Discretized fractional calculus
SIAM Journal on Mathematical Analysis
An approximate method for numerical solution of fractional differential equations
Signal Processing - Fractional calculus applications in signals and systems
Finite difference/spectral approximations for the time-fractional diffusion equation
Journal of Computational Physics
Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering
Pitfalls in fast numerical solvers for fractional differential equations
Journal of Computational and Applied Mathematics
Some noises with spectrum, a bridge between direct current and white noise
IEEE Transactions on Information Theory
Numerical treatment for solving the perturbed fractional PDEs using hybrid techniques
Journal of Computational Physics
Exponentially accurate spectral and spectral element methods for fractional ODEs
Journal of Computational Physics
Journal of Computational Physics
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In this paper we present a general technique to construct high order schemes for the numerical solution of the fractional ordinary differential equations (FODEs). This technique is based on the so-called block-by-block approach, which is a common method for the integral equations. In our approach, the classical block-by-block approach is improved in order to avoiding the coupling of the unknown solutions at each block step with an exception in the first two steps, while preserving the good stability property of the block-by-block schemes. By using this new approach, we are able to construct a high order schema for FODEs of the order @a,@a0. The stability and convergence of the schema is rigorously established. We prove that the numerical solution converges to the exact solution with order 3+@a for 01. A series of numerical examples are provided to support the theoretical claims.