- Open Access
Particular solutions of a certain class of associated Cauchy-Euler fractional partial differential equations via fractional calculus
© Lin et al.; licensee Springer 2013
- Received: 30 January 2013
- Accepted: 1 May 2013
- Published: 16 May 2013
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, B, C, M, N, α and β are arbitrary constants.
MSC: 26A33, 33C10, 34A05.
- fractional calculus
- differential equation
- partial differential equation
- generalized Leibniz rule
- analytic function
- index law
- linearity property
- principle value
- initial and boundary value
Dedicated to Professor Hari M Srivastava.
The subject of fractional calculus (that is, derivatives and integrals of any real or complex order) has gained importance and popularity during the past two decades or so, due mainly to its demonstrated applications in numerous seemingly diverse fields of science and engineering (cf. [1–15]). By applying the following definition of a fractional differential (that is, fractional derivative and fractional integral) of order , many authors have obtained particular solutions of a number of families of homogeneous (as well as nonhomogeneous) linear fractional differ-integral equations.
In this paper, we present a direct way to obtain explicit solutions of such types of the associated Cauchy-Euler fractional partial differential equation with initial and boundary values. The results are a coincidence that the solutions are obtained by the methods applying the Laplace transform with the residue theorem. In this paper, we present some useful definitions and preliminaries for the paper as follows.
First of all, we find it is worthwhile to recall here the following useful lemmas and properties associated with the fractional differ-integration which is defined above.
Lemma 1.1 (Linearity property)
for any constants and .
Lemma 1.2 (Index law)
Lemma 1.3 (Generalized Leibniz rule)
where is the ordinary derivative of of order n (), it being tacitly assumed (for simplicity) that is the polynomial part (if any) of the product .
Lemma 1.4 (Cauchy’s residue theorem)
Proof The proofs between ‘ν is not an integer’ and ‘ν is an integer’ are not coincident, so we mention the proof as follows.
for since .
Therefore we have Property 1.1 for arbitrary . □
where denotes the ordinary derivative of order n and Γ is the gamma function.
with , , and A (≠0), B, C, M, N are constants, has its solutions of the form given by
when the discriminant ;
when the discriminant , and the roots , of Equation (2.5) are repeated; that is, ;
when the discriminant , and , are the conjugate pair roots of Equation (2.5).
There are three different cases to be considered, depending on whether the roots of this quadratic equation (2.5) are distinct real roots, equal real roots (repeated real roots), or complex roots (roots appear as a conjugate pair). The three cases are due to the discriminant of the coefficients .
• Case I: Distinct real roots (when ).
where () are constants.
• Case II: Repeated real roots (when ).
where () are constants.
• Case III: Conjugate complex roots (when ).
where () are constants.
forms a fundamental solution, where , and λ are constants. □
when the discriminant , and , are the conjugate pair roots of Equation (2.10).
The analysis of three cases is similar to Theorem 2.1, we can obtain each solution of the forms as follows:
with , two distinct real roots,
with , repeated real roots, and
with the conjugate complex roots.
Remark The constant λ in Equations (2.2) and (2.7) can be solved directly by constant initial value and constant boundary values (or by the numerical methods).
when the discriminant , and , are the conjugate pair roots of Equation (2.5) with .
with , , and A (≠0), B, C, M are constants, has its solutions of the form given by
when the discriminant ;
when the discriminant , and the roots , of Equation (2.10) with are repeated; that is, ;
when the discriminant , and , are the conjugate pair roots of Equation (2.10) with .
where , and λ are constants.
by taking , , , , and in Theorem 2.1. □
where the discriminant .
The analysis of the case is similar to Theorem 2.2. □
That is, .
Thus, , .
The solution obtained by the method of Laplace transform and the residue theorem is a coincidence, which is our result above. □
The authors are deeply appreciative of the comments and suggestions offered by the referees for improving the quality and rigor of this paper. 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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