Algorithms for the fractional calculus: A selection of numerical methods

  title={Algorithms for the fractional calculus: A selection of numerical methods},
  author={Kai Diethelm and Neville J. Ford and Alan D. Freed and Yury F. Luchko},
  journal={Computer Methods in Applied Mechanics and Engineering},
Numerical approach for solving time fractional diffusion equation
In this article one of the fractional partial differential equations was solved by finite difference scheme  based on five point and three point central space method with discretization in time. We
Applications of the Fractional Calculus: On a Discretization of Fractional Diffusion Equation in One Dimension
  • Tomáš Kisela
  • Mathematics
    Communications - Scientific letters of the University of Zilina
  • 2010
Fractional calculus is a mathematical discipline dealing with derivatives and integrals of non-integer orders. Its story started around 1695 when Leibniz first suggested the idea of half-derivative
Review of methods and approaches for mechanical problem solutions based on fractional calculus
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Numerical simulations of two-dimensional fractional subdiffusion problems
The growing number of applications of fractional derivatives in various fields of science and engineering indicates that there is a significant demand for better mathematical algorithms for models
A Numerical Method for a Class of Linear Fractional Differential Equations
A numerical method is proposed for the numerical solution of initial value problems of a certain class of linear Fractional Differential Equations (FDEs) with the Jumarie’s modified Riemann-Liouville
Numerical simulations of 2D fractional subdiffusion problems
A numerical method for the distributed order time-fractional diffusion equation
This paper is devoted to the numerical approximation of the diffusion equation with distributed order in time. A numerical method is proposed in the case where the order of the time derivative is


On the Solution of Nonlinear Fractional-Order Differential Equations Used in the Modeling of Viscoplasticity
The authors have recently developed a mathematical model for the description of the behavior of viscoplastic materials. The model is based on a nonlinear differential equation of order β, where β is
The numerical solution of fractional differential equations: Speed versus accuracy
The fixed memory principle is analysed and an alternative nested mesh variant is presented that gives a good approximation to the true solution at reasonable computational cost.
Numerical solution of fractional order differential equations by extrapolation
It is concluded that the application of extrapolation is justified, and the algorithm is obtained a very efficient differential equation solver with practically no additional numerical costs.
Analysis of Fractional Differential Equations
We discuss existence, uniqueness, and structural stability of solutions of nonlinear differential equations of fractional order. The differential operators are taken in the Riemann–Liouville sense
Differential equations of fractional order:methods results and problem —I
Thc paper deals with the so-called differential equations of fractional order in which an unknown function is contained under the operation of a derivative of fractional order. A survey of the
Detailed Error Analysis for a Fractional Adams Method
The numerical method can be seen as a generalization of the classical one-step Adams–Bashforth–Moulton scheme for first-order equations and a detailed error analysis is given.
A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations
We discuss an Adams-type predictor-corrector method for the numericalsolution of fractional differential equations. The method may be usedboth for linear and for nonlinear problems, and it may be
Diffusion in a Semi-Infinite Region with Nonlinear Surface Dissipation
The title problem is posed as a linear heat equation in one space dimension $(x > 0)$ and time $(t > 0)$, with a nonlinear radiative-type boundary condition on the surface $(x = 0)$. Existence and
Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation
A solution to the more than 300-years old problem of geometric and physical interpretation of fractional integration and dieren tiation (i.e., integration and dieren tiation of an arbitrary real