A generalized fractional derivative (GFD) definition is proposed in this work. For a differentiable function expanded by a Taylor series, we show that . GFD is applied for some functions to investigate that the GFD coincides with the results from Caputo and Riemann–Liouville fractional derivatives. The solutions of the Riccati fractional differential equation are obtained via the GFD. A comparison with the Bernstein polynomial method , enhanced homotopy perturbation method , and conformable derivative is also discussed. Our results show that the proposed definition gives a much better accuracy than the well-known definition of the conformable derivative. Therefore, GFD has advantages in comparison with other related definitions. This work provides a new path for a simple tool for obtaining analytical solutions of many problems in the context of fractional calculus.

A solution of the fractionalN-dimensional radial Schrödinger equation (SE) with the Deng–Fan potential (DFP) is investigated by the generalized fractional Nikiforov–Uvarov (NU) method. The analytical formulas of energy eigenvalues and corresponding eigenfunctions for the DFP are generated. Furthermore, the current results are applied to several diatomic molecules (DMs) for the DFP as well as the shifted Deng–Fan potential (SDFP). For both the DFP and its shifted potential, the effect of the fractional parameter ( $${\delta }$$δ ) on the energy levels of various DMs is examined numerically and graphically. We found that the energy eigenvalues are gradually improved when the fractional parameter increases. The energy spectra of various DMs are also evaluated in three-dimensional space and higher dimensions. It is worthy to note that the energy spectrum raises as the number of dimensions increases. In addition, the dependence of the energy spectra of the DFP and its shifted potential on the reduced mass, screening parameter, equilibrium bond length and rotational and vibrational quantum numbers is illustrated. To validate our findings, the energy levels of the DFP and SDFP are estimated at the classical case ( $${\delta =1}$$δ=1 ) for various DMs and found that they are entirely compatible with earlier studies.Graphical In this study, a new algorithm of the generalized fractional Nikiforov–Uvarov method is employed to obtain new solutions to the fractional N-dimensional radial Schrödinger equation with the Deng–Fan potential. In addition, the results are applied to several diatomic molecules. The impact of the fractional parameter on the energy levels of various diatomic molecules is investigated. We found that the energy of the diatomic molecule is more bounded at lower fractional parameter values than in the classical case.

An analytic solution of theN-dimensional radial Schrödinger equation with the combination of vector and scalar potentials via the Laplace transformation method (LTM) is derived. The current potential is extended to encompass the spin hyperfine, spin-orbit, and tensor interactions. The energy eigenvalues and the corresponding eigenfunctions have been obtained in theN-dimensional space. The present results are employed to study the different properties of the heavy-light mesons (HLM). The masses of the scalar, vector, pseudoscalar, and pseudovector forB,B_s,D, andD_smesons have been calculated in the three-dimensional space. The effect of the dimensional number space is discussed on the masses of the HLM. We observed that the meson mass increases with increasing dimensional space. The decay constants of the pseudoscalar and vector mesons have been computed. In addition, the leptonic decay widths and branching ratio for theB⁺,D⁺, andBs+mesons have been studied. Therefore, the used method with the current potential gives good results which are in good agreement with experimental data and are improved in comparison with recent theoretical studies.

New solutions to the $D$-dimensional Schrödinger equation (SE) have been found using the improved Tietz potential (ITP) in the framework of the generalised fractional derivative (GFD). By employing the generalised fractional Nikiforov-Uvarov (GFNU) technique, the analytical formulas of the energy spectra and wave functions have been expressed in terms of the fractional parameters in $D$-dimensions. Furthermore, the derived solutions have been employed for various diatomic molecules (DMs) that have wide applications in chemical and molecular physics. With the aid of molecular constants, the potential energy curves for the chosen DMs have been generated using the ITP. To verify the efficacy of the method used in this study, the vibrational energy levels for various DMs have been predicted in both the classical and fractional forms. It has been demonstrated that the predicted values for the vibrational energy spectra agree with the observed data far better than in the previous studies. Moreover, it is discovered that the vibrational energies of different DMs calculated in the existence of fractional parameters are better in fitting the experimental Rydberg-Klein-Rees (RKR) data than those calculated in the classical case. In light of this, it can be concluded that fractional order has a considerable impact on the vibrational energy spectrum of DMs. As goodness of fit indices, the average absolute deviation (AAD) and mean absolute percentage deviation (MAPD) have been estimated. Based on the estimated AAD and MAAD results, the ITP is an excellent model for fitting the RKR data for all of the selected DMs. GRAPHICAL ABSTRACT

In this paper, quarkonium dissociation is investigated in an anisotropic plasma in the hot and dense media. For that purpose, the multidimensional Schrödinger equation is solved analytically by Nikiforov-Uvarov (NU) method for the real part of the potential in an anisotropic medium. The binding energy and dissociation temperature are calculated. In comparison with an isotropic medium, the binding energy of quarkonium is enhanced in the presence of an anisotropic medium. The present results show that the dissociation temperature increases with increasing anisotropic parameter for 1S state of the charmonium and bottomonium. We observe that the lower baryonic chemical potential has small effect in both isotropic and anisotropic media. A comparison is presented with other pervious theoretical works.