Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in various fields, such as physics, mechanics, aerodynamics, chemistry, and engineering and biological sciences, involves derivatives of fractional order. Fractional differential equations also provide an excellent tool for the description of memory and he...

Fractional differential equations have been discussed in this study. We utilize the Riemann-Liouville fractional calculus to implement it within the generalization of the well known class of differential equations. The Rayleigh differential equation has been generalized of fractional second order. The existence of periodic and positive outcome is established in a new method. The solution is des...

In this paper, we exhibit two methods to numerically solve the fractional integro differential equations and then proceed to compare the results of their applications on different problems. For this purpose, at first shifted Jacobi polynomials are introduced and then operational matrices of the shifted Jacobi polynomials are stated. Then these equations are solved by two methods: Caputo fractio...

in this paper we apply hybrid functions of general block-pulse‎ ‎functions and legendre polynomials for solving linear and‎ ‎nonlinear multi-order fractional differential equations (fdes)‎. ‎our approach is based on incorporating operational matrices of‎ ‎fdes with hybrid functions that reduces the fdes problems to‎ ‎the solution of algebraic systems‎. ‎error estimate that verifies a‎ ‎converge...

Fractional differential equations are generalizations of integer order differential equations. These generalizations are valuable tools in modelling many processes and phenomena in various fields of engineering, physics, biology and economics. Since a fractional-order derivative is nonlocal and has weakly singular kernels, it provides an excellent instrument for the description of memory and he...

Comparison results of the nonlinear scalar Riemann-Liouville fractional differential equation of order q, 0 < q ≤ 1, are presented without requiring Hölder continuity assumption. Monotone method is developed for finite systems of fractional differential equations of order q, using coupled upper and lower solutions. Existence of minimal and maximal solutions of the nonlinear fractional different...

This paper is a result of comparison of some available numerical methods for solving nonlinear fractional order ordinary differential equations. These methods are compared according to their computational complexity, convergence rate, and approximation error. The present study shows that when these methods are applied to nonlinear differential equations of fractional order, they have different ...

Regular and singular perturbations of fractional ordinary differential equations (fODEs) are considered. This is likely the first attempt to describe these problems. Similarities and differences between these cases and the analogous ones for classical (integer-order) differential equations are pointed out. Examples, including the celebrated Bagley-Torvik equations are discussed. Asymptotic-nume...

In this paper, operational matrix method based upon the Bernstein polynomials is proposed to solve the variable order fractional integro-differential equations in the Caputo derivative sense. We derive the Bernstein polynomials operational matrix of fractional order integration and introduce the product operational matrix of Bernstein polynomials. A truncated the Bernstein polynomials series to...

We prove that the electromagnetic fields in dielectric media whose susceptibility follows a fractional powerlaw dependence in a wide frequency range can be described by differential equations with time derivatives of noninteger order. We obtain fractional integro-differential equations for electromagnetic waves in a dielectric. The electromagnetic fields in dielectrics demonstrate a fractional ...