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In this paper, the examination of soliton solutions of the biofilm model with the help of a new extended direct algebraic method is expressed. Besides the exact solutions, the existence of these solutions is also discussed with the help of the Schauder fixed point theorem. The nonlinear dynamical biofilm model which we consider in this paper is basically the bistable Allen–Cahn equation with quartic potential. For more physical understanding, the 3D plots of the solutions are presented. The different types of hyperbolic, trigonometric, and rational soliton solutions are gained. In the biofilm model, different parameters belong to certain spaces and give the exact solutions by applying the technique of new extended direct algebraic method. Exact solutions of the biofilm model are highlighted along with their restriction.

The current research deals with the exact solutions of the nonlinear partial differential equations having two important difficulties, that is, the coefficient singularities and the stochastic function (white noise). There are four major contributions to contemporary research. One is the mathematical analysis where the explicit a priori estimates for the existence of solutions are constructed by Schauder’s fixed point theorem. Secondly, the control of the solution behavior subject to the singular parameter ϵ when ϵ → 0. Thirdly, the impact of noise that is present in the differential equation has been successfully handled in exact solutions. The final contribution is to simulate the exact solutions and explain the plots.

Internal Combustion (IC) engines are commonly used in the process industry, and proper Air-Fuel Ratio (AFR) regulation in their fuel system is essential for improved engine performance, fuel efficiency, and environmental protection. Since faults in the sensors of the AFR system cause the engine to shut down, fault tolerance is essentially required for them. In this paper, an advanced Active Fault-Tolerant Control System (AFTCS) for AFR control of an IC engine is proposed to prevent the engine shutdown. For analytical redundancy, the proposed AFTCS uses the Genetic Algorithm (GA) based observer model in the Fault Detection and Isolation (FDI) unit to generate estimated value for the faulty sensors using the values of other healthy sensors. The proposed framework was simulated in the MATLAB/Simulink environment to verify its effectiveness. In comparison to previous literature works, the results show that the AFR control system has superior fault tolerance output for sensor faults, especially for the MAP sensor, in terms of less oscillatory response. Finally, a comparison of the proposed GA-based AFTCS was carried out with the existing literature works and its superior performance was elaborated.

This study presents the investigations on the effect of heat transfer on droplet formation in T-type microfluidic channel. Mineral oil acts as a continuous phase, and water acts as a dispersed phase. The Volume of Fluid model is used to investigate the formation of droplets of water in oil in the microchannel. The physical properties of both fluids change with an increase in temperature of dispersed phase, of which the dominant properties are viscosities of fluids and interfacial tension. The parameters such as droplet diameter, distance between two consecutive droplets and detachment time were studied with respect to temperature in the range of 300 K to 325 K. These properties are also studied at different flow rate ratios 0.125, 0.25, and 0.5. The distance between two consecutive droplets was found to decrease with the increase in temperature due to the rapid motion of droplets in the temperature zone. The droplet detachment time also decreases, due to the decrease in interfacial forces which hold the droplets in the dispersed phase. The droplet diameter decreases initially with the increase in temperature, but it was observed that after 320 K the diameter of the droplet increases due to the coalescence of two consecutive droplets downstream in the microchannel. This temperature is reported as a limiting condition for thermally controlled water-in-oil microemulsions.

In this paper, mainly the exact traveling waves fitting the mathematical model in electro-cardio-physiology have been investigated. The source functions of noise-type are used in the proposed coupled system of nonlinear partial differential equations. The Ricatti–Bernoulli Sub-ODE method together with the Bäcklund extension is applied. The interesting plots for the obtained traveling waves and solitons are shown. The proposed model provides new applications for the considerations under the electric pulses or shocks. The application of current method and the modification of classical existence theory are first time applied to the stochastic nonlinear problems in electro-cardiac-physiology.

Measles is a highly transmissible disease in children around the world. According to the World Health Organization (WHO), 73% of deaths of children were due to measles in 2018. This study describes the physical solution of the SIQR model for measles spread under the effect of natural delay amongst different compartments. By three different numerical techniques, the efficacies of solutions of the underlying system have been compared and a clear preference of nonstandard finite-difference (NSFD) scheme over the rest has been established. It has also been observed, on principle, that the NSFD formulation recovers all the essential traits of a continuous model namely the boundedness, positivity and stability of equilibriums of populations. The numerical results have also been supported by a very strong classical analysis of the model where the existence of a solution vector in explicit subsets of the function spaces has been guaranteed which leads to optimization of fixed-point methods.

The current work is devoted to investigating the disease dynamics and numerical modeling for the delay diffusion infectious rabies model. To this end, a non-linear diffusive rabies model with delay count is considered. Parameters involved in the model are also described. Equilibrium points of the model are determined and their role in studying the disease dynamics is identified. The basic reproduction number is also studied. Before going towards the numerical technique, the definite existence of the solution is ensured with the help of the Schauder fixed point theorem. A standard result for the uniqueness of the solution is also established. Mapping properties and relative compactness of the operator are studied. The proposed finite difference method is introduced by applying the rules defined by R.E. Mickens. Stability analysis of the proposed method is done by implementing the Von–Neumann method. Taylor’s expansion approach is enforced to examine the consistency of the said method. All the important facts of the proposed numerical device are investigated by presenting the appropriate numerical test example and computer simulations. The effect of τ on infected individuals is also examined, graphically. Moreover, a fruitful conclusion of the study is submitted.

In the recent article, a reaction-advection-diffusion model of the hepatitis-B virus (HBV) is studied. Existence and uniqueness of the optimal solution for the proposed model in function spaces is analyzed. The advection and diffusion terms make the model more generic than the simple model. So, the numerical investigation plays a vital role to understand the behavior of the solutions. To find the existence and uniqueness of the optimal solutions, a closed and convex subset (closed ball) of the Banach space is considered. The explicit estimates regarding the solution of the system for the admissible auxiliary data is computed. On the other hand, for the numerical approximation of the solution, an elegant numerical technique is devised to find the approximate solutions. After constructing the discrete model, some fundamental properties must necessarily be possessed by the proposed numerical scheme. For instance, consistency, stability, and positivity of the solutions. These properties are carefully studied in the current article. To prove the positivity of the proposed scheme, M-matrix theory is used. All the above mentioned properties are verified by sketching the graph via simulations. Furthermore, these plots are helpful to understand the true behavior of the solutions. For this purpose, a fruitful discussion is included about the simulations to justify our results.

The novel coronavirus disease, coined as COVID-19, is a murderous and infectious disease initiated from Wuhan, China This killer disease has taken a large number of lives around the world and its dynamics could not be controlled so far In this article, the spatio-temporal compartmental epidemic model of the novel disease with advection and diffusion process is projected and analyzed To counteract these types of diseases or restrict their spread, mankind depends upon mathematical modeling and medicine to reduce, alleviate, and anticipate the behavior of disease dynamics The existence and uniqueness of the solution for the proposed system are investigated Also, the solution to the considered system is made possible in a well-known functions space For this purpose, a Banach space of function is chosen and the solutions are optimized in the closed and convex subset of the space The essential explicit estimates for the solutions are investigated for the associated auxiliary data The numerical solution and its analysis are the crux of this study Moreover, the consistency, stability, and positivity are the indispensable and core properties of the compartmental models that a numerical design must possess To this end, a nonstandard finite difference numerical scheme is developed to find the numerical solutions which preserve the structural properties of the continuous system The M-matrix theory is applied to prove the positivity of the design The results for the consistency and stability of the design are also presented in this study The plausibility of the projected scheme is indicated by an appropriate example Computer simulations are also exhibited to conclude the results

In this paper, nonlinear fractional order Ebola virus mathematical model is discussed for the complex transmission of the epidemic problems. We developed the fractional order Ebola virus transmission model for the treatment and control to reduce its effect on a population which play an important role for public health. Qualitative analysis has been made to verify the the steady state and uniqueness of the system is also developed for reliable results. Caputo fractional derivative operator φ i ∈ (0,1] works to achieve the fractional differential equations. Laplace with Adomian Decomposition Method successfully solved the fractional differential equations. Ultimately, numerical simulations are also developed to evaluate the effects of the device parameter on spread of disease and effect of fractional parameter φ i on obtained solution which can also be assessed by tabulated results.

The current article deals with the analysis and numerical solution of fractional order Schnakenberg (S-B) model. This model is a system of autocatalytic reactions by nature, which arises in many biological systems. This study is aiming at investigating the behavior of natural phenomena with a more realistic and practical approach. The solutions are obtained by applying the Grunwald–Letnikov (G–L) finite difference (FD) and the proposed G–L nonstandard finite difference (NSFD) computational schemes. The proposed formulation is explicit in nature, strongly structure preserving as well as it is independent of the time step size. One very important feature of our proposed scheme is that it preserves the positivity of the solution of continuous fractional order S-B model because the unknown variables involved in this system describe the chemical concentrations of different substances. The comparison of the proposed scheme with G–L FD method reflects the significance of the said method.

In this article, an implicit numerical design is formulated for finding the numerical solution of spatiotemporal nonlinear dynamical system with advection. Such type of problems arise in many fields of life sciences, mathematics, physics and engineering. The epidemic model describes the population densities that have some special types of features. These features should be maintained by the numerical design. The proposed scheme, not only solves the nonlinear physical system but also preserves the structure of the state variables. Von-Neumann criteria, M-matrix theory and Taylor’s expansion are used for proving some standard results. Basic reproduction number is evaluated and its key role in deciding the stability at the equilibrium points is also investigated. Graphical solutions are also presented against the test problem.

The gist behind this study is to extend the classical computer virus model into fractal fractional model and subsequently to solve the model by Atangana-Toufik method. This method solve nonlinear model under consideration very efficiently. We use the Mittag-Leffler kernels on the proposed model. Atangana-Baleanu integral operator is used to solve the set of fractal-fractional expressions. In this model, three types of equilibrium points are described i.e trivial, virus free and virus existing points. These fixed points are used to establish some standard results to discuss the stability of the system by calculating the Jacobian matrices at these points. Routh-Hurwitz criteria is used to verify that the system is locally asymptotically stable at all the steady states. The emphatic role of the basic reproduction number R 0 is also brought into lime light for stability analysis. Sensitivity analysis of R 0 is also discussed. Optimal existence and uniqueness of the solution is the nucleus of this study. Computer simulations and patterns and graphical patterns illustrate reliability and productiveness of the proposed method.

Starting from distinguishing boundaries for monogenic functions (see Tutschke in Adv. Appl. Clifford Algebr 25:441–451, 2015), one can solve boundary value problems for monogenic functions (see Dao in Boundary value problems for monogenic functions in higher dimensions. Ph.D Thesis, Hanoi University of Science and Technology, Vietnam, 2019). A boundary value problem for monogenic function in $${\mathbf {R}}^{3}$$ is the following: prescribe the vector components $$u_1$$ and $$u_2$$ on the whole boundary, the bi-vector component $$u_{12}$$ on a (one-dimensional) curve on the boundary and the real part $$u_0$$ at one point. The goal of the present paper is to reduce the same boundary value problems for more general (linear or fully non-linear) first order systems to fixed-point problems for an operator containing the monogenic solution of the boundary value problem under consideration (provided the given system is written in its Clifford-analytic normal form). In case of $${\mathbf {R}}^{3}$$ the existence of a uniquely determined solution of the boundary value problem is proved as fixed-point of the corresponding fixed-point problem in case the right hand side is Lipschitz-continuous with sufficiently small Lipschitz constants.

Background and objective: Epidemic models are used to describe the dynamics of population densities or population sizes under suitable physical conditions. In view that population densities and sizes cannot take on negative values, the positive character of those quantities is an important feature that must be taken into account both analytically and numerically. In particular, susceptible-infected-recovered (SIR) models must also take into account the positivity of the solutions. Unfortunately, many existing schemes to study SIR models do not take into account this relevant feature. As a consequence, the numerical solutions for these systems may exhibit the presence of negative population values. Nowadays, positivity (and, ultimately, boundedness) is an important characteristic sought for in numerical techniques to solve partial differential equations describing epidemic models. Method: In this work, we will develop and analyze a positivity-preserving nonstandard implicit finite-difference scheme to solve an advection-reaction nonlinear epidemic model. More concretely, this discrete model has been proposed to approximate consistently the solutions of a spatio-temporal nonlinear advective dynamical system arising in many infectious disease phenomena. Results: The proposed scheme is capable of guaranteeing the positivity of the approximations. Moreover, we show that the numerical scheme is consistent, stable and convergent. Additionally, our finite-difference method is capable of preserving the endemic and the disease-free equilibrium points. Moreover, we will establish that our methodology is stable in the sense of von Neumann. Conclusion: Comparisons with existing techniques show that the technique proposed in this work is a reliable and efficient structure-preserving numerical model. In summary, the present approach is a structure-preserving and efficient numerical technique which is easy to implement in any scientific language by any scientist with minimal knowledge on scientific programming.