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Open Access Research

Particular solutions of a certain class of associated Cauchy-Euler fractional partial differential equations via fractional calculus

Shy-Der Lin*, Chia-Hung Lu and Shou-Mei Su

Author Affiliations

Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, 32023, Taiwan, R.O.C

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Boundary Value Problems 2013, 2013:126  doi:10.1186/1687-2770-2013-126

Published: 16 May 2013

Abstract

In recent years, various operators of fractional calculus (that is, calculus of integrals and derivatives of arbitrary real or complex orders) have been investigated and applied in many remarkably diverse fields of science and engineering. Many authors have demonstrated the usefulness of fractional calculus in the derivation of particular solutions of a number of linear ordinary and partial differential equations of the second and higher orders. The purpose of this paper is to present a certain class of the explicit particular solutions of the associated Cauchy-Euler fractional partial differential equation of arbitrary real or complex orders and their applications as follows:

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M2">View MathML</a>; A, B, C, M, N, α and β are arbitrary constants.

MSC: 26A33, 33C10, 34A05.

Keywords:
fractional calculus; differential equation; partial differential equation; generalized Leibniz rule; analytic function; index law; linearity property; principle value; initial and boundary value