In this research, we obtain the sufficient conditions that guarantee that the set of solutions for an impulsive fractional differential inclusion involving a w-weighted $\psi$-Hilfer fractional derivative, $D_{0,t}^{\sigma ,v,\psi ,w},$of order $\mu \in (1,2),$ in infinite dimensional Banach spaces that are not empty and compact. We demonstrate the exact relation between a differential equation involving $D_{0,t}^{\sigma ,v,\psi ,w}$ of order $\mu$ $\in (1,2)$ in the presence of non-instantaneous impulses and its corresponding fractional integral equation. Then, we derive the formula for the solution for the considered problem. The desired results are achieved using the properties of the w-weighted $\psi$-Hilfer fractional derivative and appropriate fixed-point theorems for multivalued functions. Since the operator $D_{0,t}^{\sigma ,v,\psi ,w}$ includes many types of well-known fractional differential operators, our results generalize several results recently published in the literature. We give an example that illustrates and supports our theoretical results.

In this paper, we introduce and implement the optimal auxiliary function method to solve a system of fractional-order Whitham–Broer–Kaup equations, a class of nonlinear partial differential equations with broad applications in mathematical physics. This method provides a systematic and efficient approach to finding accurate solutions for complex systems of fractional-order equations. We give a full analysis using tables and figures to demonstrate the reliability and accuracy of our approach. We confirm the effectiveness of our suggested method in solving the considered equations using numerical simulations and comparisons, emphasizing its potential for applications in a variety of scientific and engineering areas.

The knowledge of fractional calculus can be useful in developing models that allow us to better understand and manage some phenomena. In the present paper, we study the topological structure of the mild solution set for a semi-linear differential inclusion containing the $\tau$-Caputo fractional derivative in the presence of non-instantaneous impulses and an infinite delay. We demonstrate that this set is non-empty and an $R_{\delta }$-set. We use a recent result regarding the existence of solutions for $\tau$-Caputo fractional semi-linear differential inclusions. Unlike many results, we do not suppose that the generating semigroup is compact. An illustrative example is given as an application of our results.

In this paper, we prove two results concerning the existence ofS-asymptoticallyω-periodic solutions for non-instantaneous impulsive semilinear differential inclusions of order $1<\alpha <2$1<α<2 and generated by sectorial operators. In the first result, we apply a fixed point theorem for contraction multivalued functions. In the second result, we use a compactness criterion in the space of bounded piecewise continuous functions defined on the unbounded interval $J=[0,\infty )$J=[0,∞) . We adopt the fractional derivative in the sense of the Caputo derivative. We provide three examples illustrating how the results can be applied.

This study introduces two innovative methods, the new transform iteration method and the residual power series transform method, to solve fractional nonlinear system Korteweg–de Vries (KdV) equations. These equations, fundamental in describing nonlinear wave phenomena, present complexities due to the involvement of fractional derivatives. In demonstrating the application of the new transform iteration method and the residual power series transform method, computational analyses showcase their efficiency and accuracy in computing solutions for fractional nonlinear system KdV equations. Tables and figures accompanying this research present the obtained solutions, highlighting the superior performance of the new transform iteration method and the residual power series transform method compared to existing methods. The results underscore the efficacy of these novel methods in handling complex nonlinear equations involving fractional derivatives, suggesting their potential for broader applicability in similar mathematical problems.

In this work, we study the evolution of relativistic gravitational collapse of hyperbolically symmetric stellar configurations through the lens of a gravitational complexity factor. This factor is originally a structure scalar function emerging from the $$3+1$$ 3 + 1 decomposition of the Riemann tensor into its constituent parts. Perfect-fluid relativistic stellar objects endowed with uniform density exhibit a zero value of the gravitational complexity factor. The degree of complexity factor can thus provide further insights into how the non-uniformity of energy density develops as the collapse evolves. The complete set of governing equations is presented and subsequently employed to evaluate the dynamics of the non-static hyperbolic stellar configuration in contrast with the standard spherically symmetric case. Furthermore, we highlight how the concept of complexity may be used to distinguish between competing models and to support their compliance with physical acceptability requirements. This study demonstrates the key role of the complexity-determining scalar function for modeling non-static dissipative stars with hyperbolically symmetric anisotropic matter distribution.