The accuracy and stability of an implicit solution method for the fractional diffusion equation
Journal of Computational Physics
Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies)
Weighted average finite difference methods for fractional diffusion equations
Journal of Computational Physics
Finite difference/spectral approximations for the time-fractional diffusion equation
Journal of Computational Physics
Compact finite difference method for the fractional diffusion equation
Journal of Computational Physics
A fully discrete difference scheme for a diffusion-wave system
Applied Numerical Mathematics
Fox H functions in fractional diffusion
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the seventh international symposium on Orthogonal polynomials, special functions and applications
A compact finite difference scheme for the fractional sub-diffusion equations
Journal of Computational Physics
Numerical method for solving diffusion-wave phenomena
Journal of Computational and Applied Mathematics
Numerical approaches to fractional calculus and fractional ordinary differential equation
Journal of Computational Physics
A characteristic difference method for the transient fractional convection-diffusion equations
Applied Numerical Mathematics
Numerical analysis and physical simulations for the time fractional radial diffusion equation
Computers & Mathematics with Applications
Compact alternating direction implicit method for two-dimensional time fractional diffusion equation
Journal of Computational Physics
Journal of Computational Physics
A high order schema for the numerical solution of the fractional ordinary differential equations
Journal of Computational Physics
Journal of Computational Physics
Exponentially accurate spectral and spectral element methods for fractional ODEs
Journal of Computational Physics
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In the present work, first, a new fractional numerical differentiation formula (called the L1-2 formula) to approximate the Caputo fractional derivative of order @a (0=2), while the linear interpolation approximation is applied on the first small interval [t"0,t"1]. As a result, the new formula can be formally viewed as a modification of the classical L1 formula, which is obtained by the piecewise linear approximation for f(t). Both the computational efficiency and numerical accuracy of the new formula are superior to that of the L1 formula. The coefficients and truncation errors of this formula are discussed in detail. Two test examples show the numerical accuracy of L1-2 formula. Second, by the new formula, two improved finite difference schemes with high order accuracy in time for solving the time-fractional sub-diffusion equations on a bounded spatial domain and on an unbounded spatial domain are constructed, respectively. In addition, the application of the new formula into solving fractional ordinary differential equations is also presented. Several numerical examples are computed. The comparison with the corresponding results of finite difference methods by the L1 formula demonstrates that the new L1-2 formula is much more effective and more accurate than the L1 formula when solving time-fractional differential equations numerically.