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 extended tomulti-term equations (involving more than one differential operator)too.

https://ntrs.nasa.gov/api/citations/20020024453/downloads/20020024453.pdf

A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations

Feedback systems: Input-output properties

We investigate a method for the numerical solution of the nonlinear fractional differential equation D
*
α
y(t)=f(t,y(t)), equipped with initial conditions y
(k)(0)=y
0
(k), k=0,1,...,⌈α⌉−1. Here α may be an arbitrary positive real number, and the differential operator is the Caputo derivative. The numerical method can be seen as a generalization of the classical one-step Adams–Bashforth–Moulton scheme for first-order equations. We give a detailed error analysis for this algorithm. This includes, in particular, error bounds under various types of assumptions on the equation. Asymptotic expansions for the error are also mentioned briefly. The latter may be used in connection with Richardson's extrapolation principle to obtain modified versions of the algorithm that exhibit faster convergence behaviour.

/pdf/detailed-error-analysis-for-a-fractional-adams-method-2odctuwibd.pdf

Detailed error analysis for a fractional Adams method

For the numerical approximation of fractional integrals $I^\alpha f(x) = \frac{1}{{\Gamma (\alpha )}}\int_0^x {(x - s)^{\alpha - 1} f(s)ds\qquad (x \geqq 0)} $ with $f(x) = x^{\beta - 1} g(x)$, g smooth, we study convolution quadratures. Here approximations to $I^\alpha f(x)$ on the grid $x = 0,h,2h, \cdots ,Nh$ are obtained from a discrete convolution with the values of f on the same grid. With the appropriate definitions, it is shown that such a method is convergent of order p if and only if it is stable and consistent of order p. We introduce fractional linear multistep methods: The $\alpha $th power of a pth order linear multistep method gives a pth order convolution quadrature for the approximation of $I^\alpha $. The paper closes with numerical examples and applications to Abel integral equations, to diffusion problems and to the computation of special functions.

Discretized fractional calculus

This brief studies the effects of fractional dynamics in chaotic systems. In particular, Chua's system is modified to include fractional order elements. By varying the total system order incrementally from 3.6 to 3.7, it is demonstrated that systems of "order" less than three can exhibit chaos as well as other nonlinear behavior. This effectively forces a clarification of the definition of order which can no longer be considered only by the total number of differentiations or by the highest power of the Laplace variable. >

Chaos in a fractional order Chua's system

We study the numerical approximation of an integro-differential equation which is intermediate between the heat and wave equations. The proposed discretization uses convolution quadrature based on the first- and second-order backward difference methods in time, and piecewise linear finite elements in space. Optimal-order error bounds in terms of the initial data and the inhomogeneity are shown for positive times, without assumptions of spatial regularity of the data.

/pdf/nonsmooth-data-error-estimates-for-approximations-of-an-4sp14bqyfk.pdf

Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term

Fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular Volterra integral equations. The proposed methods are convergent of the order of the underlying multistep method, also in the generic case of solutions which are not smooth at the origin. The stability properties (stability region, A-stability, A(a)-stability) are closely related to those of the underlying linear multistep method.

/pdf/fractional-linear-multistep-methods-for-abel-volterra-4snc4x4y96.pdf

Fractional linear multistep methods for Abel-Volterra integral equations of the second kind

We study the approximation properties of Runge-Kutta time discretizations of linear and semilinear parabolic equations, including incompressible Navier-Stokes equations. We derive asymptotically sharp error bounds and relate the temporal order of convergence, which is generally noninteger, to spatial regularity and the type of boundary conditions. The analysis relies on an interpretation of Runge-Kutta methods as convolution quadratures. In a different context, these can be used as efficient computational methods for the approximation of convolution integrals and integral equations. They use the Laplace transform of the confolution kernal via a discrete operational calculus. 23 refs., 2 tabs.

/pdf/runge-kutta-methods-for-parabolic-equations-and-convolution-1977u10g9z.pdf

Runge-Kutta methods for parabolic equations and convolution quadrature

Application des puissances fractionnaires des methodes multipas lineaires a la solution numerique des equations integrales de Volterra de premiere espece a faible singularite. Etude de la convergence et de la stabilite des methodes proposees

Fractional Linear Multistep Methods for Abel-Volterra Integral Equations of the First Kind

MEXX (short for MEXanical systems extrapolation integrator) is a FORTRAN code for time integration of constrained mechanical systems. MEXX is suited for direct integration of the equations of motion in descriptor form. It is based on extrapolation of a timestepping method that is explicit in regard to differential equations and linearly implicit regarding nonlinear constraints. It requires only the solution of well-structured systems of linear equations whose solution requires computations that increase linearly with the number of bodies, in the case of multibody systems with few closed kinematic loops. Position and velocity constraints are enforced throughout the integration interval, whereas acceleration constraints need not be formulated. MEXX has options for time-continuous solution representation (useful for graphics) and for the location of events such as impacts. The present article describes MEXX and its underlying concepts.

Numerical Integration of Constrained Mechanical Systems Using MEXX

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