Mohamed A. E. Herzallah

Fractional-Order Variational Calculus with Generalized Boundary Conditions

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.

On Fractional Order Hybrid Differential Equations

We develop the theory of fractional hybrid differential equations with linear and nonlinear perturbations involving the Caputo fractional derivative of order . 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.

On Abstract Fractional Order Telegraph Equation

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.

Continuation and maximal regularity of an arbitrary (fractional) order evolutionary integro-differential equation

The maximal regularity and continuation of the solution (and its fractional-order derivative) of the Cauchy problem of the non-homogeneous fractional order evolution equation D α u(t) = Au(t) + f(t), α ∈ (0,1) have been studied in [A.M.A. El-Sayed and Mohamed. A.E. Herzallah (2004). Continuation and maximal regularity of fractional-order evolution equation. Journal of Mathematical Analysis and Applications, 296 (1), 340–350.]. Here we study the maximal regularity, continuation of the solution (and its fractional derivative), and some other properties of the solution of the Cauchy problem of the non-homogeneous arbitrary (fractional) order evolutionary integral equation.

Comments on “Different methods for (3+1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation” [Comput. Math. Appl. 71(6) (2016) 1259–1269]

Abstract In this note, we show that the results in the above paper are inaccurate due to using the complex transform method for modified Riemann–Liouville fractional derivative which is an incorrect method.

Comments on Effect of fractional parameter on plane waves of generalized magneto-thermoelastic diffusion with reference temperature-dependent elastic medium [Comput. Math. Appl. 65 (2013) 1104-1118]

1. Main results In the above paper, the authors used Caputo fractional derivative (see [1,2]) to give their model and with using normal mode analysis they obtained the required solution, then they gave some particular cases and some applications. But really this solution depends on an incorrect property of Caputo fractional derivative. The authors used some calculations to obtain (37) and (38) from (10) and (12) then by using the relations θ(x, z, t) = θ(z) exp(wt + imx), C(x, z, t) = C(z) exp(wt + imx) (1) they obtained (42) and (43). It is obvious from (42) and (43)with the values of g3−g9 andw1 that the authors used the relation ∂ ∂tα exp(wt + iax) = w exp(wt + iax) (2) then the other results depended on the obtained system (40)–(43). In the followingwe prove that (2) is incorrect, andwhere all the following results in [3] depended on the system (40)–(43) which is resulted from this wrong relation, we obtain that the solutions and results in [3] are incorrect. Proof. From the definition of the Caputo fractional derivative we get that d dtα ewt = 1 Γ (1 − α)  t 0 (t − s) d ds  ∞ 