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Renewal Processes of Mittag-Leffler and Wright Type

Mainardi, Francesco, Gorenflo, Rudolf, Vivoli, Alessandro (2005)

Fractional Calculus and Applied Analysis

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...

Robust fractional adaptive control based on the strictly Positive Realness Condition

Samir Ladaci, Abdelfatah Charef, Jean Jacques Loiseau (2009)

International Journal of Applied Mathematics and Computer Science

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...

Set-valued fractional order differential equations in the space of summable functions

Hussein A.H. Salem (2008)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper, we study the existence of integrable solutions for the set-valued differential equation of fractional type ( D α ₙ - ∑ i = 1 n - 1 a i D α i ) x ( t ) ∈ F ( t , x ( φ ( t ) ) ) , a.e. on (0,1), I 1 - α ₙ x ( 0 ) = c , 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

Diethelm, Kai (2007)

Fractional Calculus and Applied Analysis

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.

Solution of Space-Time Fractional Schrödinger Equation Occurring in Quantum Mechanics

Saxena, R., Saxena, Ravi, Kalla, S. (2010)

Fractional Calculus and Applied Analysis

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...

Solvability of an Infinite System of Singular Integral Equations

El Borai, Mahmoud M., Abbas, Mohamed I. (2007)

Serdica Mathematical Journal

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.

Solving Fractional Diffusion-Wave Equations Using a New Iterative Method

Daftardar-Gejji, Varsha, Bhalekar, Sachin (2008)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 26A33, 31B10In the present paper a New Iterative Method [1] has been employed to find solutions of linear and non-linear fractional diffusion-wave equations. Illustrative examples are solved to demonstrate the efficiency of the method.* This work has partially been supported by the grant F. No. 31-82/2005(SR) from the University Grants Commission, N. Delhi, India.

Currently displaying 281 – 300 of 360