A generalized fractional derivative (GFD) definition is proposed in this work. For a differentiable function expanded by a Taylor series, we show that D α D β f t = D α + β f t ; 0 < α ≤ 1 ; 0 < β ≤ 1 . 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 BPM , enhanced homotopy perturbation method EHPM , and conformable derivative CD 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.

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A linear sigma model with logarithmic mesonic potential is proposed for computing nucleon properties. The logarithmic potential is based on some aspects of QCD. The field equations have been solved in mean-field approximation. Obtained results for nucleon properties are good in comparison with the original model and agree with measured data.

Thermodynamics properties of heavy mesons are calculated within the framework of the N-dimensional radial Schr ̈dinger equation. The Cornell potential is extended by including the quadratic potential plus the inverse of quadratic potential. The energy eigenvalues and the corresponding wave functions are calculated in the N-dimensional space using the Nikiforov-Uvarov (NV) method. The obtained results are applied for calculating the mass of spectra of charmonium, bottomonium, b ̅ and c ̅ mesons. The thermodynamics properties of heavy quarkonia such as, the mean energy, specific heat, free energy, and entropy are calculated. The effect of temperature and the dimensionality number on heavy mesons masses and thermodynamics properties is investigated. The obtained results are improved in comparison with other theoretical approaches and in a good agreement with experimental data. We conclude that the present potential well describes thermodynamic properties in the three-dimensional space and also the higher dimensional space.