Browsing by Author "Herzallah, Mohamed A. E."

Now showing 1 - 11 of 11

Citation - WoS: 15
Citation - Scopus: 18
Existence of A Periodic Mild Solution for A Nonlinear Fractional Differential Equation
(Pergamon-elsevier Science Ltd, 2012) Herzallah, Mohamed A. E.; Baleanu, Dumitru; Baleanu, Dumitru; 56389; Matematik
The aim of this manuscript is to analyze the existence of a periodic mild solution to the problem of the following nonlinear fractional differential equation (R)(0)D(t)(alpha)u(t) - lambda u(t) = f(t, u(t)), u(0) = u(1) = 0, 1 < alpha < 2, lambda is an element of R, where D-R(0)t(alpha), denotes the Riemann-Liouville fractional derivative. We obtained the expressions of the general solution for the linear fractional differential equation by making use of the Laplace and inverse Laplace transforms. By making use of the Banach contraction mapping principle and the Schaefer fixed point theorem, the existence results of one or at least one mild solution for a nonlinear fractional differential equation were given. (C) 2011 Elsevier Ltd. All rights reserved.
Citation - WoS: 41
Citation - Scopus: 44
Fractional Euler-Lagrange equations revisited
(Springer, 2012) Herzallah, Mohamed A. E.; Baleanu, Dumitru; Baleanu, Dumitru; Matematik
This paper presents the necessary and sufficient optimality conditions for the Euler-Lagrange fractional equations of fractional variational problems with determining in which spaces the functional must exist where the functional contains right and left fractional derivatives in the Riemann-Liouville sense and the upper bound of integration less than the upper bound of the interval of the fractional derivative. In order to illustrate our results, one example is presented.
Citation - WoS: 69
Citation - Scopus: 77
Fractional-order Euler-Lagrange equations and formulation of Hamiltonian equations
(Springer, 2009) Herzallah, Mohamed A. E.; Baleanu, Dumitru; Baleanu, Dumitru; Matematik
This paper presents the fractional order Euler-Lagrange equations and the transversality conditions for fractional variational problems with fractional integral and fractional derivatives defined in the sense of Caputo and Riemann-Liouville. A fractional Hamiltonian formulation was developed and some illustrative examples were treated in detail.
Citation - WoS: 10
Citation - Scopus: 13
Fractional-order variational calculus with generalized boundary conditions
(Springer, 2011) Herzallah, Mohamed A. E.; Baleanu, Dumitru; Baleanu, Dumitru; Matematik
This paper presents the necessary and sufficient optimality conditions for fractional variational problems involving the right and the left fractional integrals and fractional derivatives defined in the sense of Riemman-Liouville with a Lagrangian depending on the free end-points. To illustrate our approach, two examples are discussed in detail.
Citation - WoS: 22
Citation - Scopus: 31
Hamilton-Jacobi and fractional like action with time scaling
(Springer, 2011) Herzallah, Mohamed A. E.; Baleanu, Dumitru; Muslih, Sami I.; Baleanu, Dumitru; Rabei, Eqab M.; Matematik
This paper represents the Hamilton-Jacobi formulation for fractional variational problem with fractional like action written as an integration over a time scaling parameter. Also we developed the fractional Hamiltonian formulation for the fractional like action. In all the given calculations, the most popular Riemann-Liouville (RL) and Caputo fractional derivatives are employed. An example illustrates our approach.
Citation - WoS: 5
Citation - Scopus: 5
Mild and strong solutions for a fractional nonlinear neumann boundary value problem
(Eudoxus Press, Llc, 2013) Herzallah, Mohamed A. E.; Baleanu, Dumitru; El-Shahed, Moustafa; Baleanu, Dumitru; 56389; Matematik
In this paper, we investigated the following fractional Neumann boundary value problem D-C(0)alpha+u(t) - lambda u(t) = f (t, u(t)), u'(0) = u'(1) = 0, 1 < alpha < 2, lambda not equal 0, where D-C(a+)alpha is the fractional Caputo derivative. We proved the existence of at least one mild solution and we determined when this solution is unique for suitable assumptions on the function f
Citation - WoS: 3
Citation - Scopus: 4
On abstract fractional order telegraph equation
(Asme, 2010) Herzallah, Mohamed A. E.; Baleanu, Dumitru; Baleanu, Dumitru; Matematik
During the last decades, there has been a great deal of interest in fractional differential equations and their applications in various fields of science and engineering. In this paper, we give a new model of the abstract fractional order telegraph equation and we study the existence and uniqueness theorems of the strong and mild solutions as well as the continuation of this solution. To illustrate the obtained results, two examples were analyzed in detail.
On fractional order hybrid differential equations
(2014) Baleanu, Dumitru; Baleanu, Dumitru; 56389; Matematik
We develop the theory of fractional hybrid differential equations with linear and nonlinear perturbations involving the Caputo fractional derivative of order 0 << 1. Using some fixed point theorems we prove the existence of mild solutions for two types of hybrid equations. Examples are given to illustrate the obtained results.
Citation - WoS: 58
Citation - Scopus: 66
On Fractional Order Hybrid Differential Equations
(Hindawi Publishing Corporation, 2014) Herzallah, Mohamed A. E.; Baleanu, Dumitru; Baleanu, Dumitru; 56389; Matematik
We develop the theory of fractional hybrid differential equations with linear and nonlinear perturbations involving the Caputo fractional derivative of order 0 < alpha < 1. Using some fixed point theorems we prove the existence of mild solutions for two types of hybrid equations. Examples are given to illustrate the obtained results.
Citation - WoS: 33
Citation - Scopus: 37
On the fractional-order diffusion-wave process
(Editura Acad Romane, 2010) Herzallah, Mohamed A. E.; Baleanu, Dumitru; El-Sayed, Ahmed M. A.; Baleanu, Dumtru; Matematik
One of the main applications of the fractional calculus, integration and differentiation of arbitrary orders is the modelling of the intermediate physical processes. Here we formulate a more general model which represents the diffusion wave process in all its cases, and give some examples discussing these different cases.
Citation - WoS: 2
Perturbation for fractional-order evolution equation
(Springer, 2010) Herzallah, Mohamed A. E.; Baleanu, Dumitru; El-Sayed, Ahmed M. A.; Baleanu, Dumitru; Matematik
Fractional calculus has gained a lot of importance during the last decades, mainly because it has become a powerful tool in modeling several complex phenomena from various areas of science and engineering. This paper gives a new kind of perturbation of the order of the fractional derivative with a study of the existence and uniqueness of the perturbed fractional-order evolution equation for CD)+alpha-epsilon u(t) = A (C)D(0+)(delta)u(t) + f(t), u(0) = u(o), alpha is an element of (0, 1), and 0 <= epsilon, delta < alpha under the assumption that A is the generator of a bounded C-o-semigroup. The continuation of our solution in some different cases for alpha, epsilon and delta is discussed, as well as the importance of the obtained results is specified.