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

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

From the Publisher:
This book is for people who need to solve ordinary differential equations (ODEs), both initial value problems (IVPs) and boundary value problems (BVPs) as well as delay differential equations (DDEs). These topics are usually taught in separate courses of length one semester each, but solving ODEs with Matlab provides a sound treatment of all three in about 250 pages. The chapters on each of these topics begin with a discussion of "the facts of life" for the problem, mainly by means of examples. Numerical methods for the problem are then developed - but only the methods most widely used. Although the treatment of each method is brief and technical issues are minimized, the issues important in practice and for understanding the codes are discussed. Often solving a real problem is much more than just learning how to call a code. The last part of each chapter is a tutorial that shows how to solve problems by means of small but realistic examples.

/pdf/solving-odes-with-matlab-1yxxtyf9zz.pdf

Solving ODEs with MATLAB

https://ntrs.nasa.gov/api/citations/20030112277/downloads/20030112277.pdf

Algorithms for the fractional calculus: A selection of numerical methods

A recently developed method for solving optimal trajectory problems uses a piecewise-polynomial representation of the state and control variables, enforces the equations of motion via a collocation procedure, and thus approximates the original calculus-of-variations problem with a nonlinear-programming problem, which is solved numerically. This paper identifies this method as a direct transcription method and proceeds to investigate the relationship between the original optimal-control problem and the nonlinear-programming problem. The discretized adjoint equation of the collocation method is found to have deficient accuracy, and an alternate scheme which discretizes the equations of motion using an explicit Runge-Kutta parallel-shooting approach is developed. Both methods are applied to finite-thrust spacecraft trajectory problems, including a low-thrust escape spiral, a three-burn rendezvous, and a low-thrust transfer to the moon.

Discrete approximations to optimal trajectories using direct transcription and nonlinear programming

The application of certain difference schemes (box, trapezoidal, Euler and backward Euler) to the numerical solution of boundary value problems for nonlinear first order systems of ordinary differential equations with a singularity of the first kind is examined. The solution of the linear eigenvalue problem is also considered.

Difference Methods for Boundary Value Problems with a Singularity of the First Kind

The solution of a nonlinear system of first order differential equations with nonlinear boundary conditions by implicit Runge-Kutta methods based on interpolatory quadrature formulae is examined. An equivalence between implicit Runge-Kutta and collocation schemes is established. It is shown that the difference equations obtained have a unique solution in a neighbourhood of an isolated solution of the continuous problem, that this solution can be computed by Newton iteration and that it converges to the isolated solution. The order of convergence is equal to the degree of precision of the related quadra- ture formula plus one. The efficient implementation of the methods is discussed and numerical examples are given. 1. Introduction. We investigate the application of certain implicit Runge- Kutta methods (cf. Butcher (2)) to the numerical solution of nonlinear boundary- value problems of the form

/pdf/the-application-of-implicit-runge-kutta-and-collection-lceaah9gvz.pdf

The application of implicit Runge-Kutta and collection methods to boundary-value problems

The application of collocation methods based on piecewise polynomials to the numerical solution of boundary value problems for systems of ordinary differential equations with a singularity of the first kind is examined. The schemes are shown to be stable and convergent. Enhanced accuracy at the nodes for suitably chosen collocation points (superconvergence) is established for a class of important problems.

Collocation Methods for Singular Boundary Value Problems

A generalized Euler-Maclaurin sum formula is established for product integration based on piecewise Lagrangian interpolation. The integrands considered may have algebraic or logarithmic singularities. The results are used to obtain accurate con- vergence rates of numerical methods for Fredholm and Volterra integral equations with singular kernels.

/pdf/asymptotic-expansions-for-product-integration-3rqrv89ffg.pdf

Asymptotic Expansions for Product Integration

A Fredholm theory for linear boundary value problems $t^\alpha y' = G(t)y + g(t),\,0 < t \leqq 1,\,y \in C[0,1] \cap C^1 (0,1],\,\alpha \geqq 1$; $B_0 y(0) + B_1y(1) = \gamma $ is established, together with existence and regularity results for continuous solutions of nonlinear systems of ordinary differential equations $t^\alpha y' = f(t,y)$. This theory is applied to boundary value problems on infinite intervals and is illustrated by two examples. Finally, some fundamental properties of the generalized linear eigenvalue problem $t^\alpha y' - (G(t) - \lambda H(t))y = 0,\,0 < t \leqq 1,\,y \in C[0,1] \cap C^1 (0,1]$; $B_0 y(0) + B_1 y(1) = 0$, are derived.

Richard Weiss

Papers

Difference Methods for Boundary Value Problems with a Singularity of the First Kind

The application of implicit Runge-Kutta and collection methods to boundary-value problems

Collocation Methods for Singular Boundary Value Problems

Asymptotic Expansions for Product Integration

On the Boundary Value Problem for Systems of Ordinary Differential Equations with a Singularity of the Second Kind