Fractional Sturm–Liouville problems for Weber fractional derivatives

Abstract

In this paper, we introduce the regular and singular fractional Sturm–Liouville problem (SLP) ${\mathcal{D}}^{\alpha }p\left(x\right){\mathcal{D}}^{\alpha }y+q\left(x\right)y\left(x\right)=\lambda {\omega }_{\alpha }\left(x\right)y\left(x\right),0<\alpha \le 1,$ where the operator ${\mathcal{D}}^{\alpha }$ 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.

KEYWORDS:

• Sturm–Liouville problem
• finite transform
• Weyl fractional derivative
• Weber fractional derivative

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 34B12
• 35R11
• 34A08

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.