In this article, we solve fractional Integro differential equations (FIDEs) through a well-known technique known as the Chebyshev Pseudospectral method. In the Caputo manner, the fractional derivative is taken. The main advantage of the proposed technique is that it reduces such types of equations to linear or nonlinear algebraic equations. The acquired results demonstrate the accuracy and reliability of the current approach. The results are compared to those obtained by other approaches and the exact solution. Three test problems were used to demonstrate the effectiveness of the proposed technique. For different fractional orders, the results of the proposed technique are plotted. Plotting absolute error figures and comparing results to some existing solutions reveals the accuracy of the proposed technique. The comparison with the exact solution, hybrid Legendre polynomials, and block-pulse functions approach, Reproducing Kernel Hilbert Space method, Haar wavelet method, and Pseudo-operational matrix method confirm that Chebyshev Pseudospectral method is more accurate and straightforward as compared to other methods.

In this manuscript, a (3+1)-dimensional nonlinear evolution equation (EE), which is a generalization of the (3+1)-dimensional Hirota bilinear equation and the (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation, is explored. The Hirota direct method, along with the Cole-Hopf transformation, is utilized to achieve the bilinearization of the considered equation. Via a suitable auxiliary function (AF) and bilinear form, multiple soliton and lump solutions and their interaction with kink solitons are derived. Moreover, the chaotic behavior of the acquired soliton solutions is demonstrated in the light of the Duffing chaotic system. All the results are portrayed using 3D and density plots to show the behavior of the obtained solutions.

This paper presents some oscillation criteria for second-order half-linear dynamic equations defined on unbounded above arbitrary time scales. These criteria offer sufficient conditions for all solutions of the equations to display oscillatory behavior. We investigate both delay and advanced cases of these equations, and our approach encompasses a broader class of dynamic equations than previously considered in the literature. The results of this study not only generalize well-known oscillation criteria used in differential equations but also significantly broaden their applicability to arbitrary time scales. Additionally, we provide illustrative examples to demonstrate the relevance and accuracy of our findings.

For a graph $G$ with edge set $E$, let $d(u)$ denote the degree of a vertex $u$ in $G$. The diminished Sombor (DSO) index of $G$ is defined as $DSO(G)=\sum_{uv\in E}\sqrt{(d(u))^2+(d(v))^2}(d(u)+d(v))^{-1}$. The cyclomatic number of a graph is the smallest number of edges whose removal makes the graph acyclic. A connected graph of maximum degree at most $4$ is known as a molecular graph. The primary motivation of the present study comes from a conjecture concerning the minimum DSO index of fixed-order connected graphs with cyclomatic number $3$, posed in the recent paper [F. Movahedi, I. Gutman, I. Redžepović, B. Furtula, Diminished Sombor index, MATCH Commun. Comput. Chem. 95 (2026) 141--162]. The present paper gives all graphs minimizing the DSO index among all molecular graphs of order $n$ with cyclomatic number $\ell$, provided that $n\ge 2(\ell-1)\ge4$.