4,566 research outputs found
Fractional differential equations solved by using Mellin transform
In this paper, the solution of the multi-order differential equations, by
using Mellin Transform, is proposed. It is shown that the problem related to
the shift of the real part of the argument of the transformed function, arising
when the Mellin integral operates on the fractional derivatives, may be
overcame. Then, the solution may be found for any fractional differential
equation involving multi-order fractional derivatives (or integrals). The
solution is found in the Mellin domain, by solving a linear set of algebraic
equations, whose inverse transform gives the solution of the fractional
differential equation at hands.Comment: 19 pages, 2 figure
Fractional differential equations and numerical methods
The increasing use of Fractional Calculus demands more accurate arid efficient methods for the numerical solution of fractional differential equations. We introduce the concepts of Fractional Calculus and give the definitions of fractional integrals and derivatives in the Riemann-Liouville and Caputo forms. We explore three existing Numerical Methods of solution of Fractional Differential Equations. 1. Diethelm's Backward Difference Form (BDF) method. 2. Lubich's Convolution Quadrature method. 3. Luchko and Diethelm's Operational Calculus (using the Mittag-Lefner function) method. We present useful recursive expressions we developed to compute the Taylor Series coefficients in the Operational Calculus method. These expressions are used in the calculation of the convolution and starting weights. We compare their accuracy and performance of the numerical methods, and conclude that the more complex methods produce the more accurate results
The Variable-Order Fractional Calculus of Variations
This book intends to deepen the study of the fractional calculus, giving
special emphasis to variable-order operators. It is organized in two parts, as
follows. In the first part, we review the basic concepts of fractional calculus
(Chapter 1) and of the fractional calculus of variations (Chapter 2). In
Chapter 1, we start with a brief overview about fractional calculus and an
introduction to the theory of some special functions in fractional calculus.
Then, we recall several fractional operators (integrals and derivatives)
definitions and some properties of the considered fractional derivatives and
integrals are introduced. In the end of this chapter, we review integration by
parts formulas for different operators. Chapter 2 presents a short introduction
to the classical calculus of variations and review different variational
problems, like the isoperimetric problems or problems with variable endpoints.
In the end of this chapter, we introduce the theory of the fractional calculus
of variations and some fractional variational problems with variable-order. In
the second part, we systematize some new recent results on variable-order
fractional calculus of (Tavares, Almeida and Torres, 2015, 2016, 2017, 2018).
In Chapter 3, considering three types of fractional Caputo derivatives of
variable-order, we present new approximation formulas for those fractional
derivatives and prove upper bound formulas for the errors. In Chapter 4, we
introduce the combined Caputo fractional derivative of variable-order and
corresponding higher-order operators. Some properties are also given. Then, we
prove fractional Euler-Lagrange equations for several types of fractional
problems of the calculus of variations, with or without constraints.Comment: The final authenticated version of this preprint is available online
as a SpringerBrief in Applied Sciences and Technology at
[https://doi.org/10.1007/978-3-319-94006-9]. In this version some typos,
detected by the authors while reading the galley proofs, were corrected,
SpringerBriefs in Applied Sciences and Technology, Springer, Cham, 201
Cauchy Problem for Fractional Ricatti Differential Equations Type with Alpha Order Caputo Fractional Derivatives
In this paper, we investigate solution of the fractional Ricatti differential equations (FRDEs) with alpha order Caputo fractional derivatives. In fact, FRDEs are analogous of the Ricatti ordinary differential equations. The multi power series method is used to obtain a useful formula that is implemented to find an explicit solution of Cauchy problem for FRDEs without solving any integral. This formula is explicit and easy to compute by using Maple software to get explicit solution. Also, it is shown that the proposed formula can be used to solve the Cauchy problem for Ricatti ordinary differential equations
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