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This paper proposes novel analytical solutions of the mass-spring-damper systems described by certain generalized fractional derivatives. The Liouville–Caputo left generalized fractional derivative and the left generalized fractional derivative were used. The behaviors of the analytical solutions of the mass-spring-damper systems described by the left generalized fractional derivative and the Liouville–Caputo left generalized fractional derivative were represented graphically and the effect of the orders of the fractional derivatives analyzed. We finish by analyzing the global asymptotic stability and the converging-input-converging-state of the unforced mass-damper system, the unforced spring-damper, the spring-damper system, and the mass-damper system.

Here we present weighted fractional Iyengar type inequalities with respect to L p norms, with 1 ≤ p ≤ ∞ . Our employed fractional calculus is of Caputo type defined with respect to another function. Our results provide quantitative estimates for the approximation of the Lebesgue–Stieljes integral of a function, based on its values over a finite set of points including at the endpoints of its interval of definition. Our method relies on the right and left generalized fractional Taylor’s formulae. The iterated generalized fractional derivatives case is also studied. We give applications at the end.

This study introduces some remarks on generalized fractional integral and differential operators, which generalize some familiar fractional integral and derivative operators, with respect to a given function. We briefly explain how to formulate representations of generalized fractional operators. Then, mainly, we propose a predictor–corrector algorithm for the numerical simulation of initial value problems involving generalized Caputo-type fractional derivatives with respect to another function. Numerical solutions of some generalized Caputo-type fractional derivative models have been introduced to demonstrate the applicability and efficiency of the presented algorithm. The proposed algorithm is expected to be widely used and utilized in the field of simulating fractional-order models.

In this study, Sardar sub-equation method is employed to obtain the solitary wave solutions for generalized fractional Tzitzéica type equations. By utilizing this method, novel solutions are derived for Tzitzéica, Tzitzéica Dodd–Bullough–Mikhailov and Tzitzéica–Dodd–Bullough equations in terms of fractional derivatives. The benefit of proposed method is that it offers a wide variety of soliton solutions, consisting of dark, bright, singular, periodic singular as well as combined dark-singular and combined dark–bright solitons. These solutions provide valuable insights into the intricate dynamics of generalized fractional Tzitzéica type evolution equations. The fractional wave and Painlevé transformation are utilized to transform the governing equation. The outcomes of our study are presented in a manner that highlights the practical utility and adeptness of fractional derivatives, along with the effectiveness of the proposed approach, in addressing a spectrum of nonlinear equations. Our findings reveal that the proposed method presents a comprehensive and efficient approach to explore exact solitary wave solutions for generalized fractional Tzitzeica type evolution equations.

In this paper, we obtain the Gronwall type inequality for generalized fractional operators unifying Riemann-Liouville and Hadamard fractional operators. We apply this inequality to the dependence of the solution of differential equations, involving generalized fractional derivatives, on both the order and the initial conditions. More properties for the generalized fractional operators are formulated and the solutions of initial value problems in certain new weighted spaces of functions are established as well.

In this paper, the generalized type k -fractional derivatives are introduced and their semi-group, commutative and inverse properties are presented. These derivatives can be reduced to other fractional derivatives by substituting the values of the parameters involved. The Mellin transform of generalized Caputo type k -fractional derivative is also found.

In this article, we consider two inverse problems with a generalized fractional derivative. The first problem, IP1, is to reconstruct the function u based on its value and the value of its fractional derivative in the neighborhood of the final time. We prove the uniqueness of the solution to this problem. Afterwards, we investigate the IP2, which is to reconstruct a source term in an equation that generalizes fractional diffusion and wave equations, given measurements in a neighborhood of final time. The source to be determined depends on time and all space variables. The uniqueness is proved based on the results for IP1. Finally, we derive the explicit solution formulas to the IP1 and IP2 for some particular cases of the generalized fractional derivative.

Background Fractional calculus has gained significant attention due to its applications in modeling complex systems. This paper introduces several fundamental fractional operators, including the Riemann–Liouville, Caputo, and Hadamard operators. From these, a novel α-generalized fractional operator is derived, which plays a crucial role in solving fractional partial differential equations (FPDEs). Methods The α-generalized operator is analytically constructed and its properties are rigorously proven. It is then integrated with the Daftardar–Jafari iterative method (DJM) to solve both linear and nonlinear FPDEs. The convergence of the proposed method is established using the Lipschitz condition. Two benchmark equations—the linear fractional Burger’s equation and the nonlinear fractional Heat-type equation—are used to demonstrate the method’s applicability. MATLAB is employed to implement the numerical schemes and visualize the results. Results The DJM combined with the α-generalized operator yields superior numerical results compared to the Adomian decomposition method. This superiority is evident in both accuracy and convergence, especially for the nonlinear fractional Heat-type equation. Tabulated results and graphical comparisons confirm the effectiveness of the proposed approach across different values of α. Conclusions The integration of the α-generalized operator with the Daftardar–Jafari method provides a powerful tool for solving FPDEs. Its numerical advantages make it a promising alternative to existing methods. The study recommends extending this framework to other fractional operators, iterative schemes, and potential transformation techniques, with applications in quantum physics, fluid dynamics, and beyond.

The quark-gluon plasma analysis relies on the heavy quark potential, which is influenced by the anisotropic plasma parameter ( ξ ) , temperature (t), and baryonic chemical potential (μ). Employing the generalized fractional derivative Nikiforov-Uvarov (GFD-NU) method, we solved the topologically-fractional Schrödinger equation. Two scenarios were explored: the classical model (α = β = 1) and the fractional model (α, β < 1). This allowed us to obtain the binding energy of charmonium ( c c ¯ ) and bottomonium ( b b ¯ ) in the 1p state. The presence of the topological defect leads to a splitting between the np and nd states. While increasing the temperature reduces the binding energy, increasing the anisotropic parameter has the opposite effect. Compared to the classical model, the fractional model yields lower binding energies. Additionally, the binding energy further decreases with increasing topological defect parameter, and the influence of the baryonic chemical potential is negligible. We also obtained the wave function for the p-state of charmonium and bottomonium. Here, increasing the anisotropic parameter shifts the wave function to higher values. Moreover, the wave function is lower in the fractional model compared to the classical model. Increasing the topological defect parameter again increases the wave function, while the baryonic chemical potential has no discernible effect.

In this paper, we propose two new approximation methods on a general mesh for the generalized Caputo fractional derivative of order α ∈ ( 0 , 1 ) . The accuracy of these two methods is shown to be of order ( 3 - α ) which improves some previous work done to date. To demonstrate the accuracy and usefulness of the proposed approximations, we carry out experiment on test examples and apply these approximations to solve generalized fractional sub-diffusion equations. The numerical results indicate that the proposed methods perform well in practice. Our contributions lie in two aspects: (i) we propose high order approximations that work on a general mesh; (ii) we establish the well-posedness of generalized fractional sub-diffusion equations and develop numerical schemes using the new high order approximations.