In this paper, an efficient numerical method is considered for solving the fractional wave equation (FWE). The fractional derivative is described in the Caputo sense. The method is based on Laguerre approximations. The properties of Laguerre polynomials are utilized to reduce FWE to a system of ordinary differential equations, which is solved by the finite difference method. An approximate form...

Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. We present in detail the existence of the coupled-solutions using the fixed-point theorem. A detailed analysis of the uniqueness of the coupl...

This paper is concerned with a kind of fractional order systems with Caputo-Hadamard derivative (of order q ∈ C and <(q) ∈ (1, 2)) and not instantaneous impulses. The obtained result uncovers that there exists a general solution for these impulsive systems, which means that the state trajectory of these impulsive systems is non-unique, and it is expounded by a numerical example. c ©2017 All rig...

In this paper, we discuss existence and uniqueness of solutions to nonlinear fractional order ordinary differential equations with boundary conditions in an ordered Banach space. We use the Caputo fractional differential operator and the nonlinearity depends on the fractional derivative of an unknown function. Schauder’s fixed point Theorem is the main tool used here to establish the existence ...

In this paper, we study periodic boundary value problems for a class of linear fractional evolution equations involving the Caputo fractional derivative. Utilizing compactness of the constructed evolution operators and Fredholm alternative theorem, some interesting alternative results for the mild solutions are presented. Periodic motion controllers that are robust to parameter drift are also d...

*Correspondence: [email protected] 1Institute of Mathematics and Computer Science, Jan Długosz University, Czȩstochowa, 42-200, Poland Full list of author information is available at the end of the article Abstract The one-dimensional time-fractional advection-diffusion equation with the Caputo time derivative is considered in a line segment. The fundamental solution to the Dirichlet pro...

This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involv...

The time-fractional diffusion-wave equation is considered in a half-plane. The Caputo fractional derivative of the order 0 < α < 2 is used. Several examples of problems with Dirichlet and Neumann boundary conditions are solved using the Laplace integral transform with respect to time and the Fourier transforms with respect to spatial coordinates. The solution is written in terms of the Mittag-L...

A fractional order HIV model with both virus-tocell and cell-to-cell transmissions and therapy effect is investigated. The conditions for the existence of the equilibria are determined. Local stability analysis of the HIV model is studied by using the fractional Routh-Hurwitz stability conditions. We have generalized the integer LaSalle’s invariant theorem into fractional system and given some ...

In this paper, we consider a nonhomogeneous space-time fractional telegraph equation defined in a bounded space domain, which is obtained from the standard telegraph equation by replacing the firstor second-order time derivative by the Caputo fractional derivative Dt , α > 0; and the Laplacian operator by the fractional Laplacian (−∆) , β ∈ (0, 2]. We discuss and derive the analytical solutions...