Abstract
In many recent works, many authors have demonstrated the usefulness of fractional calculus in the derivation of particular solutions of a significantly large number of linear ordinary and partial differential equations of the second and higher orders. The main objective of the present paper is to show how this simple fractional calculus method to the solutions of some families of fractional differential equations would lead naturally to several interesting consequences, which include (for example) a generalization of the classical Frobenius method. The methodology presented here is based chiefly upon some general theorems on (explicit) particular solutions of some families of fractional differential equations with the Laplace transform and the expansion coefficients of binomial series.
MSC:26A33, 33C10, 34A05.
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1 Introduction, definitions and preliminaries
In the past two decades, the widely investigated subject of fractional calculus has remarkably gained importance and popularity due to its demonstrated applications in numerous diverse fields of science and engineering. These contributions to the fields of science and engineering are based on the mathematical analysis. It covers the widely known classical fields such as Abel’s integral equation and viscoelasticity. Also, including the analysis of feedback amplifiers, capacitor theory, generalized voltage dividers, fractional-order Chua-Hartley systems, electrode-electrolyte interface models, electric conductance of biological systems, fractional-order models of neurons, fitting of experimental data, and the fields of special functions, etc. (see, for example, [1–4]).
In this paper, we apply the Laplace of the fractional derivative and the expansion coefficients of binomial series to derive the explicit solutions to homogeneous fractional differential equations.
We present some useful definitions and preliminaries as follows.
Definitions
where the Euler gamma function is defined by
-
2.
The Laplace transform of a function , is defined by
-
3.
The Mittag-Leffler function (cf. [5, 6]) is defined by
-
4.
The simplest Wright function (cf. [7, 8]) is defined by
-
5.
The general Wright function (cf. [7, 8]) is defined for , complex , and real (; ) by the series
where , , and .
-
6.
The Riemann-Liouville fractional derivatives and of order () are defined by
(1.1)
and
respectively, where means the integral part of .
-
7.
The Pochhammer symbol (or the shifted factorial, since for ) (cf. [9]) given by
-
8.
The binomial coefficients are defined by
where λ and n are integers. Observe that , then
Preliminaries
-
1.
(, ; ).
-
2.
The Laplace transform of the generalized Wright function is given by
-
3.
(cf. [10]), where , (), , for any .
Remark 1.1 By appropriately appealing to Definition 2, it is not difficult to prove Preliminary 3 by the technique of integral transform as follows
The interchange of the order of integration in the above derivation can be justified by applying Fubini’s theorem.
2 Solutions of the fractional differential equations
Throughout this section, we let be such that for some value of the parameter s, the Laplace transform converges.
Theorem 2.1 Let and . Then the fractional differential equation
with the initial conditions and has its solution given by
Proof Applying the Laplace transform (see Preliminary 3) and taking into account, we have
Equation (2.3) yields
since
Thus, from Equation (2.4), we derive the following solution by the inverse Laplace transform to Equation (2.2):
□
Example 2.1 The fractional differential equation of a generalized viscoelastic free damping oscillation (cf. [1])
with the initial conditions and has its solution given by
In particular, if and , then the equation
with the initial conditions and has its solution given by
Theorem 2.2 Let and . Then the fractional differential equation
with the initial conditions and has its solution given by
Proof Applying the Laplace transform (see Preliminary 3) and taking into account, we have
That is,
Equation (2.12) yields
since
Thus, from Equation (2.13), we derive the following solution by the inverse Laplace transform to Equation (2.11):
This solution can be expressed by the Wright function as
□
Example 2.2 If we let , and in Theorem 2.2, then the equation
has a solution
Theorem 2.3 Let and . Then the equation
with the initial condition has its solution given by
Proof Applying the Laplace transform to Equation (2.15), that is,
we have
□
Remark 2.1 If in Equation (2.10), then the equation
with the initial conditions and has its solution given by
Theorem 2.4 A nearly simple harmonic vibration equation (cf. [1])
with the initial conditions and has its solution given by
Proof We complete this proof by putting in Equation (2.17). □
In fact, by applying the Laplace transform to a linear fractional differential equation with the initial conditions, we can easily derive its solutions as the previous forms in this paper.
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
The present investigation was supported, in part, by the National Science Council of the Republic of China under Grant NSC-101-2115-M-033-002.
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S-DL carried out the molecular genetic studies, participated in the sequence alignment and drafted the manuscript. C-HL participated in the sequence alignment.
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Lin, SD., Lu, CH. Laplace transform for solving some families of fractional differential equations and its applications. Adv Differ Equ 2013, 137 (2013). https://doi.org/10.1186/1687-1847-2013-137
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DOI: https://doi.org/10.1186/1687-1847-2013-137