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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]

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]

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]

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 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]

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, 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, 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, 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: 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: 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]

Abstract: The object of this article is to present the solution of a fractional generalization of the Schrödinger equation of quantum mechanics in one dimension. The method followed in deriving the solution is that of joint Laplace and Fourier transforms. The solution is obtained in a compact and computational form in terms of the H-function. A result given earlier by Debnath for the solution of a generalized Schrödinger equation is obtained in an explicit form in terms of the H-function, as a special case of our findings. [Copyright &y& Elsevier]