This article investigates the proper existence conditions and uniqueness results for a class of fuzzy fractional Caputo Volterra-Fredholm integro-differential equations (FFCV-FIDE) with initial conditions. The findings are based on Banach's contraction principle and Schaefer's fixed point theorem. Furthermore, the solution to the posed problem is found using the Adomian decomposition technique (ADT). We support the concept with several examples. The relationship between the upper and lower reduced approximation of the fuzzy solutions was demonstrated numerically and graphically using MATLAB.

This research explores the energy of the prime ideal graph of a commutative ring. The study demonstrates the energy formula of the graph associated with transmission-based matrices. Through research, the findings highlight the distance, Wiener-Hosoya, and distance signless Laplacian matrices. It should be noted that the distance and Wiener-Hosoya energies are always twice their spectral radius, meanwhile, it does not hold for distance signless Laplacian energy.

In this paper, the streamline diffusion weak Galerkin finite element method is proposed and analyzed for solving unsteady time convection diffusion problem in two dimension. The v-elliptic property and the stability of this scheme are proved in terms of some conditions. We derive an error estimate in L2(μ) and H1(μ) norm. Numerical experiments have demonstrated the effectiveness of the method in solving convection propagation problems, and the theoretical analysis has been validated.

This paper outlines a numerical method called the Bernstein operational matrix of derivative (BOMD) of order two and order three with the approach of the Chebyshev collocation technique to solve boundary value problems (BVP). BOMD with suitable collocation points is implemented to solve the BVP using the linear combination of Bernstein polynomials with unknown coefficients to approximate the solutions. The derivatives featured in the problem sets will be approximated by utilizing the matrix. The subsequent examination involves a mathematical analysis of the proposed method, including evaluating its order, absolute error metrics and comparative assessments with alternative methodologies. Four problems involving linear and non-linear equations and systems, along with practical real-world problems, are addressed to assess the reliability of the proposed method.

Effective analysis of water quality is crucial for the well-being of ecosystems and human populations, notably in rivers like Thirumanimutharu. This study advances current methodologies by integrating the picture fuzzy PROMETHEE II approach with the analytic hierarchy process (AHP), offering a robust solution to the challenges posed by data ambiguity and imprecision. Born from the fusion of PROMETHEE II with picture fuzzy sets, this innovative method adeptly represents qualitative criteria and linguistic notions inherent in water quality assessment. A practical case study, utilising real water quality data, demonstrates the superior accuracy and durability of the combined picture fuzzy PROMETHEE II and AHP approach compared to traditional methods. These results significantly contribute to our understanding of water quality assessment, providing essential insights for environmental experts and decision-makers. The approach�s efficacy is particularly highlighted in the management and conservation of water resources in the Thirumanimutharu river, emphasising the need for advanced methodologies in environmental conservation and resource management. This research underscores the importance of adopting sophisticated techniques for more informed decision-making, ultimately contributing to the water resources and the preservation of ecological balance in the Thirumanimutharu river and similar water bodies.

The diamond product of a graph G(V,E) with a graph H(V′,E′) denoted by G⋄H is a graph whose a vertex set V(G⋄H) is a Hom(G,H) and an edge set E(G⋄H)={{f,g}|f,g∈Hom(G,H) and {f(x),g(x)}∈E′, for all x∈V(G)}. A round robin tournament problem involves creating a schedule where each participant plays against every other participant exactly once. This research represents the application of the diamond product of path graph and complete graph to 2n-participants round robin tournament problem. Moreover, the research also represents an algorithm to find a solution of 2n-participants round robin tournament problem.

Let G be a finite group. The non-commuting graph of G is a simple graph Γ(G) whose vertices are elements of G∖Z(G), where Z(G) is the center of G, and two distinct vertices a and b are joint by an edge if ab≠ba. In this paper, we study the non-commuting graph of the group U6n. The independent number, clique and chromatic numbers of the non-commuting graph of the group U6n, Γ(U6n), are determined. Additionally, the resolving polynomial, total eccentricity and independent polynomials of Γ(U6n) are computed. Finally, the detour and eccentric connectivity indices of Γ(U6n) are found.

Let G be a simple graph. A function ϕ:V(G)→{1,2,…,k} a vertex k-labeling which assigns labels to the vertices of G. For any edge xy in G, we define the weight of this edge as wϕ(xy)=ϕ(x)+ϕ(y). If all the edge weights are distinct, then ϕ is termed as an edge irregular k -labeling of G. The smallest possible value of k for which the graph G possesses an edge irregular k-labeling is denoted as the edge irregularity strength of G and is represented as es(G). In this paper, we investigate the edge irregular k-labeling of some classes of grid graphs, namely rhombic graph Rmn, triangular graph Lmn and octagonal graph Omn. As by-product, we obtain their precise value of edge irregularity strength.

The study involves a mathematical analysis of the Brusselator system on a convex bounded three-dimensional open domain, considering Neumann boundary conditions. We establish the global existence and uniqueness of the strong solution for this system. Achieving high regularity for the strong solution requires stringent conditions on the initial data. The study demonstrates the continuous dependence of the solution on the initial conditions.

This paper examines the theoretical, analytical, and approximate solutions of the Caputo fractional Volterra-Fredholm integro-differential equations (FVFIDEs). Utilizing Schaefer's fixed-point theorem, the Banach contraction theorem and the Arzel\`{a}-Ascoli theorem, we establish some conditions that guarantee the existence and uniqueness of the solution. Furthermore, the stability of the solution is proved using the Hyers-Ulam stability and Gronwall-Bellman's inequality. Additionally, the Laplace Adomian decomposition method (LADM) is employed to obtain the approximate solutions for both linear and non-linear FVFIDEs. The method's efficiency is demonstrated through some numerical examples.