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Original Article

Fractional Sturm–Liouville problems for Weber fractional derivatives

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Pages 217-237 | Received 14 Mar 2017, Accepted 21 Nov 2017, Published online: 29 Jan 2018
 

Abstract

In this paper, we introduce the regular and singular fractional Sturm–Liouville problem (SLP) Dαp(x)Dαy+q(x)y(x)=λωα(x)y(x),0<α1, where the operator Dα is the Weber fractional derivative of order α. We show then the eigenvalues of fractional SLP are real and the eigenfunctions corresponding to distinct eigenvalues are orthogonal. We also consider a Jacobi type singular fractional SLP and treat the behaviours of its eigenvalues. We further introduce the finite fractional Fourier transform and use this transform for solving the space-fractional diffusion equation problem.

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

Acknowledgements

The authors would like to thank referees for their careful reading and constructive comments. They also thank Professor N. Ahanjideh for reading the last version of manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

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