Recent applications of fractional calculus to science and engineering.
Relations between weighted Orlicz and spaces through fractional integrals
We characterize the class of weights, invariant under dilations, for which a modified fractional integral operator maps weak weighted Orlicz spaces into appropriate weighted versions of the spaces , where . This generalizes known results about boundedness of from weak into Lipschitz spaces for and from weak into . It turns out that the class of weights corresponding to acting on weak for of lower type equal or greater than , is the same as the one solving the problem for weak...
Relativistic wave equations with fractional derivatives and pseudodifferential operators.
Renewal Processes of Mittag-Leffler and Wright Type
2000 MSC: 26A33, 33E12, 33E20, 44A10, 44A35, 60G50, 60J05, 60K05.After sketching the basic principles of renewal theory we discuss the classical Poisson process and offer two other processes, namely the renewal process of Mittag-Leffler type and the renewal process of Wright type, so named by us because special functions of Mittag-Leffler and of Wright type appear in the definition of the relevant waiting times. We compare these three processes with each other, furthermore consider corresponding...
Riesz potential operators and inverses via fractional centred derivatives.
Robust fractional adaptive control based on the strictly Positive Realness Condition
This paper presents a new approach to robust adaptive control, using fractional order systems as parallel feedforward in the adaptation loop. The problem is that adaptive control systems may diverge when confronted with finite sensor and actuator dynamics, or with parasitic disturbances. One of the classical robust adaptive control solutions to these problems makes use of parallel feedforward and simplified adaptive controllers based on the concept of positive realness. The proposed control scheme...
Rozšíření derivace pomocí teorie stability
Semilinear fractional order integro-differential equations with infinite delay in Banach spaces
This paper concerns the existence of mild solutions for fractional order integro-differential equations with infinite delay. Our analysis is based on the technique of Kuratowski’s measure of noncompactness and Mönch’s fixed point theorem. An example to illustrate the applications of main results is given.
Semilinear functional differential equations of fractional order with state-dependent delay.
Series solutions of time-fractional PDEs by homotopy analysis method.
Set-valued fractional order differential equations in the space of summable functions
In this paper, we study the existence of integrable solutions for the set-valued differential equation of fractional type , a.e. on (0,1), , αₙ ∈ (0,1), where F(t,·) is lower semicontinuous from ℝ into ℝ and F(·,·) is measurable. The corresponding single-valued problem will be considered first.
Smoothness Properties of Solutions of Caputo-Type Fractional Differential Equations
Mathematics Subject Classification: 26A33, 34A25, 45D05, 45E10We 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 characterization of the situations where smooth solutions exist. The results can be extended to a class of weakly singular Volterra integral equations.
Solitary wave solutions for a time-fraction generalized Hirota-Satsuma coupled KdV equation by a new analytical technique.
Solution of an extraordinary differential equation by Adomian decomposition method.
Solution of some weight problems for the Riemann-Liouville and Weyl operators.
Solution of Space-Time Fractional Schrödinger Equation Occurring in Quantum Mechanics
Dedicated to Professor A.M. Mathai on the occasion of his 75-th birthday. Mathematics Subject Classi¯cation 2010: 26A33, 44A10, 33C60, 35J10.The object of this article is to present the computational solution of one-dimensional space-time fractional Schrödinger equation occurring in quantum mechanics. The method followed in deriving the solution is that of joint Laplace and Fourier transforms. The solution is derived in a closed and computational form in terms of the H-function. It provides an elegant...
Solution of Volterra-type integro-differential equations with a generalized Lauricella confluent hypergeometric function in the kernels.
Solutions to boundary-value problems for nonlinear differential equations of fractional order.
Solvability of an Infinite System of Singular Integral Equations
2000 Mathematics Subject Classification: 45G15, 26A33, 32A55, 46E15.Schauder's fixed point theorem is used to establish an existence result for an infinite system of singular integral equations in the form: (1) xi(t) = ai(t)+ ∫t0 (t − s)− α (s, x1(s), x2(s), …) ds, where i = 1,2,…, α ∈ (0,1) and t ∈ I = [0,T]. The result obtained is applied to show the solvability of an infinite system of differential equation of fractional orders.
Solvability of nonautonomous fractional integrodifferential equations with infinite delay.