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Fractional Calculus
(2016)
This book will give readers the possibility of finding very important mathematical tools for working with fractional models and solving fractional differential equations, such as a generalization of Stirling numbers in the framework of fractional calculus and a set of efficient numerical methods. Moreover, we will introduce some applied topics, in particular fractional variational methods which are used in physics, engineering or economics. We will also discuss the relationship between semi-Markov continuous-time random walks and the space-time fractional diffusion equation, which generalizes the usual theory relating random walks to the diffusion equation. These methods can be applied in finance, to model tick-by-tick (log)-price fluctuations, in insurance theory, to study ruin, as well as in macroeconomics as prototypical growth models.
All these topics are complementary to what is dealt with in existing books on fractional calculus and its applications. This book will keep in mind the trade-off between full mathematical rigor and the needs of readers coming from different applied areas of science and engineering. In particular, the numerical methods listed in the book are presented in a readily accessible way that immediately allows the readers to implement them on a computer in a programming language of their choice.
The second edition of the book has been expanded and now includes a discussion of additional, newly developed numerical methods for fractional calculus and a chapter on the application of fractional calculus for modeling processes in the life sciences.
Traditional methods for the numerical approximation of fractional derivatives have a number of drawbacks due to the non-local nature of the fractional differential operators. The main problems are the arithmetic complexity and the potentially high memory requirements when they are implemented on a computer. In a recent paper, Yuan and Agrawal have proposed an approach for operators of order α ∈ (0,1) that differs substantially from the standard methods. We extend the method to arbitrary α > 0, α∉N, and give an analysis of the main properties of this approach. In particular it turns out that the original algorithm converges rather slowly. Based on our analysis we are able to identify the source of this slow convergence and propose some modifications leading to a much more satisfactory behaviour. Similar results are obtained for a closely related method proposed by Chatterjee.
We show that, iff∈Ck[−1,1] (k≥2), the error term of every modified positive interpolatory quadrature rule for Cauchy principal value integrals of the type∫−1−1w(x)f(x)x−λdx ,λ∈(−1,1), fulfills Rn[f;λ]=O(n−klnn) uniformly for allλ∈(−1,1), and hence it is of optimal order of magnitude in the classesCk[−1,1] (k=2,3,4,…). Here, w is a weight function with the property0≤w(x)1−x2−−−−−√≤C . We give explicit upper bounds for the Peano-type error constants of such rules. This improves and completes earlier results by Criscuolo and Mastroianni (Calcolo 22 (1985), 391–441 and Numer. Math. 54 (1989), 445–461) and Ioakimidis (Math. Comp. 44 (1985), 191–198). For the special case of the Gaussian rule, we show that the restrictionk≥2 can be dropped. The results are based on a new representation of the Peano kernels of these formulae via the Peano kernels of the underlying classical quadrature formulae. This representation may also be useful in connection with some different problems.
We prove convergence results and error estimates for interpolatory product quadrature formulas for Cauchy principal value integrals on the real line with Freud–type weight functions. The formulas are based on polynomial interpolation at the zeros of orthogonal polynomials associated with the weight function under consideration. As a by–product, we obtain new bounds for the derivative of the functions of the second kind for these weight functions.
Boundedness and uniform numerical approximation of the weighted Hilbert transform on the real line
(2001)
We establish the uniform boundedness of the weighted Hilbert transform in function spaces associated with a class of even weights on the real line with varying rates of smooth decay near∞. We then consider the numerical approximation of the weighted Hilbert transform and to this end we establish convergence results and error estimates which we prove are sharp. Our formulae are based on polynomial interpolation at the zeros of orthogonal polynomials associated with the weight function under consideration, augmented by two carefully chosen extra points. Typical examples of weights that are studied are: (a) w α (x) : = exp(− | x|α , α> 1, x ∈ R; (b) w k,β x: =exp (−expk (| x 7verbar;β)), β> 0, k > 1, x ∈ R.
We consider the numerical solution of (possibly nonlinear) fractional differential equations of the form y(α)(t)=f(t,y(t),y(β1)(t),y(β2)(t),…,y(βn)(t)) with α>βn>βn−1>⋯>β1 and α−βn⩽1, βj−βj−1⩽1, 0<β1⩽1, combined with suitable initial conditions. The derivatives are understood in the Caputo sense. We begin by discussing the analytical questions of existence and uniqueness of solutions, and we investigate how the solutions depend on the given data. Moreover we propose convergent and stable numerical methods for such initial value problems.
Modern computational simulation's increasing and mainly speed-oriented use of HPC systems often conflicts with the goal of making research reproducible. Indeed, the simulations that result from HPC use often behave reproducibly in only a limited way. As a discussion of this phenomenon's technical background describes, the problems entailed will be very difficult to overcome.
Increasing the efficiency of shooting methods for terminal value problems of fractional order
(2015)
Shooting methods are a well established tool for the numerical solution of terminal value problems of fractional order. However, they can be computationally quite expensive because of their iterative nature in which (a) each single iteration may be costly, and (b) the number of iterations can be large. In this paper we propose algorithmic strategies for improving the efficiency of such methods. Our strategies are aimed at simultaneously reducing the cost of each iteration and reducing the number of required iterations.
The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee, if possible, the future evolution, or, at least, the challenges identified in the scope of advanced research works. This paper gives a vision about the directions for further research as well as some open problems of FC. A number of topics in mathematics, numerical algorithms and physics are analyzed, giving a systematic perspective for future research.
Upper and lower estimates for the separation of solutions to fractional differential equations
(2022)
Given a fractional differential equation of order α∈(0,1] with Caputo derivatives, we investigate in a quantitative sense how the associated solutions depend on their respective initial conditions. Specifically, we look at two solutions x1 and x2, say, of the same differential equation, both of which are assumed to be defined on a common interval [0, T], and provide upper and lower bounds for the difference x1(t)−x2(t) for all t∈[0,T] that are stronger than the bounds previously described in the literature.