For decades, understanding the dynamics of infectious diseases and halting their spread has been a major focus of mathematical modelling and epidemiology. The stochastic SIRS (susceptible–infectious–recovered–susceptible) reaction–diffusion model is a complicated but crucial computational scheme due to the combination of partial immunity and an incidence rate. Considering the randomness of individual interactions and the spread of illnesses via space, this model is a powerful instrument for studying the spread and evolution of infectious diseases in populations with different immunity levels. A stochastic explicit finite difference scheme is proposed for solving stochastic partial differential equations. The scheme is comprised of predictor–corrector stages. The stability and consistency in the mean square sense are also provided. The scheme is applied to diffusive epidemic models with incidence rates and partial immunity. The proposed scheme with space’s second-order central difference formula solves deterministic and stochastic models. The effect of transmission rate and coefficient of partial immunity on susceptible, infected, and recovered people are also deliberated. The deterministic model is also solved by the existing Euler and non-standard finite difference methods, and it is found that the proposed scheme forms better than the existing non-standard finite difference method. Providing insights into disease dynamics, control tactics, and the influence of immunity, the computational framework for the stochastic SIRS reaction–diffusion model with partial immunity and an incidence rate has broad applications in epidemiology. Public health and disease control ultimately benefit from its application to the study and management of infectious illnesses in various settings.

Engineering and technological research groups are becoming interested in neural network techniques to improve productivity, business strategies, and societal development. In this paper, an explicit numerical scheme is given for both linear and nonlinear differential equations. The scheme is correct to second order. Additionally, the scheme’s consistency and stability are guaranteed. Backpropagation of Levenberg–Marquardt, the effect of including an induced magnetic field in a mathematical model for electrical boundary layer nanofluid flow on a flat plate, is quantitatively investigated using artificial neural networks. Later, the model is reduced into a set of boundary value problems, which are then resolved using the suggested scheme and a shooting strategy. The outcomes are also contrasted with earlier studies and the MATLAB solver bvp4c for validation purposes. In addition, neural networking is also employed for mapping input to outputs for velocity, temperature, and concentration profiles. These results prove that artificial neural networks can make precise forecasts and optimizations. Using a neural network to optimize the fluid flow in an electrical boundary layer while subjected to an induced magnetic field is a promising application of the suggested computational scheme. Fluid dynamics benefits greatly from combining numerical methods and artificial neural networks, which could lead to new developments in various fields. Results from this study may aid in optimizing fluid systems, leading to greater productivity and effectiveness in numerous technical fields.

An explicit computational scheme is proposed for solving fractal time-dependent partial differential equations (PDEs). The scheme is a three-stage scheme constructed using the fractal Taylor series. The fractal time order of the scheme is three. The scheme also ensures stability. The approach is utilized to model the time-varying boundary layer flow of a non-Newtonian fluid over both stationary and oscillating surfaces, taking into account the influence of heat generation that depends on both space and temperature. The continuity equation of the considered incompressible fluid is discretized by first-order backward difference formulas, whereas the dimensionless Navier–Stokes equation, energy, and equation for nanoparticle volume fraction are discretized by the proposed scheme in fractal time. The effect of different parameters involved in the velocity, temperature, and nanoparticle volume fraction are displayed graphically. The velocity profile rises as the parameter I grows. We primarily apply this computational approach to analyze a non-Newtonian fluid’s fractal time-dependent boundary layer flow over flat and oscillatory sheets. Considering spatial and temperature-dependent heat generation is a crucial factor that introduces additional complexity to the analysis. The continuity equation for the incompressible fluid is discretized using first-order backward difference formulas. On the other hand, the dimensionless Navier–Stokes equation, energy equation, and the equation governing nanoparticle volume fraction are discretized using the proposed fractal time-dependent scheme.

In this paper, we introduce the concept of orthogonal convex structure contraction mapping and prove some fixed point theorems on orthogonal ♭-metric spaces. We adopt an example to highlight the utility of our main result. Finally, we apply our result to examine the existence and uniqueness of the solution for the spring-mass system via an integral equation with a numerical example.

This contribution addresses a fractal numerical scheme that can be employed for handling fractal time-dependent parabolic equations. The numerical scheme presented in this contribution can be used to discretize integer order and fractal derivatives in a given differential equation. Therefore, the scheme and results can be used for both cases. The proposed finite difference scheme is based on two stages. Fractal time derivatives are discretized by employing the proposed approach. For the scalar convection–diffusion equation, we derive the stability condition of the proposed fractal scheme. Using a nonlinear chemical reaction, the approach is also used to solve the Quantum Calculus model of a Williamson nanofluid’s unsteady Darcy–Forchheimer flow over flat and oscillatory sheets. The findings indicate a negative correlation between the velocity profile and the porosity parameter and inertia coefficient, with an increase in these factors resulting in a drop in the velocity profile. Additionally, the fractal scheme under consideration is being compared to the fractal Crank–Nicolson method, revealing that the proposed scheme exhibits a superior convergence speed compared to the fractal Crank–Nicolson method. Several problems involving the motion of non-Newtonian nanofluids through magnetic fields and porous media can be investigated with the help of the proposed numerical scheme. This research has implications for developing more efficient heat transfer and energy conversion devices based on nanofluids.

Consider a complex Hilbert space $\left(\mathbb{X},⟨·,·⟩\right)$ equipped with a positive (semidefinite) bounded linear operator $\mathcal{A}$ on $\mathbb{X}$. The $\mathcal{A}$-joint numerical radius for two $\mathcal{A}$-bounded operators $\mathcal{T}$ and $\mathcal{S}$ is defined as ${\omega }_{\mathcal{A},e}(\mathcal{T},\mathcal{S})={sup}_{{\parallel x\parallel }_{\mathcal{A}}=1}\sqrt{{\left|{⟨\mathcal{T}x,x⟩}_{\mathcal{A}}\right|}^2+{\left|{⟨\mathcal{S}x,x⟩}_{\mathcal{A}}\right|}^2}.$ Among other results, in this study we demonstrate that ${\omega }_{e,\mathcal{A}}^2\left(\mathcal{T},\mathcal{S}\right)\le 2^{1-{\displaystyle \frac{1}{r}}}{\left[max\left\{{∥\mathcal{T}∥}_{\mathcal{A}}^{2r},{∥\mathcal{S}∥}_{\mathcal{A}}^{2r}\right\}+{\omega }_{\mathcal{A}}^r\left({\mathcal{S}}^{{♯}_{\mathcal{A}}}\mathcal{T}\right)\right]}^{{\displaystyle \frac{1}{r}}}$ for $r\ge 1$ and that ${\omega }_{e,\mathcal{A}}^2\left(\mathcal{T},\mathcal{S}\right)\ge \frac{1}{2}{\omega }_{\mathcal{A}}\left({\mathcal{T}}^2+{\mathcal{S}}^2\right)+\frac{1}{2}max\left\{{\omega }_{\mathcal{A}}\left(\mathcal{S}+\mathcal{T}\right),{\omega }_{\mathcal{A}}\left(\mathcal{S}-\mathcal{T}\right)\right\}\left|{\omega }_{\mathcal{A}}\left(\mathcal{S}\right)-{\omega }_{\mathcal{A}}\left(\mathcal{T}\right)\right|$. Additionally, we provide several inequalities related to the $\mathcal{A}$-numerical radius and $\mathcal{A}$-seminorm.