We consider the relative group presentation P = G, x|r where x = {x} and r = {xg 1 xg 2 xg 3 x −1 g 4 }. We show modulo a small number of exceptional cases exactly when P is aspherical. If H = g −1 1 g 2 , g −1 1 g 3 g 1 , g 4 ≤ G then the exceptional cases occur when H is isomorphic to one of C 5 , C 6 , C 8 or C 2 × C 4 . Mathematics Subject Classification: 20F05, 57M05.

Various scholars have lately employed a wide range of strategies to resolve specific types of symmetrical fractional differential equations. This paper introduces a new implicit finite difference method with variable-order time-fractional Caputo derivative to solve semi-linear initial boundary value problems. Despite its extensive use in other areas, fractional calculus has only recently been applied to physics. This paper aims to find a solution for the fractional diffusion equation using an implicit finite difference scheme, and the results are displayed graphically using MATLAB and the Fourier technique to assess stability. The findings show the unconditional stability of the implicit time-fractional finite difference method. This method employs a variable-order fractional derivative of time, enabling greater flexibility and the ability to tackle more complicated problems.

This study employing a non-standard computational method for a stochastic anthrax epidemic model can enhance accuracy, evaluate control measures, and identify critical factors. The mathematical modeling of an anthrax disease includes the four-compartment of the population as susceptible animals (s), infected animals (i), carcasses animals (c), and grams spores of animals in the environment (a). The continuous model analysis (equilibria, reproduction number, and local stability of equilibria) is studied rigorously. The stochastic model is based on transition probabilities and parametric perturbation techniques. The fundamental properties of the model with standard computational methods such as Euler Maruyama, stochastic Euler, and stochastic Runge Kutta are studied. Unfortunately, these methods are time-dependent and even valid for short-period analysis of the disease. In comparison, the non-standard computational method, like the non-standard finite difference method nonstandard finite difference in the sense of stochastic, is designed for the given model. The non-standard computational method and its dynamical properties (positivity, boundedness, and dynamical consistency) are studied thoroughly. In the end, numerical results of the non-standard computational method with the existing standard computational methods are provided. These benefits contribute to a more comprehensive understanding of anthrax epidemiology and support effective decision-making in public health interventions.

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