Variational methods for fractional qq-Sturm--Liouville Problems

In this paper, we formulate a regular $q$-fractional Sturm--Liouville problem (qFSLP) which includes the left-sided Riemann--Liouville and the right-sided Caputo $q$-fractional derivatives of the same order $\alpha$, $\alpha\in (0,1)$. We introduce the essential $q$-fractional variational analysis needed in proving the existence of a countable set of real eigenvalues and associated orthogonal eigenfunctions for the regular qFSLP when $\alpha>1/2$ associated with the boundary condition $y(0)=y(a)=0$. A criteria for the first eigenvalue is proved. Examples are included. These results are a generalization of the integer regular $q$-Sturm--Liouville problem introduced by Annaby and Mansour in [1]

A matrix method for fractional Sturm-Liouville problems on bounded domain

A matrix method for the solution of direct fractional Sturm-Liouville problems on bounded domain is proposed where the fractional derivative is defined in the Riesz sense. The scheme is based on the application of the Galerkin spectral method of orthogonal polynomials. The order of convergence of the eigenvalue approximations with respect to the matrix size is studied. Some numerical examples that confirm the theory and prove the competitiveness of the approach are finally presented

Applied Mathematics and Fractional Calculus

In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia