Fractional variational calculus with classical and combined Caputo derivatives

We give a proper fractional extension of the classical calculus of variations by considering variational functionals with a Lagrangian depending on a combined Caputo fractional derivative and the classical derivative. Euler-Lagrange equations to the basic and isoperimetric problems as well as transversality conditions are proved.     

HILBERT SPACES OF FUNCTIONALS ASSOCIATED WITH NONLINEAR OPERATORS

Abstract

In [7] the subject of reproducing kernel Hilbert spaces (RKHSs) of linear functionals associated with linear operators and, in particular, with second-order generalized stochastic processes (GSPs), is pursued. In this work these ideas are extended to nonlinear operators. As an example the characteristic operator of a GSP is pursued. The so-called nonlinear space of the process associated with the characteristic operator is investigated and the RKHS of functionals isometrically isomorphic to it is constructed. Unlike the linear space, the nonlinear analysis is not limited to second order GSPs.     

Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative

In this paper, we shall discuss the properties of the well-known Mittag-Leffler function, and consider the existence and uniqueness of the solution of the periodic boundary value problem for a fractional differential equation involving a Riemann-Liouville fractional derivative by using the monotone iterative method.     

Fractional order differential equations on an unbounded domain

We are concerned with the existence of bounded solutions of a boundary value problem on an unbounded domain for differential equations involving the Caputo fractional derivative. Our results are based on a fixed point theorem of Schauder combined with the diagonalization method.     

General fractional variational problem depending on indefinite integrals

In this report, we consider two kind of general fractional variational problem depending on indefinite integrals include unconstrained problem and isoperimetric problem. These problems can have multiple dependent variables, multiorder fractional derivatives, multiorder integral derivatives and boundary conditions. For both problems, we obtain the Euler-Lagrange type necessary conditions which must be satisfied for the given functional to be extremum. Also, we apply the Rayleigh-Ritz method for solving the unconstrained general fractional variational problem depending on indefinite integrals. By this method, the given problem is reduced to the problem for solving a system of algebraic equations using shifted Legendre polynomials basis functions. An approximate solution for this problem is obtained by solving the system. We discuss the analytic convergence of this method and finally by some examples will be showing the accurately and applicability for this technique.     

The monotonic property and stability of solutions of fractional differential equations

In this paper we improve on the monotone property of Lemma 1.7.3 in Lakshmikantham et al. (2009) [5] for the case g(t,u)=λu with a nonnegative real number λ. We also investigate the Mittag-Leffler stability of solutions of fractional differential equations by using the fractional comparison principle.     

The profile of blowing-up solutions to a nonlinear system of fractional differential equations

We investigate the profile of the blowing-up solutions to the nonlinear nonlocal system (FDS):

     

Higher-order Hahn’s quantum variational calculus

We prove a necessary optimality condition of Euler-Lagrange type for quantum variational problems involving Hahn’s derivatives of higher-order.     

The Euler and Weierstrass conditions for nonsmooth variational problems

Necessary conditions are developed for a general problem in the calculus of variations in which the Lagrangian function, although finite, need not be Lipschitz continuous or convex in the velocity argument. For the first time in such a broadly nonsmooth, nonconvex setting, a full subgradient version of Euler's equation is derived for an arc that furnishes a local minimum in the classical weak sense, and the Weierstrass inequality is shown to accompany it when the arc gives a local minimum in the strong sense. The results are achieved through new techniques in nonsmooth analysis.This research was supported in part by funds from the U.S.-Israel Science Foundation under grant 90-00455, and also by the Fund for the Promotion of Research at the Technion under grant 100-954 and by the U.S. National Science Foundation under grant DMS-9200303.This article was processed by the author using the style filepljourlm from Springer-Verlag.     

Fractional differential equations and Lyapunov functionals

We consider a scalar fractional differential equation, write it as an integral equation, and construct several Lyapunov functionals yielding qualitative results about the solution. It turns out that the kernel is convex with a singularity and it is also completely monotone, as is the resolvent kernel. While the kernel is not integrable, the resolvent kernel is positive and integrable with an integral value of one. These kernels give rise to essentially different types of Lyapunov functionals. It is to be stressed that the Lyapunov functionals are explicitly given in terms of known functions and they are differentiated using Leibniz’s rule. The results are readily accessible to anyone with a background of elementary calculus.     

Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions

In this paper, the fractional differential transform method is developed to solve fractional integro-differential equations with nonlocal boundary conditions. The method is described and illustrated with numerical examples. The results reveal that the method is accurate and easy to apply.     

On the approximate solutions of constrained inverse Lipschitz function problems and inverse function theorems

We present a principle of approximate solutions of constrained inverse Lipschitz function problems. As corollaries and applications of the principle, we obtain a result of convergence of an approximate solutions sequence for the constrained problems, a conclusion relating direct and inverse images of upper and lower limits of a sequence of subsets, and several versions of inverse Lipschitz function theorems. Finally we give local uniqueness criteria for solutions to constrained nonlinear problems in finite dimension spaces.     

Discontinuous control problems for non-convex dynamics and near viability for singularly perturbed control systems

The aim of this paper is to study two classes of discontinuous control problems without any convexity assumption on the dynamics. In the first part we characterize the value function for the Mayer problem and the supremum cost problem using viscosity tools and the notion of ε-viability (near viability). These value functions are given with respect to discontinuous cost functionals. In the second part we obtain results describing the ε-viability (near viability) of singularly perturbed control systems.     

On two-dimensional parametric variational problems

Let be a two-dimensional parametric variational integral the Lagrangian F(x,z) of which is positive definite and elliptic, and suppose that is a closed rectifiable Jordan curve in . We then prove that there is a conformally parametrized minimizer of in the class of surfaces of the type of the disk B which are bounded by . An immediate consequence of this theorem is that the Dirichlet integral and the area functional have the same infima, a result whose proof usually requires a Lichtenstein-type mapping theorem or else Morrey's lemma on -conformal mappings. In addition we show that the minimizer of is H?lder continuous in B, and even in if satisfies a chord-arc condition. In Section 1 it is described how our results are related to classical investigations, in particular to the work of Morrey. Without difficulty our approach can be carried over to two-dimensional surfaces of codimension greater than one. Received July 20, 1998 / Accepted October 23, 1998     

Sharp discrete inequalities and applications to discrete variational problems

Some new discrete inequalities involving monotonic or convex functions are obtained. While these are interesting inequalities in their own right, they can be applied to solving certain types of discrete variational problems effectively.     

Fundamental systems for linear differential equations smoothly depending on a parameter

Homogeneous linear differential equations in Banach spaces are considered, which depend smoothly on a parameter. A fundamental system of solutions is generated by eigenvalues and eigenvectors of the corresponding operator pencil. The dependence on the parameter of these solutions in canonical form is, in general, not smooth because of branching eigenvalues. This means, the eigenvalues, their number and multiplicities may change in a nonsmooth way with respect to the parameter. We construct a new fundamental system depending smoothly on the parameter, wich can be represented by linear combinations of the solutions in canonical form.     

Controllability of nonlinear fractional dynamical systems

In this paper we establish a set of sufficient conditions for the controllability of nonlinear fractional dynamical systems. The results are obtained by using the recently derived formula for solution representation of systems of fractional differential equations and the application of the Schauder fixed point theorem. Examples are provided to illustrate the results.     

Numerical method for solving diffusion-wave phenomena

We find solutions for the diffusion-wave problem in 1D with n-term time fractional derivatives whose orders belong to the intervals (0,1),(1,2) and (0,2) respectively, using the method of the approximation of the convolution by Laguerre polynomials in the space of tempered distributions. This method transfers the diffusion-wave problem into the corresponding infinite system of linear algebraic equations through the coefficients, which are uniquely solvable under some relations between the coefficients with index zero.The method is applicable for nonlinear problems too.     

Computing ODE symmetries as abnormal variational symmetries

We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [P.D.F. Gouveia, D.F.M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control, Comput. Methods Appl. Math. 5 (4) (2005) 387-409], and is based on the resolution of a first order linear PDE that arises as a necessary and sufficient condition of invariance for abnormal optimal control problems. A computer algebra procedure is developed, which permits one to obtain ODE symmetries by the proposed method. Examples are given, and results compared with those obtained by previous available methods.