Xiaoyun Jiang

Exact solutions of fractional Schrödinger-like equation with a nonlocal term

We study the time-space fractional Schrodinger equation with a nonlocal potential. By the method of Fourier transform and Laplace transform, the Green function, and hence the wave function, is expressed in terms of H-functions. Graphical analysis demonstrates that the influence of both the space-fractal parameter α and the nonlocal parameter ν on the fractional quantum system is strong. Indeed, the nonlocal potential may act similar to a fractional spatial derivative as well as fractional time derivative.

Parameter estimation for the fractional fractal diffusion model based on its numerical solution

In this paper, we mainly consider a problem of parameter estimation for the fractional fractal diffusion model used to describe the anomalous diffusion in porous media. The Bayesian method is proposed to estimate parameters for the fractional fractal diffusion model based on its numerical solution. Firstly, the center Box difference method is employed to solve the fractional fractal diffusion equation subject to the initial-boundary conditions. Then we apply the Bayesian method to estimate three parameters for the model simultaneously, that is the fractional order α , the fractal dimension d f and the structure parameter ? . To testify the validity of the results for the models parameter estimation, experimental data from the fast desorption process of methane in coal are used. It is shown that the numerical results modeled with parameters given by the Bayesian method coincide well with the desorption experimental data, indicating that the Bayesian method is efficient and valid in identifying multi-parameter for the fractional fractal diffusion model. Besides, the fractional fractal diffusion model is demonstrated to be more capable than the classical Ficks model to describe the anomalous diffusion behavior of gas in coal. To clarify the parameters effect on the relative diffusion amount of methane for the fractional fractal diffusion model, parameter sensitivity analysis is also carried out. This paper provides a specific and efficient parameter estimation method for the fractional fractal diffusion model.

Creep constitutive models for viscoelastic materials based on fractional derivatives

To describe the time-dependent creep behavior of viscoelastic material, fractional constitutive relation models which are represented by the fractional element networks are studied. Three sets of creep experimental data for polymer and rock are employed to demonstrate the effectiveness of these fractional derivative models. The corresponding constrained problem of nonlinear optimization is solved with an interior-point algorithm to obtain best fitting parameters of these fractional derivative models. The comparison results of measured values and calculated values versus time are displayed through graphics. The results demonstrate that the fractional PoyntingThomson model is optimal in simulating the creep behavior of viscoelastic materials. And it also shows that the interior-point method is effective in the inverse problem to estimate parameters of fractional viscoelastic models.

Parameters estimation for a new anomalous thermal diffusion model in layered media

In this paper, we study an inverse problem of parameters estimation for a new time-fractional heat conduction model in multilayered medium. In the anomalous thermal diffusion model, we consider the fractional derivative boundary conditions and the conduction obeys modified Fourier law with RiemannLiouville fractional operator of different order in each layer. For the direct problem, we construct an effective finite difference scheme by using the balance method to deal with the discontinuity interface. For the inverse problem, we apply the nonlinear conjugate gradient (NCG) method with different conjugated coefficients to simultaneously identify the fractional exponent in each layer. Finally, we use experimental data to verify the effectiveness of the proposed technique, in which the Jacobian matrix is achieved by a derivative-free approach. We analyze the sensitivity coefficients and the convergence behaviors of the NCG algorithm. The simulation results confirm that the fractional heat conduction model with estimated parameters gives a more accurate fitting than the classical counterpart and the NCG method is a feasible and effective technique for the inverse problem of parameters estimation in fractional model.

Parameter estimation for the fractional Schrödinger equation using Bayesian method

In this paper, the fractional Schrodinger equation is studied. The Bayesian method is put forward to estimate some relevant parameters of the equation. Results show that the estimated values can fit well with the exact solution. The varying initial values and maximum iterations have little effect on the estimated results. It indicates that the Bayesian method is efficient for the multi-parameter estimation for the fractional Schrodinger equation. This method can also be used to estimate parameters for the fractional Schrodinger equation in other potential field.

Galerkin-Legendre spectral schemes for nonlinear space fractional Schrödinger equation

In the paper, we first propose a Crank-Nicolson Galerkin-Legendre (CN-GL) spectral scheme for the one-dimensional nonlinear space fractional Schrödinger equation. Convergence with spectral accuracy is proved for the spectral approximation. Further, a Crank-Nicolson ADI Galerkin-Legendre spectral method for the two-dimensional nonlinear space fractional Schrödinger equation is developed. The proposed schemes are shown to be efficient with second-order accuracy in time and spectral accuracy in space which are higher than some recently studied methods. Moreover, some numerical results are demonstrated to justify the theoretical analysis.

Crank–Nicolson Fourier spectral methods for the space fractional nonlinear Schrödinger equation and its parameter estimation

ABSTRACT In this paper, the Crank–Nicolson Fourier spectral approximations for solving the space fractional nonlinear Schrödinger equation are proposed. Firstly, the numerical formats of the Crank–Nicolson Fourier Galerkin and Fourier collocation methods are established. The fast Fourier transform technique is applied to practical computation. Secondly, Convergence with spectral accuracy in space and second-order accuracy in time is verified for both Galerkin and collocation approximations. Moreover, a rigorous analysis of the conservation for the Crank–Nicolson Fourier Galerkin fully discrete system is derived. Thirdly, the Bayesian method is presented to estimate the fractional derivative order and the coefficient of nonlinear term based on the spectral format of the direct problem. Finally, some numerical examples are given to confirm the theoretical analysis.

A Crank–Nicolson ADI Galerkin–Legendre spectral method for the two-dimensional Riesz space distributed-order advection–diffusion equation

Abstract In the paper, a Crank–Nicolson alternating direction implicit (ADI) Galerkin–Legendre spectral scheme is presented for the two-dimensional Riesz space distributed-order advection–diffusion equation. The Gauss quadrature has a higher computational accuracy than the mid-point quadrature rule, which is proposed to approximate the distributed order Riesz space derivative so that the considered equation is transformed into a multi-term fractional equation. Moreover, the transformed equation is solved by discretizing in space by the ADI Galerkin–Legendre spectral scheme and in time using the Crank–Nicolson difference method. Stability and convergence analysis are verified for the numerical approximation. A lot of numerical results are demonstrated to justify the theoretical analysis.

Spectral method for solving the time fractional Boussinesq equation

Abstract In this paper, Fourier spectral approximation for the time fractional Boussinesq equation with periodic boundary condition is considered. The space is discretized by the Fourier spectral method and the Crank–Nicolson scheme is used to discretize the Caputo time fractional derivative. Stability and convergence analysis of the numerical method are proven. Some numerical examples are included to testify the effectiveness of our given method. Based on the presented numerical results, the Fourier spectral method is shown to be effective for solving the time fractional Boussinesq equation.

L1 Fourier spectral methods for a class of generalized two-dimensional time fractional nonlinear anomalous diffusion equations

Abstract In this paper, L1 Fourier spectral methods are derived to obtain the numerical solutions for a class of generalized two-dimensional time-fractional nonlinear anomalous diffusion equations involving Caputo fractional derivative. Firstly, we establish the L1 Fourier Galerkin full discrete and L1 Fourier collocation schemes with Fourier spectral discretization in spatial direction and L1 difference method in temporal direction. Secondly, stability and convergence for both Galerkin and collocation approximations are proved. It is shown that the proposed methods are convergent with spectral accuracy in space and ( 2 − α ) order accuracy in time. For implementation, the equivalence between pseudospectral method and collocation method is discussed. Furthermore, a numerical algorithm of Fourier pseudospectral method is developed based on two-dimensional fast Fourier transform (FFT2) technique. Finally, numerical examples are provided to test the theoretical claims. As is shown in the numerical experiments, Fourier spectral methods are powerful enough with excellent efficiency and accuracy.