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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.
Fractional Calculus
(2012)
The subject of fractional calculus and its applications (that is, convolution-type pseudo-differential operators including integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, mainly due to its applications in diverse fields of science and engineering. These operators have been used to model problems with anomalous dynamics, however, they also are an effective tool as filters and controllers, and they can be applied to write complicated functions in terms of fractional integrals or derivatives of elementary functions, and so on.
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 was written with a trade-off in mind 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. Numerical code is also provided.
We show that the error term of every modified compound quadrature rule for Cauchy principal value integrals with degree of exactnesss is of optimal order of magnitude in the classesC k[−1,1],k=1,2,...,s, but not inC s+1[−1,1]. We give explicit upper bounds for the error constants of the modified midpoint rule, the modified trapezoidal rule and the modified Simpson rule. Furthermore, the results are generalized to analogous rules for Hadamard-type finite part integrals.
For the numerical evaluation of Cauchy principal value integrals of the form , λε(−1, 1), f εCs[− 1, 1], we consider a quadrature method based on spline interpolation of odd degree 2k + 1,k ∈N0. We show that these rules converge uniformly for λ ∈ (− 1, 1). In particular, we calculate the exact order of magnitude of the error and show that it is equal to the order of the optimal remainder in the class of functions with bounded sth derivative if s ε s;;2k + 1, 2k + 2};. Finally, we compare the rule to the well-known quadrature rule of Elliott and Paget which only converges pointwise.
The order of convergence of modified interpolatory quadratures for singular integrals of Cauchy type
(1995)
We investigate the error term of the dth degree compound quadrature formulae for finite-part integrals of the form ∫10x−pf(x) dx where p∈ and p ≥1. We are mainly interested in error bounds of the form |R[f]|≤c∥∥f(s)∥∥∞ with best possible constants c. It is shown that, for p∉ and n uniformly distributed nodes, the error behaves as O(np–s–1 for f∈Cs[0,1], p–1 <s ≤d+1. In a previous paper we have shown that this is not true for p∈
As an improvement, we consider the case of non-uniformly distributed nodes. Here, we show that for all p ≥ I and f∈Cs[0,1], an O(n–s) error estimate can be obtained in theory by a suitable choice of the nodes. A set of nodes with this property is staled explicitly. In practice, this graded mesh causes stability problems which are computationally expensive to overcome.
For the numerical approximation of Cauchy principal value integrals, we consider the so-called modified quadrature formulas, i.e. formulas obtained by first subtracting out the singularity and then applying a classical quadrature formula. We are interested in error bounds holding uniformly for all possible positions of the singular point. The standard error bounds are based on suprema of derivatives, but they often overestimate the true errors by a factor that grows with the number of nodes of the quadrature formula. We give new bounds involving the total variation Var -(s) and LP-normst|-(s)t|p of some derivative of the integrand function. These bounds give additional possibilities for sharper estimations of the error.
Differenetial equations involving deriatives of non-integer order have shown to be adequate models for various physical phenomena in areas like damping laws, diffusion processes, etc. A small number of algorithms for the numerical solution of these equations has been suggested, but mainly without any error estimates. In this paper, we propose an implicit algorithm for the approximate solution of an important class of these equations. The algorithm is based on a quadrature formula approach. Error estimates and numerical examples are given.
We consider the representation of error functionals in numerical quadrature by the Peano kernel method. It is easily observed that the usual expressions for Peano kernels of order s still make sense if s is not a natural number. In this paper, we discuss how to interpret these Peano kernels, we state their main properties, and we compare them to the (classical) Peano kernels of integer order.
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 a material constant typically in the range 0 < β < 1. This equation is coupled with a first-order differential equation. In the present paper, we introduce and discuss a numerical scheme for the numerical solution of these equations. The algorithm is based on a PECE-type approach.
For the numerical evaluation of finite-part integrals with singularities of order p ⩾ 1, we give error bounds for quadrature methods based on spline approximation. These bounds behave in the same way as the optimal ones. The ideas of the proof are also useful for methods based on other approximation processes.
The FracPECE subroutine for the numerical solution of differential equations of fractional order
(1999)
We consider quadrature formulae for Cauchy principal value integrals Iw,ζ[f]=∫abf(x)x−ζw(x)dx, a<ζ<b. The quadrature formulae considered here are so-called modified formulae, which are obtained by first subtracting the singularity, and then applying some standard quadrature formula Qn. The aim of this paper is to determine the asymptotic behaviour of the constants ki,n in error estimates of the form |Rnmod[f;ζ]|<ki,n(ζ)||f(i)||∞ for fixed i and n→∞, where Rnmod[f;ζ] is the quadrature error. This is done for quadrature formulae Qn for which the Peano kernels Ki,n of fixed order i behave in a certain regular way, including, e.g., many interpolatory quadrature formulae as Gauss–Legendre and Clenshaw–Curtis formulae, as well as compound quadrature formulae. It turns out that essentially all the interpolatory formulae behave in a very similar way.
The numerical approximation of integrals containing strongly singular integrals, in particular Cauchy principal value integrals, is a major issue connected, e.g., to the boundary integral approach for many types of partial differential equations. Whereas the one-dimensional problem has been addressed very intensively in recent years, much less attention has been paid to multidimensional problems. In the present paper, we investigate two possible approaches to this problem, corresponding to generalizations of two approaches known in the 1-D case. In principle, both methods can be applied to integration domains of arbitrary shape, although we find that certain combinations of algorithms and domains are more useful than others. In particular, we discuss error estimates and show that the methods are highly competitive. Moreover, in contrast to most of the previously discussed methods, the approaches are very efficient when integrals have to be calculated for various locations of the singularity.
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.
We investigate strategies for the numerical solution of the initial value problem with initial conditions where 0<α1<α2<⋯<αν. Here y (α j ) denotes the derivative of order α j >0 (not necessarily α j ∈ ℕ) in the sense of Caputo. The methods are based on numerical integration techniques applied to an equivalent nonlinear and weakly singular Volterra integral equation. The classical approach leads to an algorithm with very high arithmetic complexity. Therefore we derive an alternative that leads to lower complexity without sacrificing too much precision.
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 and the initial conditions are specified according to Caputo's suggestion, thus allowing for interpretation in a physically meaningful way. We investigate in particular the dependence of the solution on the order of the differential equation and on the initial condition, and we relate our results to the selection of appropriate numerical schemes for the solution of fractional differential equations.
This paper is devoted to studying the asymptotic behaviour of solutions to generalized noncommensurate fractional systems. To this end, we first consider fractional systems with rational orders and introduce a criterion that is necessary and sufficient to ensure the stability of such systems. Next, from the fractional-order pseudospectrum definition proposed by Sanca et al., we formulate the concept of a rational approximation for the fractional spectrum of a noncommensurate fractional systems with general, not necessarily rational, orders. Our first important new contribution is to
show the equivalence between the fractional spectrum of a noncommensurate linear system and its rational approximation. With this result in hand, we use ideas developed in our earlier work to demonstrate the stability of an equilibrium point to nonlinear systems in arbitrary finite-dimensional spaces. A second novel aspect of our work is the fact that the approach is constructive. Finally, we give numerical simulations to illustrate the merit of the proposed theoretical results.
Revisiting diffusive representations for enhanced numerical approximation of fractional integrals
(2023)
This study reexamines diffusive representations for fractional integrals with the goal of pioneering new variants of such representations. These variants aim to offer highly efficient numerical algorithms for the approximate computation of fractional integrals. The approach seamlessly aligns with established techniques used in addressing problems involving integer-order operators, contributing to a unified framework for numerical solutions.
Sard's classical generalization of the Peano kernel theorem provides an extremely useful method for expressing and calculating sharp bounds for approximation errors. The error is expressed in terms of a derivative of the underlying function. However, we can apply the theorem only if the approximation is exact on a certain set of polynomials.
In this paper, we extend the Peano-Sard theorem to the case that the approximation is exact for a class of generalized polynomials (with non-integer exponents). As a result, we obtain an expression for the remainder in terms of a fractional derivative of the function under consideration. This expression permits us to give sharp error bounds as in the classical situation. An application of our results to the classical functional (vanishing on polynomials) gives error bounds of a new type involving weighted Sobolev-type spaces. In this way, we may state estimates for functions with weaker smoothness properties than usual.
The standard version of the Peano-Sard theory is contained in our results as a special case.
In this paper we provide numerical evidence, via graphics generated with the help of computer simulations, that switching the control parameter of a dynamical system belonging to a class of fractional-order systems in a deterministic way, one obtains an attractor which belongs to the class of all admissible attractors of the considered system. For this purpose, while a multistep numerical method for fractional-order differential equations approximates the solution to the mathematical model, the control parameter is switched periodically every few integration steps. The switch is made inside of a considered set of admissible parameter values. Moreover, the synthesized attractor matches the attractor obtained with the control parameter replaced with the averaged switched parameter values. The results are verified in this paper on a representative system, the fractional-order Lü system. In this way we were able to extend the applicability of the algorithm presented in earlier papers using a numerical method for fractional differential equations.
An Improvement of a Nonclassical Numerical Method for the Computation of Fractional Derivatives
(2009)
Standard methods for the numerical calculation of fractional derivatives can be slow and memory consuming due to the nonlocality of the differential operators. Yuan and Agrawal (2002, “A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives,” ASME J. Vibr. Acoust., 124, pp. 321–324) have proposed a more efficient approach for operators whose order is between 0 and 1 that differs substantially from the traditional concepts. It seems, however, that the accuracy of the results can be poor. We modify the approach, adapting it better to the properties of the problem, and show that this leads to a significantly improved quality. Our idea also works for operators of order greater than 1.
Fractional calculus was first developed by pure mathematicians in the middle of the 19th century. Some 100 years later, engineers and physicists have found applications for these concepts in their areas. However there has traditionally been little interaction between these two communities. In particular, typical mathematical works provide extensive findings on aspects with comparatively little significance in applications, and the engineering literature often lacks mathematical detail and precision. This book bridges the gap between the two communities. It concentrates on the class of fractional derivatives most important in applications, the Caputo operators, and provides a self-contained, thorough and mathematically rigorous study of their properties and of the corresponding differential equations. The text is a useful tool for mathematicians and researchers from the applied sciences alike. It can also be used as a basis for teaching graduate courses on fractional differential equations.
Consider two different solutions of a first-order differential equation. Under rather general conditions we know that these two functions are separated from each other, i.e. their graphs never meet or even cross each other. We ask whether such a result is true for Caputo-type fractional differential equations as well. We can give a partial answer that is positive in some situations and negative under different assumptions. For the remaining cases we state a conjecture and explain why we believe in it. A key ingredient of the analysis is a result concerning the existence of zeros of the solutions of a class of Volterra equations.
We consider the numerical solution of the Bagley-Torvik equation Ay″(t) + BD * 3/2 y(t) + Cy(t) = f(t), as a prototype fractional differential equation with two derivatives. Approximate solutions have recently been proposed in the book and papers of Podlubny in which the solution obtained with approximate methods is compared to the exact solution. In this paper we consider the reformulation of the Bagley-Torvik equation as a system of fractional differential equations of order 1/2. This allows us to propose numerical methods for its solution which are consistent and stable and have arbitrarily high order. In this context we specifically look at fractional linear multistep methods and a predictor-corrector method of Adams type.
We consider ordinary fractional differential equations with Caputo-type differential operators with smooth right-hand sides. In various places in the literature one can find the statement that such equations cannot have smooth solutions. We prove that this is wrong, and we give a full charac-terization of the situations where smooth solutions exist. The results can be extended to a class of weakly singular Volterra integral equations.
In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new.
We propose a new mathematical model for the simulation of the dynamics of a dengue fever outbreak. Our model differs from the classical model in that it involves nonlinear differential equations of fractional, not integer, order. Using statistics from the 2009 outbreak of the disease in the Cape Verde islands, we demonstrate that our model is capable of providing numerical results that agree very well with the real data.
The popular elastic law of Fung that describes the non-linear stress-strain behavior of soft biological tissues is extended into a viscoelastic ma-terial model that incorporates fractional derivatives in the sense of Caputo. This one-dimensional material model is then transformed into a three-dimensional constitutive model that is suitable for general analysis. The model is derived in a configuration that differs from the current, or spatial, configuration by a rigid-body rotation; it being the polar config-uration. Mappings for the fractional-order operators of integration and differentiation between the polar and spatial configurations are presented as a theorem. These mappings are used in the construction of the proposed viscoelastic model.
Eine Gruppe, deren Mitglieder sich zwischen mehreren zur Wahl stehenden Alternativen entscheiden müssen, hat eine große Anzahl von Möglichkeiten, aus den Präferenzen der Einzelnen eine von der Gemeinschaft getragene Entscheidung zu ermitteln. Wie lässt sich sicherstellen, dass diese gemeinschaftliche Entscheidung den Willen der Gruppe sinnvoll widerspiegelt?
Die Untersuchung von Methoden, die die vielen individuellen Meinungen zu einer einzigen Entscheidung für die gesamte Gruppe zusammenfassen, und die Darstellung der wichtigsten Eigenschaften dieser Verfahren sind Inhalt dieses Buches. Neben theoretischen Überlegungen steht dabei gleichberechtigt die Betrachtung zahlreicher Beispiele, die oft unerwartete Eigenschaften erkennen lassen.
A viscoelastic model of the K-BKZ (Kaye, Technical Report 134, College of Aeronautics, Cranfield 1962; Bernstein et al., Trans Soc Rheol 7: 391–410, 1963) type is developed for isotropic biological tissues and applied to the fat pad of the human heel. To facilitate this pursuit, a class of elastic solids is introduced through a novel strain-energy function whose elements possess strong ellipticity, and therefore lead to stable material models. This elastic potential – via the K-BKZ hypothesis – also produces the tensorial structure of the viscoelastic model. Candidate sets of functions are proposed for the elastic and viscoelastic material functions present in the model, including two functions whose origins lie in the fractional calculus. The Akaike information criterion is used to perform multi-model inference, enabling an objective selection to be made as to the best material function from within a candidate set.
Fractional differential equations containing only one fractional derivative are a well understood and frequently used tool for the mathematical description of many physical processes, but they are not always sufficient to reflect all the relevant phenomena. It is sometimes necessary to use models with more than one fractional derivative. We describe two important mathematical ways to use this concept: multi-term equations and multi-order systems. First we show the relations between these two concepts. Then we investigate their most important analytical properties, and finally we look at numerical methods for their approximate solution.
We propose an algorithm for the numerical evaluation of convolution integrals of the form ∫0xk(x−y)f(y,x) dy, for x∈[0,X]. Our method is especially suitable in situations where the fundamental interval [0, X] is very long and the kernel function k is expensive to calculate. Separate versions are provided where the forcing function f is known in advance, and where it must be determined step-by-step along the solution path. These methods are efficient with respect to both run time and memory requirements.
The rapidly growing number of cores on modern supercomputers imposes scalability demands not only on applications but also on the software tools needed for their development. At the same time, increasing application and system complexity makes the optimization of parallel codes more difficult, creating a need for scalable performance-analysis technology with advanced functionality. However, delivering such an expensive technology can hardly be accomplished by single tool developers and requires higher degrees of collaboration within the HPC community. The unified performance-measurement system Score-P is a joint effort of several academic performance-tool builders, funded under the BMBF program HPC-Software für skalierbare Parallelrechner in the SILC project (Skalierbare Infrastruktur zur automatischen Leistungsanalyse paralleler Codes). It is being developed with the objective of creating a common basis for several complementary optimization tools in the service of enhanced scalability, improved interoperability, and reduced maintenance cost.
An efficient parallel algorithm for the numerical solution of fractional differential equations
(2011)
The numerical solution of differential equations of fractional order is known to be a computationally very expensive problem due to the nonlocal nature of the fractional differential operators. We demonstrate that parallelization may be used to overcome these difficulties. To this end we propose to implement the fractional version of the second-order Adams-Bashforth-Moulton method on a parallel computer. According to many recent publications, this algorithm has been successfully applied to a large number of fractional differential equations arising from a variety of application areas. The precise nature of the parallelization concept is discussed in detail and some examples are given to show the viability of our approach.
The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus
(2012)
We generalize the classical mean value theorem of differential calculus by allowing the use of a Caputo-type fractional derivative instead of the commonly used first-order derivative. Similarly, we generalize the classical mean value theorem for integrals by allowing the corresponding fractional integral, viz. the Riemann-Liouville operator, instead of a classical (firstorder) integral. As an application of the former result we then prove a uniqueness theorem for initial value problems involving Caputo-type fractional differential operators. This theorem generalizes the classical Nagumo theorem for first-order differential equations.
This paper gives an overview about the Score-P performance measurement infrastructure which is being jointly developed by leading HPC performance tools groups. It motivates the advantages of the joint undertaking from both the developer and the user perspectives, and presents the design and components of the newly developed Score-P performance measurement infrastructure. Furthermore, it contains first evaluation results in comparison with existing performance tools and presents an outlook to the long-term cooperative development of the new system.
Recently, Trefethen (SIAM Rev., 50 (2008), pp. 67--87) and Xiang and Bornemann (SIAM J. Numer. Anal., 50 (2012), pp. 2581--2587) investigated error bounds for n-point Gauss and Clenshaw--Curtis quadrature for the Legendre weight with integrands having limited smoothness properties. Putting their results into the context of classical quadrature theory, we find that the observed behavior is by no means surprising and that it can essentially be proved for a very large class of quadrature formulas with respect to a broad set of weight functions.
Energy efficiency is an important aspect of future exascale systems, mainly due to rising energy cost. Although High performance computing (HPC) applications are compute centric, they still exhibit varying computational characteristics in different regions of the program, such as compute-, memory-, and I/O-bound code regions. Some of today’s clusters already offer mechanisms to adjust the system to the resource requirements of an application, e.g., by controlling the CPU frequency. However, manually tuning for improved energy efficiency is a tedious and painstaking task that is often neglected by application developers. The European Union’s Horizon 2020 project READEX (Runtime Exploitation of Application Dynamism for Energy-efficient eXascale computing) aims at developing a tools-aided approach for improved energy efficiency of current and future HPC applications. To reach this goal, the READEX project combines technologies from two ends of the compute spectrum, embedded systems and HPC, constituting a split design-time/runtime methodology. From the HPC domain, the Periscope Tuning Framework (PTF) is extended to perform dynamic auto-tuning of fine-grained application regions using the systems scenario methodology, which was originally developed for improving the energy efficiency in embedded systems. This paper introduces the concepts of the READEX project, its envisioned implementation, and preliminary results that demonstrate the feasibility of this approach.
Traditional mathematical models for many phenomena in various different fields of science are based on the use of integer order derivatives. These models are usually well understood from an analytic point of view, in particular regarding the qualitative behaviour of their solutions. The availability of such information is important for evaluating whether the mathematical model really reflects the actual properties that the process in question has, and thus for showing that the set of equations is indeed a suitable model for the concrete process. In many cases, fractional order generalizations of the integer order models allow to obtain better quantitative agreement with experimental data, but the knowledge about qualitative properties is frequently lacking. Thus, the question whether the fractional order model is in fact able to correctly reproduce the behaviour that the underlying process must exhibit frequently remains unanswered. In this paper we want to use a specific example from the life sciences, viz. a model describing a fermentation process, in order to initiate a discussion of this matter.
Score-P is a measurement infrastructure originally designed for the analysis and optimization of the performance of HPC codes. Recent extensions of Score-P and its associated tools now also allow the investigation of energy-related properties and support the user in the implementation of corresponding improvements. Since it would be counterproductive to completely ignore performance issues in this connection, the focus should not be laid exclusively on energy. We therefore aim to optimize software with respect to an objective function that takes into account energy and run time. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)
A note on the well-posedness of terminal value problems for fractional differential equations
(2018)
This note is intended to clarify some important points about the well-posedness of terminal value problems for fractional differential equations. It follows the recent publication of a paper by Cong and Tuan in this journal, in which a counter-example calls into question the earlier results in a paper by this note's authors. Here, we show in the light of these new insights, that a wide class of terminal value problems of fractional differential equations is well posed, and we identify those cases where the well-posedness question must be regarded as open.
In this paper, we investigate some aspects of the qualitative theory for multi-order fractional differential equation systems. First, we obtain a fundamental result on the existence and uniqueness for multi-order fractional differential equation systems. Next, a representation of solutions of homogeneous linear multi-order fractional differential equation systems in series form is provided. Finally, we give characteristics regarding the asymptotic behavior of solutions to some classes of linear multi-order fractional differential equation systems.
It is well known that a continuously differentiable function is monotone in an interval [a, b] if and only if its first derivative does not change its sign there. We prove that this is equivalent to requiring that the Caputo derivatives of all orders α ∈ (0, 1) with starting point a of this function do not have a change of sign there. In contrast to what is occasionally conjectured, it is not sufficient if the Caputo derivatives have a constant sign for a few values of α ∈ (0, 1) only.
As in the embedded systems domain, energy efficiency has recently become one of the main design criteria in high performance computing. The European Union Horizon 2020 project READEX (Run-time Exploitation of Application Dynamism for Energy-efficient eXascale computing) has developed a tools-aided auto-tuning methodology inspired by system scenario based design. Applying similar concepts as those presented in earlier chapters of this book, the dynamic behavior of HPC applications is exploited to achieve improved energy efficiency and performance. Driven by a consortium of European experts from academia, HPC resource providers, and industry, the READEX project has developed the first generic framework of its kind for split design-time and run-time tuning while targeting heterogeneous systems at the Exascale level. Using a real-life boundary element application, energy savings of more than 30% can be shown.
The solution of fractional-order differential problems requires in the majority of cases the use of some computational approach. In general, the numerical treatment of fractional differential equations is much more difficult than in the integer-order case, and very often non-specialist researchers are unaware of the specific difficulties. As a consequence, numerical methods are often applied in an incorrect way or unreliable methods are devised and proposed in the literature. In this paper we try to identify some common pitfalls in the use of numerical methods in fractional calculus, to explain their nature and to list some good practices that should be followed in order to obtain correct results.
Diffusive representations of fractional differential and integral operators can provide a convenient means to construct efficient numerical algorithms for their approximate evaluation. In the current literature, many different variants of such representations have been proposed. Concentrating on Riemann-Liouville integrals whose order is in (0,1), we here present a general approach that comprises most of these variants as special cases and that allows a detailed investigation of the analytic properties of each variant. The availability of this information allows to choose concrete numerical methods for handling the representations that exploit the specific properties, thus allowing to construct very efficient overall methods.
Given a function f ∈ C^3[0, 1] and some q ∈ (0, 1), we look at the approximation for the Hadamard finite-part integral = R1/0x^−q−1f(x)dx based on a piecewise linear interpolant for f at n equispaced nodes (i.e., the product trapezoidal rule). The main purpose of this paper is to give sufficient conditions for the sequence of approximations to converge against the correct value of the integral in a monotonic way. An application of the results yields detailed information on the error term of a backward differentiation formula for a fractional differential equation.
In this paper, a new algorithm for the numerical solution of the initial value problems for general linear multi-term differential equations of frac-tional order with constant coefficients and fractional derivatives defined in the Caputo sense is presented. The algorithm essentially uses some ideas from the convolution quadrature and discretized operational calculus. An-other basic element of the method is the formulas for analytical solution of the problem under consideration given in terms of the Mittag-Leffler type functions. Error estimates and numerical examples are presented. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.
The midpoint rectangle formula for Riemann-Stieltjes integrals is known to provide an O(h2) error bound if the integrator is linear, but only an O(h) bound in the general case. Recently Xie et al. (J. Stat. Comput. Simulation 73 (2003), 59-70) have provided sufficient conditions for O(h2) bounds also for nonlinear integrators. We now provide additional insight by giving alternative sufficient conditions that may be used in cases where those of Xie et al. are not applicable. Specifically, our bounds have a simpler structure.
Tiefziehprozesse lassen sich heute durch viele der auf dem Markt befindlichen Umformsimulationsprogramme mit ausreichender Genauigkeit simulieren. Im Hinblick auf Rückfederungsprognosen muß die Qualität der Simulationssoftware jedoch weiter verbessert werden, um auch hier zuverlässige Ergebnisse zu ermöglichen. In der Industrie und bei Anbietern kommerzieller Finite Element Programme werden daher auf diesem Gebiet erhebliche Anstregungen unternommen. Aufgrund der großen Anzahl von Einflussfaktoren ist die Problemstellung überaus komplex. Entsprechend groß ist die Anzahl von Untersuchungen zu diesem Thema. Diese Untersuchungen beleuchten aber zumeist nur Teilaspekte des Problems. Dies ist verständlich, da Betrachtungen über das Zusammenwirken aller Einflussfaktoren einen enormen Aufwand bedeuten und durch einzelne Softwarehäuser nicht geleistet werden können. Darüber hinaus verhindern auch bestimmte Vorgaben durch die verwendete kommerzielle Software detailliertere Untersuchungen. So werden z.B. Untersuchungen zu Materialmodellen, mit denen eine verbesserte Beschreibung des Verfestigungsverhaltens von Metallen erreicht werden soll, häufig mit Elementtypen durchgeführt, deren Eignung für die Anforderungen einer zuverlässigen Rückfederungssimulation durchaus in Frage gestellt werden muss.
GNS nimmt derzeit mit dem Umformsimulationsprogramm INDEED neben anderen Softwarehäusern an einem Projekt zur Verbesserung von Rückfederungsprogosen teil. Einige der dabei gewonnenen Erkenntnisse sollen im folgenden am Beispiel eines dabei aufgetretenen Problems vorgestellt werden. Insbesondere soll die große Bedeutung der verwendeten Elementtypen für die Rückfederungssimulation betont werden.
Many recently developed models in areas like viscoelasticity, electrochemistry, diffusion processes, etc. are formulated in terms of derivatives (and integrals) of fractional (non-integer) order. In this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context. We believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way.
An extension of the well-posedness concept for fractional differential equations of Caputo’s type
(2014)
It is well known that, under standard assumptions, initial value problems for fractional ordinary differential equations involving Caputo-type derivatives are well posed in the sense that a unique solution exists and that this solution continuously depends on the given function, the initial value, and the order of the derivative. Here, we extend this well-posedness concept to the extent that we also allow the location of the starting point of the differential operator to be changed, and we prove that the solution depends on this parameter in a continuous way too if the usual assumptions are satisfied. Similarly, the solution to the corresponding terminal value problems depends on the location of the starting point and of the terminal point in a continuous way too.
This article describes the fundamentals of the theory of ordinary fractional differential equations of Caputo’s type. Starting from the existence and uniqueness of solutions and the well-posedness in general, the flow of topics continues via a derivation of explicit solution formulas for certain important classes of problems and the discussion of their smoothness properties to the stability properties of these solutions. The main focus is on initial value problems, but terminal value problems are briefly considered as well. In addition to dealing with standard single-order problems, the presentation also contains a short discussion of multiterm equations and multiorder systems.
Mathematical models based on differential operators of fractional order have proven to be very useful for describing the properties of viscoelastic materials. However, the associated differential equations can usually not be solved analytically. In this article, we provide a survey of the most important numerical methods. We restrict our attention to those types of fractional differential equations that are most important in the context of viscoelasticity, i.e., we discuss numerical methods for ordinary fractional differential equations and for certain types of time-fractional partial differential equations. Space-fractional partial differential equations are not discussed.
This article describes fundamental approaches for the numerical handling of problems arising in fractional calculus. This includes, in particular, methods for approximately computing fractional integrals and fractional derivatives, where the emphasis is placed on Caputo operators, as well as solvers for the associated differential and integral equations.
Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known variants of this approach is that they require the numerical approximation of some integrals over an unbounded integral whose integrand decays rather slowly, which implies that their numerical handling is difficult and costly. We present a novel variant of such a diffusive representation. This form also requires the numerical approximation of an integral over an unbounded domain, but the integrand decays much faster. This property allows to use well established quadrature rules with much better convergence properties.
Fractional order models have proven to be a very useful tool for the modeling of the mechanical behaviour of viscoelastic materials. Traditional numerical solution methods exhibit various undesired properties due to the non-locality of the fractional differential operators, in particular regarding the high computational complexity and the high memory requirements. The infinite state representation is an approach on which one can base numerical methods that overcome these obstacles. Such algorithms contain a number of parameters that influence the final result in nontrivial ways. Based on numerical experiments, we initiate a study leading to good choices of these parameters.
We study fractional differential equations of Riemann–Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on R, we define fractional operators by means of a functional calculus using the Fourier transform. Main tools are extrapolation- and interpolation spaces. Main results are the existence and uniqueness of solutions and the causality of solution operators for non-linear fractional differential equations.
This article concerns an analytic and numerical analysis of a class of weighted singular Cauchy integrals with exponential weights w:= exp (−Q) with finite moments and with smooth external fields Q:R→[0,∞), with varying smooth convex rate of increase for large argument. Our analysis relies in part on weighted polynomial interpolation at the zeros of orthonormal polynomials with respect to w2. We also study bounds for the first derivatives of a class of functions of the second kind for w2.
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately evident that the method is fast and memory-efficient. Moreover, the method’s design is such that good convergence properties may be expected. In this paper, we commence a systematic investigation of these convergence properties.
This paper is devoted to studying non-commensurate fractional order planar systems. Our contributions are to derive sufficient conditions for the global attractivity of non-trivial solutions to fractional-order inhomogeneous linear planar systems and for the Mittag-Leffler stability of an equilibrium point to fractional order nonlinear planar systems. To achieve these goals, our approach is as follows. Firstly, based on Cauchy’s argument principle in complex analysis, we obtain various explicit sufficient conditions for the asymptotic stability of linear systems whose coefficient matrices are constant. Secondly, by using Hankel type contours, we derive some important estimates of special functions arising from a variation of constants formula of solutions to inhomogeneous linear systems. Then, by proposing carefully chosen weighted norms combined with the Banach fixed point theorem for appropriate Banach spaces, we get the desired conclusions. Finally, numerical examples are provided to illustrate the effect of the main theoretical results.