In this paper, we introduce a general definition of the fractal derivative with respect to a function $\psi$, in the context of the order $0<\alpha \le 1$ and the function $\psi (\mathrm{\Theta })$. This novel definition generalizes the classical fractal derivative, which is recovered when $\psi (\mathrm{\Theta })=\mathrm{\Theta }$, as described in previous works by Chen et al. ^[1,2]. We explored key properties of the $\psi$-fractal derivative, including the $\psi$-fractal Laplace transform, which provides a powerful tool for solving complex differential equations in fractal domains. We also derived a generalized $\psi$-chain rule, extending classical calculus into the fractal domain, and presented fundamental operations related to this unique derivative. We give some applications.

In this manuscript, we formulated Hermite–Hadamard inequalities for convex functions by employing cotangent integrals. Additionally, we extended these Hermite–Hadamard inequalities to encompass cotangent integrals and give the application.

This article presents the cotangent fractional Gronwall inequality, a novel understanding of the Gronwall inequality within the context of the cotangent fractional derivative. We furnish an explanation of the cotangent fractional derivative and emphasize a selection of its distinct characteristics before delving into the primary findings. We present the cotangent fractional Gronwall inequality (Lemma 3.1) and a Corollary 3.2 using the Mittag-Leffler function, we establish singularity and compute an upper limit employing the Mittag-Leffler function for solutions in a nonlinear delayed cotangent fractional system, illustrating its practical utility. To underscore the real-world relevance of the theory, a tangible instance is given.

This paper introduces a numerical method based on series polynomials and collocation techniques for the solution of first-order linear matrix differential equations. The proposed framework reformulates the original problem into a system of algebraic equations through structured matrix operations, including the use of Kronecker products. A rigorous error analysis is conducted to establish the accuracy and stability of the methods. Comprehensive numerical experiments are presented, comparing the performance of the series-based collocation approach with the Bernstein polynomial method. The results demonstrate notable improvements in accuracy, particularly for higher approximation orders, thereby validating the theoretical findings and confirming the superior precision of the proposed series-based techniques.