Volume 21, Issue 4

Motivated by recently proposed generalizations of the diffusion-wave equation with the Caputo time fractional derivative of order α ∈ (1, 2), in the present survey paper a class of generalized time-fractional diffusion-wave equations is introduced. Its definition is based on the subordination principle for Volterra integral equations and involves the notion of complete Bernstein function. Various members of this class are surveyed, including the distributed-order time-fractional diffusion-wave equation and equations governing wave propagation in viscoelastic media with completely monotone relaxation moduli.

A one-dimensional fractional one-phase Stefan problem with a temperature boundary condition at the fixed face is considered by using the Riemann–Liouville derivative. This formulation is more convenient than the one given in Roscani and Santillan ( Fract. Calc. Appl. Anal. , 16 , No 4 (2013), 802–815) and Tarzia and Ceretani ( Fract. Calc. Appl. Anal. , 20 , No 2 (2017), 399–421), because it allows us to work with Green’s identities (which does not apply when Caputo derivatives are considered). As a main result, an integral relationship between the temperature and the free boundary is obtained which is equivalent to the fractional Stefan condition. Moreover, an exact solution of similarity type expressed in terms of Wright functions is also given.

In this work we extend a variational method to study the approximate controllability and finite dimensional exact controllability (finite-approximate controllability) for the fractional semilinear evolution equations with nonlocal conditions in Hilbert spaces. Assuming the approximate controllability of the corresponding linear equation we obtain sufficient conditions for the finite-approximate controllability of the fractional semilinear evolution equation under natural conditions. The obtained results are generalization and continuation of the recent results on this issue. Applications to heat equations are treated.

The order of fractional differential equations (FDEs) has been proved to be of great importance in an accurate simulation of the system under study. In this paper, the orders of some classes of linear FDEs are determined by using the asymptotic behaviour of their solutions. Specifically, it is demonstrated that the decay rate of the solutions is influenced by the order of fractional derivatives. Numerical investigations are conducted into the proven formulas.

We investigate the problem of a high-energy source localized within a one-dimensional superdiffusive medium of finite length. The problem is modeled by a fractional diffusion equation with a nonlinear source term. For the boundary conditions, we consider both the case of homogeneous Dirichlet conditions and the case of homogeneous Neumann conditions. We investigate this model to determine whether or not blow-up occurs. It is demonstrated that a blow-up may or may not occur for the Dirichlet case. On the other hand, a blow-up is unavoidable for the Neumann case.

This paper discusses the existence and controllability of a class of fractional order evolution inclusions with time-varying delay. In the weak topology setting we establish the existence of solutions. Then the controllability of this system with a nonlocal condition is established by applying the Glicksberg-Ky Fan fixed point theorem. As an application, nonlocal problems of a fractional reaction-diffusion equation with a discontinuous nonlinear term is examined.

In this paper, we show that the price of an European call option, whose underlying asset price is driven by the space-time fractional diffusion, can be expressed in terms of rapidly convergent double-series. This series formula is obtained from the Mellin-Barnes representation of the option price with help of residue summation in ℂ 2 . We also derive the series representation for the associated risk-neutral factors, obtained by Esscher transform of the space-time fractional Green functions.

We present PLC-based fractional-order controller design for an industrial-oriented water tank volume control application. The system comprises input delay which is a typified characteristic in such industrial process control applications. The particular contribution of this work is on discrete fractional-order PID implementation via PLC and its application to the aforementioned realistic water tank test bed. Stability and robustness properties of fractional-order discrete PID feedback-loops for different approximation methods and orders are also shown. Fractional-order controllers are obtained for a variety of stability margin choices, and benefits of the non-integer-order controllers compared to the integer-order PID control are illustrated via simulation and experimental runs on a realistic test-bed.

This article deals with some existence results for a class of Caputo–Hadamard fractional differential equations. The results are based on the Mönch’s fixed point theorem associated with the technique of measure of noncompactness. Two illustrative examples are presented.

In this article, we propose an implicit-explicit scheme combining with the fast solver in space to solve two-dimensional nonlinear time-fractional subdiffusion equation. The applications of implicit-explicit scheme and fast solver will smartly enhance the computational efficiency. Due to the non-smoothness (or low regularities) of solutions to fractional differential equations, correction terms are introduced in the proposed scheme to improve the accuracy of error. The stability and convergence of the present scheme are also investigated. Numerical examples are carried out to demonstrate the efficiency and applicability of the derived scheme for both linear and nonlinear fractional subdiffusion equations with non-smooth solutions.

In this paper, we consider the application of the finite difference method for a class of novel multi-term time fractional viscoelastic non-Newtonian fluid models. An important contribution of the work is that the new model not only has a multi-term time derivative, of which the fractional order indices range from 0 to 2, but also possesses a special time fractional operator on the spatial derivative that is challenging to approximate. There appears to be no literature reported on the numerical solution of this type of equation. We derive two new different finite difference schemes to approximate the model. Then we establish the stability and convergence analysis of these schemes based on the discrete H 1 norm and prove that their accuracy is of O ( τ + h 2 ) and O ( τ min{3– γ s ,2– α q ,2– β } + h 2 ), respectively. Finally, we verify our methods using two numerical examples and apply the schemes to simulate an unsteady magnetohydrodynamic (MHD) Couette flow of a generalized Oldroyd-B fluid model. Our methods are effective and can be extended to solve other non-Newtonian fluid models such as the generalized Maxwell fluid model, the generalized second grade fluid model and the generalized Burgers fluid model.

A new lumped capacitance model that employs fractional order operators is proposed for use on transient heat conduction problems. Details and implications of the fractional lumped capacitance model’s development and application are discussed. The model is shown to agree with observed heating and cooling temperature profiles of laser aiming paper being heated by a laser under various conditions.

In this paper, we study the existence and stability of Hilfer-type fractional differential equations (dynamic equations) on time scales. We obtain sufficient conditions for existence and uniqueness of solutions by using classical fixed point theorems such as Schauder's fixed point theorem and Banach fixed point theorem. In addition, Ulam stability of the proposed problem is also discussed. As in application, we provide an example to illustrate our main results.