In this paper, we have proved some common fixed point theorems for a pair of mappings satisfying certain rational contraction condition in the frame work of bicomplex valued metric space (X, d). We have also defined the max function for the partial order £ in bicomplex valued metric d.

This work purposes to establish two small numerical modifications for the Fractional Euler method (FEM) and the Modified Fractional Euler Method (MFEM) to deal with fractional initial value problems. Two such modifications, which are named Improved Modified Fractional Euler Method 1 (IMFEM 1) and Improved Modified Fractional Euler Method 2 (IMFEM 2), endeavor to further enhance FEM and MFEM in terms of attaining more accuracy. By utilizing certain theoretical results, the resultant error bounds of the proposed methods are analyzed and estimated. Several numerical comparisons are carried out to validate the efficiency of our proposed methods.

“Extremely High Frequency (EHF)” and “Very high frequency (VHF)” bands are mainly utilized with “Underwater Wireless Sensor Networks (UWSNs)” for communication purposes. However, due to the mobility of underwater sensors in water because of the water tide, the EHF/VHF signals may attenuate, lose or fade depending on the condition of the water. Therefore, it is a challenging stint of finding the optimal parameters of UWSN topology planning. In this paper, three “Multi-Objective Optimization Algorithms (MOOAs)” have been utilized to mitigate this problem, namely MOSFP, SPEA2 and NSGA-II. This work also intends to minimize path loss. On the other hand, it intends to maximize the power density of the network. Various network configurations, such as distance between sender and receiver, water conductivity and water permeability, are considered to evaluate the proposed objective models. Qualitative and quantitative tests have been conducted to analyze the results. From the analysis of the intersection point of Pareto-front of the objective functions, it is shown that all the algorithms find the optimal distance between transmitter and receiver, which balances the aforementioned maximization and minimization objective functions. This value is 36 m.

This work proposes a new numerical approach for dealing with fractional stochastic differential equations. In particular, a novel three-point fractional formula for approximating the Riemann–Liouville integrator is established, and then it is applied to generate approximate solutions for fractional stochastic differential equations. Such a formula is derived with the use of the generalized Taylor theorem coupled with a recent definition of the definite fractional integral. Our approach is compared with the approximate solution generated by the Euler–Maruyama method and the exact solution for the purpose of verifying our findings.

This paper establishes a novel fractional-order version of a recently expanded form of the Susceptible-Exposed-Infectious-Recovery (SEIR) Mpox model. This model is investigated by means of demonstrating some significant findings connected with the stability analysis and the vaccination impact, as well. In particular, we analyze the fractional-order Mpox model in terms of its invariant region, boundedness of solution, equilibria, basic reproductive number, and its elasticity. In accordance with an effective vaccine, we study the progression and dynamics of the Mpox disease in compliance with various scenarios of the vaccination ratio through the proposed fractional-order Mpox model. Accordingly, several numerical findings of the proposed model are depicted with the use of two numerical methods; the Fractional Euler Method (FEM) and Modified Fractional Euler Method (MFEM). Such findings demonstrate the influence of the fractional-order values coupled with the vaccination rate on the dynamics of the established disease model.