting method depends on many factors such as the availability of historical data, the time horizon, the lag period, and the time available for the forecast. Persistence based methods are particularly tailored for real time forecasting which require fast information and are typically a good trade-off choice when dealing with real time operation of microgrids. Their accuracy, however, could be not satisfactory in some cases such as when it appears critical to consider the trend of the power output in the last few minutes rather than only the last measured value. Derivatives help reach this goal, but fractional derivatives seem to be a more accurate choice in order to take into account the history of the variable to be forecasted as they are a promising tool for describing memory phenomena. In this paper, a novel intra-hour forecasting method is proposed based on the Caputo derivative. Numerical applications are carried out to show the efficacy of the proposed approach. Also, the accuracy of the proposed approach is tested through comparison with three models namely, persistence, derivative-persistence and auto regressive moving average models. The strength of the proposed forecasting tool is strictly related to its low computational burden without compromising accuracy. This makes of it an interesting means for real time grid operation strategies and can be of interest for the grid operators especially in vision of the changes distribution grids are witnessing with the transition to the smart grid paradigm. [ABSTRACT FROM AUTHOR]

• The time-fractional telegraph equation with moving time-harmonic source is considered on a real line. • Two characteristic versions of this equation: the "wave-type" with the second and Caputo fractional time-derivatives as well as the "heat-type" with the first and Caputo fractional time-derivatives are investigated. • The solution to the "wave-type" equation contains wave fronts and describes the Doppler effect contrary to the solution for the "heat-type" equation. • For the time-fractional telegraph equation it is impossible to consider the quasi-steady-state corresponding to the solution being a product of a function of the spatial coordinate and the time-harmonic term. • The derived solutions can be successfully used when the source term can be expanded into a Fourier series. The time-fractional telegraph equation with moving time-harmonic source is considered on a real line. We investigate two characteristic versions of this equation: the "wave-type" with the second and Caputo fractional time-derivatives as well as the "heat-type" with the first and Caputo fractional time-derivatives. In both cases the order of fractional derivative 1 < α < 2. For the time-fractional telegraph equation it is impossible to consider the quasi-steady-state corresponding to the solution being a product of a function of the spatial coordinate and the time-harmonic term. The considered problem is solved using the integral transforms technique. The solution to the "wave-type" equation contains wave fronts and describes the Doppler effect contrary to the solution for the "heat-type" equation. Numerical results are illustrated graphically for different values of nondimensional parameters. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the asymptotics of the solutions to the anomalous diffusion equations. The fractional anomalous diffusion equations are obtained from the existing anomalous diffusion and typical diffusion equations by replacing the first-order time derivative with fractional derivatives of order for the sub-diffusion and for the super-diffusion respectively. In most situations, fractional derivatives mean Riemann–Liouville derivative or Caputo derivative. In this paper, we use these two kinds of fractional derivatives. Using Laplace transform and Fourier transform, we obtain the asymptotics estimates of solutions to the anomalous diffusion equations. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to investigate the collocation method based on Multiquadric (MQ) radial basis functions (RBFs) for fractional differential equations (FDEs). In the process of doing this, a scattered data RBF interpolation is considered. A posterior error bound for the smooth solutions of FDEs in the proposed method is established in the norm, using an alternate characterization of the native space. Numerical results are presented, which confirm the theoretical prediction of the convergence behavior of the proposed method. [Copyright &y& Elsevier]

Abstract: The flow through porous media can be better described by fractional models than the classical ones since they include inherently memory effects caused by obstacles in the structures. The variational iteration method was extended to find approximate solutions of fractional differential equations with the Caputo derivatives, but the Lagrange multipliers of the method were not identified explicitly. In this paper, the Lagrange multiplier is determined in a more accurate way and some new variational iteration formulae are presented. [Copyright &y& Elsevier]

Abstract: In this paper, a new formula of Caputo fractional-order derivatives of shifted Jacobi polynomials of any degree in terms of shifted Jacobi polynomials themselves is proved. We discuss a direct solution technique for linear multi-order fractional differential equations (FDEs) subject to nonhomogeneous initial conditions using a shifted Jacobi tau approximation. A quadrature shifted Jacobi tau (Q-SJT) approximation is introduced for the solution of linear multi-order FDEs with variable coefficients. We also propose a shifted Jacobi collocation technique for solving nonlinear multi-order fractional initial value problems. The advantages of using the proposed techniques are discussed and we compare them with other existing methods. We investigate some illustrative examples of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques. [Copyright &y& Elsevier]

Abstract: In this article, Haar wavelets have been employed to obtain solutions of boundary value problems for linear fractional partial differential equations. The differential equations are reduced to Sylvester matrix equations. The algorithm is novel in the sense that it effectively incorporates the aperiodic boundary conditions. Several examples with numerical simulations are provided to illustrate the simplicity and effectiveness of the method. [Copyright &y& Elsevier]

Abstract: The aim of present paper is to obtain the approximate analytical solution of time fractional Harry Dym equation by using homotopy perturbation method (HPM). The beauty of the paper is error analysis which shows that our approximate solution converges very rapidly to the exact solution and the numerical solution is compared with the known analytical solution which is nearly identical with the exact solution. The results show that the solution of HPM is good agreement with the exact solution. The fractional derivatives are described in the Caputo sense. The results reveal that the method is very effective and simple. [Copyright &y& Elsevier]

Abstract: In this article, the approximate solutions of the non-linear Swift Hohenberg equation with fractional time derivative in the presence of dispersive term have been obtained. The fractional derivative is described in Caputo sense. Time fractional nonlinear partial differential equations in the presence of dispersion and bifurcation parameters have been computed numerically to predict hydrodynamic fluctuations at convective instability for different particular cases and results are depicted through graphs. [Copyright &y& Elsevier]

Abstract: In this paper, we study the numerical solution of the time–space fractional order (fractional for simplicity) telegraph equation, which can be used in signal analysis for transmission and propagation of electrical signals, also the modeling of the reaction diffusion and the random walk of suspension flows and so on. The semi-discrete and fully discrete numerical approximations are both analyzed, where the Galerkin finite element method for the spatial Riemann–Liouville fractional derivative with order and the finite difference schemes for the temporal Caputo derivatives with orders and are analyzed respectively. Results on the existence and uniqueness of the solution, the numerical stability, and the error estimates are displayed in details. Numerical examples are included to confirm the theoretical analysis. [Copyright &y& Elsevier]

Abstract: In this paper, the existence of positive solutions for the nonlinear Caputo fractional functional differential equation in the form is studied. By constructing a special cone and using Krasnosel’skii’s fixed point theorem, various results on the existence of at least one or two positive solutions to the fractional functional differential equation are established. The main results improve and generalize the existing results. [Copyright &y& Elsevier]

Abstract: This paper discusses the existence of solutions of the fractional differential equations satisfying the boundary conditions . Here is the Caputo fractional derivative, ), F is a continuous operator, are bounded and continuous functionals and . The existence results are proved by the Leray–Schauder degree theory. [Copyright &y& Elsevier]

Abstract: In this paper, we derived the shifted Jacobi operational matrix (JOM) of fractional derivatives which is applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs). A new approach implementing shifted Jacobi operational matrix in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The proposed methods are applied for solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: We consider a class of boundary value problems for linear multi-term fractional differential equations which involve Caputo-type fractional derivatives. Using an integral equation reformulation of the boundary value problem, some regularity properties of the exact solution are derived. Based on these properties, the numerical solution of boundary value problems by piecewise polynomial collocation methods is discussed. In particular, we study the attainable order of convergence of proposed algorithms and show how the convergence rate depends on the choice of the grid and collocation points. Theoretical results are verified by two numerical examples. [Copyright &y& Elsevier]

Abstract: We focus on a numerical scheme applied for a fractional oscillator equation in a finite time interval. This type of equation includes a complex form of left- and right-sided fractional derivatives. Its analytical solution is represented by a series of left and right fractional integrals and therefore is difficult in practical calculations. Here we elaborated two numerical schemes being dependent on a fractional order of the equation. The results of numerical calculations are compared with analytical solutions. Then we illustrate convergence and stability of our schemes. [Copyright &y& Elsevier]