Time-Fractional Derivatives in Relaxation Processes: A Tutorial Survey

2000 Mathematics Subject Classification: 26A33, 33E12, 33C60, 44A10, 45K05, 74D05,

Generalized Convolution Transforms and Toeplitz Plus Hankel Integral Equations

Mathematics Subject Classification: 44A05, 44A35

Solvability of an Infinite System of Singular Integral Equations

2000 Mathematics Subject Classification: 45G15, 26A33, 32A55, 46E15.

Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case

In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator $\Delta_+^{(\alpha,\beta,\gamma)}:= D_{x_0^+}^{1+\alpha} +D_{y_0^+}^{1+\beta} +D_{z_0^+}^{1+\gamma},$ where $(\alpha, \beta, \gamma) \in \,]0,1]^3$, and the fractional derivatives $D_{x_0^+}^{1+\alpha}$, $D_{y_0^+}^{1+\beta}$, $D_{z_0^+}^{1+\gamma}$ are in the Riemann-Liouville sense. Applying operational techniques via two-dimensional Laplace transform we describe a complete family of eigenfunctions and fundamental solutions of the operator $\Delta_+^{(\alpha,\beta,\gamma)}$ in classes of functions admitting a summable fractional derivative. Making use of the Mittag-Leffler function, a symbolic operational form of the solutions is presented. From the obtained family of fundamental solutions we deduce a family of fundamental solutions of the fractional Dirac operator, which factorizes the fractional Laplace operator. We apply also the method of separation of variables to obtain eigenfunctions and fundamental solutions.

Clifford-Hermite polynomials in fractional Clifford analysis

In this paper we generalize radial and standard Clifford-Hermite polynomials to the new framework of fractional Clifford analysis with respect to the Riemann-Liouville derivative in a symbolic way. As main consequence of this approach, one does not require an a priori integration theory. Basic properties such as orthogonality relations, differential equations, and recursion formulas, are proven.

Eigenfunctions and fundamental solutions of the Caputo fractional Laplace and Dirac operators

In this paper, by using the method of separation of variables, we obtain eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator defined via fractional Caputo derivatives. The solutions are expressed using the Mittag-Leffler function and we show some graphical representations for some parameters. A family of fundamental solutions of the corresponding fractional Dirac operator is also obtained. Particular cases are considered in both cases.

Existence Results for Abstract Partial Neutral Integro-differential Equation with Unbounded Delay

In this paper we study the existence and regularity of mild solutions for a class of abstract partial neutral integro-differential equations with unbounded delay.

Generalized Coupled Algebraic Riccati Equations for Discrete-time Markov Jump with Multiplicative Noise Systems

In this paper we consider the existence of the maximal and mean square stabilizing solutions for a set of generalized coupled algebraic Riccati equations (GCARE for short) associated to the infinite-horizon stochastic optimal control problem of discrete-time Markov jump with multiplicative noise linear systems. The weighting matrices of the state and control for the quadratic part are allowed to be indefinite. We present a sufficient condition, based only on some positive semi-definite and kernel restrictions on some matrices, under which there exists the maximal solution and a necessary and sufficient condition under which there exists the mean square stabilizing solution fir the GCARE. We also present a solution for the discounted and long run average cost problems when the performance criterion is assumed be composed by a linear combination of an indefinite quadratic part and a linear part in the state and control variables. The paper is concluded with a numerical example for pension fund with regime switching.

EXISTENCE OF S-ASYMPTOTICALLY omega-PERIODIC SOLUTIONS FOR ABSTRACT NEUTRAL EQUATIONS

A bounded continuous function it u : [0, infinity) -> X is said to be S-asymptotically omega-periodic if lim(t ->infinity)[u(t + omega) - u(t)] = 0. This paper is devoted to study the existence and qualitative properties of S-asymptotically omega-periodic mild solutions for some classes of abstract neutral functional differential equations with infinite delay, Furthermore, applications to partial differential equations are given.

Difference equations in Banach spaces

Difference equations which discretely approximate boundary value problems for second-order ordinary differential equations are analysed. It is well known that the existence of solutions to the continuous problem does not necessarily imply existence of solutions to the discrete problem and, even if solutions to the discrete problem are guaranteed, they may be unrelated and inapplicable to the continuous problem. Analogues to theorems for the continuous problem regarding a priori bounds and existence of solutions are formulated for the discrete problem. Solutions to the discrete problem are shown to converge to solutions of the continuous problem in an aggregate sense. An example which arises in the study of the finite deflections of an elastic string under a transverse load is investigated. The earlier results are applied to show the existence of a solution; the sufficient estimates on the step size are presented. (C) 2003 Elsevier Science Ltd. All rights reserved.

Observability of nonlinear fractional dynamical systems

We study the observability of linear and nonlinear fractional differential systems of order 0 < α < 1 by using the Mittag-Leffler matrix function and the application of Banach’s contraction mapping theorem. Several examples illustrate the concepts.

Constraint reasoning for differential models

The basic motivation of this work was the integration of biophysical models within the interval constraints framework for decision support. Comparing the major features of biophysical models with the expressive power of the existing interval constraints framework, it was clear that the most important inadequacy was related with the representation of differential equations. System dynamics is often modelled through differential equations but there was no way of expressing a differential equation as a constraint and integrate it within the constraints framework. Consequently, the goal of this work is focussed on the integration of ordinary differential equations within the interval constraints framework, which for this purpose is extended with the new formalism of Constraint Satisfaction Differential Problems. Such framework allows the specification of ordinary differential equations, together with related information, by means of constraints, and provides efficient propagation techniques for pruning the domains of their variables. This enabled the integration of all such information in a single constraint whose variables may subsequently be used in other constraints of the model. The specific method used for pruning its variable domains can then be combined with the pruning methods associated with the other constraints in an overall propagation algorithm for reducing the bounds of all model variables. The application of the constraint propagation algorithm for pruning the variable domains, that is, the enforcement of local-consistency, turned out to be insufficient to support decision in practical problems that include differential equations. The domain pruning achieved is not, in general, sufficient to allow safe decisions and the main reason derives from the non-linearity of the differential equations. Consequently, a complementary goal of this work proposes a new strong consistency criterion, Global Hull-consistency, particularly suited to decision support with differential models, by presenting an adequate trade-of between domain pruning and computational effort. Several alternative algorithms are proposed for enforcing Global Hull-consistency and, due to their complexity, an effort was made to provide implementations able to supply any-time pruning results. Since the consistency criterion is dependent on the existence of canonical solutions, it is proposed a local search approach that can be integrated with constraint propagation in continuous domains and, in particular, with the enforcing algorithms for anticipating the finding of canonical solutions. The last goal of this work is the validation of the approach as an important contribution for the integration of biophysical models within decision support. Consequently, a prototype application that integrated all the proposed extensions to the interval constraints framework is developed and used for solving problems in different biophysical domains.

A fractional approach for the motion planning of redundant and hyper-redundant manipulators

The trajectory planning of redundant robots through the pseudoinverse control leads to undesirable drift in the joint space. This paper presents a new technique to solve the inverse kinematics problem of redundant manipulators, which uses a fractional differential of order α to control the joint positions. Two performance measures are defined to examine the strength and weakness of the proposed method. The positional error index measures the precision of the manipulator's end-effector at the target position. The repeatability performance index is adopted to evaluate if the joint positions are repetitive when the manipulator execute repetitive trajectories in the operational workspace. Redundant and hyper-redundant planar manipulators reveal that it is possible to choose in a large range of possible values of α in order to get repetitive trajectories in the joint space.

Introduction to fractional linear systems. Part 2: discrete-time case

IEE Proceedings - Vision, Image, and Signal Processing, Vol. 147, nº 1

Soliton-type and other travelling wave solutions for an improved class of nonlinear sixth-order Boussinesq equations

An improved class of nonlinear bidirectional Boussinesq equations of sixth order using a wave surface elevation formulation is derived. Exact travelling wave solutions for the proposed class of nonlinear evolution equations are deduced. A new exact travelling wave solution is found which is the uniform limit of a geometric series. The ratio of this series is proportional to a classical soliton-type solution of the form of the square of a hyperbolic secant function. This happens for some values of the wave propagation velocity. However, there are other values of this velocity which display this new type of soliton, but the classical soliton structure vanishes in some regions of the domain. Exact solutions of the form of the square of the classical soliton are also deduced. In some cases, we find that the ratio between the amplitude of this wave and the amplitude of the classical soliton is equal to 35/36. It is shown that different families of travelling wave solutions are associated with different values of the parameters introduced in the improved equations.