Laplace transform for solving some families of fractional differential equations and its applications
© Lin and Lu; licensee Springer 2013
Received: 30 January 2013
Accepted: 27 April 2013
Published: 13 May 2013
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.
Keywordsfractional-order differential equations Riemann-Liouville fractional integrals gamma function Mittag-Leffler function Laplace transform of the fractional derivative Wright function
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.
- 2.The Laplace transform of a function , is defined by
- 6.The Riemann-Liouville fractional derivatives and of order () are defined by(1.1)
- 7.The Pochhammer symbol (or the shifted factorial, since for ) (cf. ) given by
- 8.The binomial coefficients are defined by
(, ; ).
- 2.The Laplace transform of the generalized Wright function is given by
(cf. ), where , (), , for any .
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.
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.
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.
- Gorenflo R, Mainardi F, Srivastava HM: Special functions in fractional relaxation-oscillation and fractional diffusion-wave phenomena. In Proceedings of the Eighth International Colloquium on Differential Equations. Edited by: Bainov D. VSP Publishers, Utrecht; 1998:195-202. Presented at the Eighth International Colloquium on Differential Equations held at Plovdiv, Bulgaria, 18-23 August 1997Google Scholar
- Hilfer R: Applications of Fractional Calculus in Physics. World Scientific, Singapore; 2000.MATHView ArticleGoogle Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.MATHGoogle Scholar
- Mainardi F: The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 1996, 9(6):23-28. 10.1016/0893-9659(96)00089-4MATHMathSciNetView ArticleGoogle Scholar
- Mittag-Leffler G:Sur la nouvelle fonction . C. R. Acad. Sci. 1903, 137: 554-558.Google Scholar
- Tomovski Z, Hilfer R, Srivastava HM: Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions. Integral Transforms Spec. Funct. 2010, 21: 797-814. 10.1080/10652461003675737MATHMathSciNetView ArticleGoogle Scholar
- Erdélyi A, Magnus W, Oberhettinger F, Tricomi F 3. In Higher Transcendental Functions. Krieger, Melbourne; 1981.Google Scholar
- Srivastava HM, Parmar RK, Chopra P: A class of extended fractional derivative operators and associated generating relations involving hypergeometric functions. Axioms 2012, 1(3):238-258. 10.3390/axioms1030238MATHView ArticleGoogle Scholar
- Lin S-D, Chang L-F, Srivastava HM: Some families of series identities and associated fractional differintegral formulas. Integral Transforms Spec. Funct. 2009, 20(10):737-749. 10.1080/10652460902782354MATHMathSciNetView ArticleGoogle Scholar
- Caputo M: Elasticita e Dissipazione. Zanichelli, Bologna; 1969.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.