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In this paper, we define a fractional singular Sturm-Liouville operator having Coulomb potential of type A x . Our main issue is to investigate the spectral properties for the operator. Furthermore, we prove new results according to the fractional singular Sturm-Liouville problem. MSC: 26A33; 34A08

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In this study, we consider fractional Sturm-Liouville (S-L) problems within non-singular operators. A fractional S-L problem with exponential and Mittag-Leffler kernels is given with different versions in the Riemann-Liouville and Caputo sense. Also, we obtain representation of solutions for S-L problems by the Laplace transform and find analytical solutions of the problems. Finally, we compare the solutions of the problem with these different versions, and we also compare the solutions of the problem with exponential and Mittag-Leffler kernels together by simulation under different potentials, different orders, and different eigenvalues.

In the current study, we investigate and analyze the fractional version of the heating and cooling model for buildings with energy efficiency. We apply the Caputo fractional derivative, Caputo-Fabrizio, and Atangana-Baleanu in the Caputo sense in the analysis and investigation of the governing model. We derive some novel analytical solutions by means of Laplace’s transform. Simulation analysis is carried out in order to shed more light on the physical features of the governing models. To believe the results obtained, the fractional order has been allowed to vary between (0,1], whereupon the physical observations match those obtained in the classical case, but the fractional model has persisted all the memory effects making the model much more suitable when presented in the structure of fractional derivatives.