Volume 18, Issue 5

In this paper we analyze non-linear multi-term fractional delay differential Equation denotes the Caputo fractional derivative of order α. The Schaefer fixed point theorem and Banach contraction principle are used to investigate the existence and uniqueness of solutions for above equation with periodic/ anti-periodic boundary conditions.

In this paper we investigate the boundedness of fractional integral operators on predictable martingale Hardy spaces with variable exponents defined on a probability space. More precisely, let f = (f n ) n≥0 be a martingale on probability space (Ω,F, ℙ), and let I α f, α > 0 be the fractional integral operator associated with f. Under some reasonable assumptions, it is proved that I α f is bounded on martingale Hardy spaces with variable exponents. Our method is an extension of atomic decomposition theorem to predicable martingale Hardy spaces of variable exponents.

We derive criteria governing two-weight estimates for multilinear fractional integrals and appropriate maximal functions. The two and one weight problems for multi(sub)linear strong fractional maximal operators are also studied; in particular, we derive necessary and sufficient conditions guaranteeing the trace type inequality for this operator. We also establish the Fefferman-Stein type inequality, and obtain one-weight criteria when a weight function is of product type. As a consequence, appropriate results for multilinear Riesz potential operator with product kernels follow.

In this paper some fractional analogues of classical Pearson differential equations are explicitly solved. Limit transitions between the solutions are analyzed, providing a generalization of well-known transitions between the beta and gamma, and between the gamma and normal distributions. Finally, quasi-polynomials orthogonal with respect to these fractional analogues of the classical distributions are introduced, and some conjectures about their zeros are posed.

The fractional Schrödinger equation has recently received substantial attention. We generalize the fractional Schrödinger equation to the Hilfer time derivative and the Caputo space derivative, and solve this equation for an infinite potential by using the Adomian decomposition method. The infinite domain solution of the space-time fractional Schrödinger equation in the case of Riesz space fractional derivative is obtained in terms of the Fox H-functions. We interpret our results for the fractional Schrödinger equation by introducing a complex effective potential in the standard Schrödinger equation, which can be used to describe quantum transport in quantum dots.

We propose a new definition for the fractional order Bessel operator which is based on Liouville’s first formula for a fractional derivative. From this definition we construct an operational calculus and solve fractional order Bessel equations.

We consider fractional relaxation and fractional oscillation equations involving Erdélyi-Kober integrals. In terms of the Riemann-Liouville integrals, the equations we analyze can be understood as equations with time-varying coefficients. Replacing the Riemann-Liouville integrals with Erdélyi-Kober-type integrals in certain fractional oscillation models, we obtain some more general integro-differential equations. The corresponding Cauchy-type problems can be solved numerically, and, in some cases analytically, in terms of the Saigo-Kilbas Mittag-Leffler functions. The numerical results are obtained by a treatment similar to that developed by K. Diethelm and N.J. Ford to solve the Bagley-Torvik equation. Novel results about the numerical approach to the fractional damped oscillator equation with time-varying coefficients are also presented.

The axial vibrations of a viscoelastic rod with a body attached to its end are investigated. The problem is modelled by the constitutive equations with fractional derivatives as well as with the perturbations involving a bounded noise and a white noise process. The weak solutions for the equations given below in two cases of constitutive equations as well as their stochastic moments are determined.

We obtain two-weighted estimates for the Hardy type operators from local generalized Morrey spaces L p,φ loc (X,w 1 ) defined on an arbitrary underlying quasi-metric measure space (X, μ, ϱ) with the growth condition, to L q,ψ loc (X,w 2 ), where w 1 = w 1 [ϱ(x, x 0 )], x 0 ∈ X is a weight of radial type, while w 2 = w 2 (x) may be an arbitrary weight. The proof allows to simultaneously treat a similar boundedness V L p,φ loc (X,w 1 ) → V L q,ψ loc (X,w 2 ) for vanishing Morrey spaces. We obtain sufficient conditions for such estimates in terms of some integral inequalities imposed on φ, ψ and w 1 .w 2 . We also specially treat the one weight case where w 2 (x) is also of radial type. We do not impose doubling condition on the measure μ, but base our result on the growth condition. The obtained results show the explicit dependence of the mapping properties of the Hardy type operators on the fractional dimension of the set (X, μ, ϱ). An application to spherical Hardy type operators is also given.

We discuss the existence and uniqueness of solutions to the periodic fractional problem u′ = A c D α u + f(t, u), u(0) = u(T). Here, α ∈ (0, 1), A ∈ ℝ, f is continuous and c D stands for the Caputo fractional derivative. Existence results are proved by a combination of the maximum principle with the Leray-Schauder degree method.

The present article proposes a simple tuning technique aimed to produce a robust non-integer order PID controller exhibiting iso-damping property during reparameterization of a plant. The required robustness property is achieved by letting the fractional PID control system to imitate the dynamics of a reference system having Bode’s ideal transfer function in its forward path. The objective of designing robust controller by tracking the dynamics of reference control system is defined mathematically as an H ∞ - optimal control problem. Fractional differential systems are transformed into algebraic equations by the use of triangular strip operational matrices. The H ∞ -optimal control problem is then changed to an ∞-norm minimization of a parameter (K c ,K I ,K d , λ, μ) varying square matrix. Global optimization techniques, Luus-Jaakola direct search and particle swarm optimization are employed to find the optimum values of fractional PID controller parameters. The proposed method of control system design is implemented in heating furnace temperature control, automatic voltage regulator system and some integer and fractional order process models. Fractional PI, fractional PD and various versions of fractional PID controllers are designed as optimal controllers using the triangular strip operational matrix based control design method.