26 research outputs found
The finite difference methods for fractional ordinary differential equations
No full textFractional finite difference methods are useful to solve the fractional differential equations. The aim of this article is to prove the stability and convergence of the fractional Euler method, the fractional Adams method and the high order methods based on the convolution formula by using the generalized discrete Gronwall inequality. Numerical experiments are also presented, which verify the theoretical analysis
Numerical simulation of the fractional Langevin equation
No full textIn this paper, we study the fractional Langevin equation, whose derivative is in Caputo sense. By using the derived numerical algorithm, we obtain the displacement and the mean square displacement which describe the dynamic behaviors of the fractional Langevin equation
A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations
No full textWe generalize existing Jacobi--Gauss--Lobatto collocation methods for variable-order fractional differential equations using a singular approximation basis in terms of weighted Jacobi polynomials of the form (1 ± <i>ϰ</i>)<i><sup>μ</sup>P<sub>j</sub><sup>a,b</sup></i>(<i>ϰ</i>), where <i>μ</i> > -1. In order to derive the differentiation matrices of the variable-order fractional derivatives, we develop a three-term recurrence relation for both integrals and derivatives of these weighted Jacobi polynomials, hence extending the three-term recurrence relationship of Jacobi polynomials. The new spectral collocation method is applied to solve fractional ordinary and partial differential equations with endpoint singularities. We demonstrate that the singular basis enhances greatly the accuracy of the numerical solution by properly tuning the parameter , even for cases where we do not know explicitly the form of singularity in the solution at the boundaries
Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy
No full textThis article aims to fill in the gap of the second-order accurate schemes for the time-fractional subdiffusion equation with unconditional stability. Two fully discrete schemes are first proposed for the time-fractional subdiffusion equation with space discretized by finite element method and time discretized by the fractional linear multistep methods. These two methods are unconditionally stable with maximum global convergence order of <b>O</b> ( <i>τ</i> + h <sup>r +1</sup> )in the L 2 norm, where τ and h are the step sizes in time and space, respectively, and r is the degree of the piecewise polynomial space. The average convergence rates for the two methods in time are also investigated, which shows that the average convergence rates of the two methods are <b>O</b> ( <i>τ</i> 1 . 5 + h <sup>r +1</sup> ). Furthermore, two improved algorithms are constructed, and they are also unconditionally stable and convergent of order <b>O</b> ( <i>τ</i> 2 + h <sup>r +1</sup> ). Numerical examples are provided to verify the theoretical analysis. Comparisons between the present algorithms and the existing ones are included, showing that our numerical algorithms exhibit better performances than the known ones
Implicit-explicit difference schemes for nonlinear fractional differential equations with nonsmooth solutions
No full textWe propose second-order implicit-explicit (IMEX) time-stepping schemes for nonlinear fractional differential equations with fractional order 0 < <i>β</i> <1. From the known structure of the nonsmooth solution and by introducing corresponding correction terms, we can obtain uniformly second-order accuracy from these schemes. We prove the convergence and linear stability of the proposed schemes. Numerical examples illustrate the flexibility and efficiency of the IMEX schemes and show that they are effective for nonlinear and multirate fractional differential systems as well as multiterm fractional differential systems with nonsmooth solutions
The use of finite difference/element approaches for solving the time-fractional subdiffusion equation
No full textSpectral approximations to the fractional integral and derivative
No full textIn this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value\ud
problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods
A new Crank–Nicolson finite element method for the time-fractional subdiffusion equation
No full textNumerical analysis of linear and nonlinear time-fractional subdiffusion equations
No full textIn this paper, a new type of the discrete fractional Gr{ö}nwall inequality is developed, which is applied to analyze the stability and convergence of a Galerkin spectral method for a linear time-fractional subdiffusion equation. Based on the temporal-spatial error splitting argument technique, the discrete fractional Gr{ö}nwall inequality is also applied to prove the unconditional convergence of a semi-implicit Galerkin spectral method for a nonlinear time-fractional subdiffusion equation.16 page
