In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

Numerical methods for solving the multi-term time-fractional wave-diffusion equation

Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation

Multiphoton fluorescence photobleaching recovery (MP-FPR) is a technique for measuring the three-dimensional (3D) mobility of fluorescent molecules with 3D spatial resolution of a few microns. A brief, intense flash of mode-locked laser light pulses excites fluorescent molecules via multiphoton excitation in an ellipsoidal focal volume and photobleaches a fraction. Because multiphoton excitation of fluorophores is intrinsically confined to the high-intensity focal volume of the illuminating beam, the bleached region is restricted to a known, three-dimensionally defined volume. Fluorescence in this focal volume is measured with multiphoton excitation, using the attenuated laser beam to measure fluorescence recovery as fresh unbleached dye diffuses in. The time course of the fluorescence recovery signal after photobleaching can be analyzed to determine the diffusion coefficient of the fluorescent species. The mathematical formulas used to fit MP-FPR recovery curves and the techniques needed to properly utilize them to acquire the diffusion coefficients of fluorescently labeled molecules within cells are presented here. MP-FPR is demonstrated on calcein in RBL-2H3 cells, using an anomalous subdiffusion model, as well as in aqueous solutions of wild-type green fluorescent protein, yielding a diffusion coefficient of 8.7 x 10(-7) cm(2)s(-1) in excellent agreement with the results of other techniques.

Measurement of molecular diffusion in solution by multiphoton fluorescence photobleaching recovery

A Crank-Nicolson-type difference scheme is proposed for solving the subdiffusion equation with fractional derivative, and the truncation error is analyzed in detail. At each temporal level, only a tridiagonal linear system needs to be solved and the Thomas algorithm may be used. The solvability, unconditional stability, and $H^1$ norm convergence are proved. The convergence order is ${\rm min}\{2-{\gamma}/{2},\;1+\gamma\}$ in the temporal direction and two in the spatial direction. By the Sobolev embedding inequality, we obtain the maximum norm error estimate. A spatial compact scheme based on the Crank-Nicolson-type difference scheme is also presented, and similar results are given. The convergence order is $\mathcal{O}(\tau^{{\rm min}\{2-{\gamma}/{2},\;1+\gamma\}}+h^4)$. Numerical experiments are included to support the theoretical results, and comparisons with the related works are presented to show the effectiveness of our method.

Error Estimates of Crank-Nicolson-Type Difference Schemes for the Subdiffusion Equation

This paper aims to develop an implicit meshless approach based on the radial basis function (RBF) for numerical simulation of time fractional diffusion equations. The meshless RBF interpolation is firstly briefed. The discrete equations for two-dimensional time fractional diffusion equation (FDE) are obtained by using the meshless RBF shape functions and the strong-forms of the time FDE. The stability and convergence of this meshless approach are discussed and theoretically proven. Numerical examples with different problem domains and different nodal distributions are studied to validate and investigate accuracy and efficiency of the newly developed meshless approach. It has proven that the present meshless formulation is very effective for modeling and simulation of fractional differential equations.

An implicit RBF meshless approach for time fractional diffusion equations

Recently, the numerical modelling and simulation for anomalous subdiffusion equation (ASDE), which is a type of fractional partial differential equation( FPDE) and has been found with widely applications in modern engineering and sciences, are attracting more and more attentions. The current dominant numerical method for modelling ASDE is Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings. This paper aims to develop an implicit meshless approach based on the radial basis functions (RBF) for numerical simulation of the non-linear ASDE. The discrete system of equations is obtained by using the meshless shape functions and the strong-forms. The stability and convergence of this meshless approach are then discussed and theoretically proven. Several numerical examples with different problem domains are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. The results obtained by the meshless formulations are also compared with those obtained by FDM in terms of their accuracy and efficiency. It is concluded that the present meshless formulation is very effective for the modeling and simulation of the ASDE. Therefore, the meshless technique should have good potential in development of a robust simulation tool for problems in engineering and science which are governed by the various types of fractional differential equations.

An advanced implicit meshless approach for thenon-linear anomalous subdiffusion equation

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An advanced implicit meshless approach for thenon-linear anomalous subdiffusion equation