System Of Nonlinear Differential Equations Research Articles

Analog quantum simulation of partial differential equations

Abstract Quantum simulators were originally proposed for simulating one partial differential equation in particular -- Schrodinger's equation. Can quantum simulators also efficiently simulate other partial differential equations? While most computational methods for partial differential equations -- both classical and quantum -- are digital (they must be discretised first), partial differential equations have continuous degrees of freedom. This suggests that an analog representation can be more natural. While digital quantum degrees of freedom are usually described by qubits, the analog or continuous quantum degrees of freedom can be captured by qumodes. Based on a method called Schrodingerisation, we show how to directly map D-dimensional linear partial differential equations onto a (D+1)-qumode quantum system where analog or continuous-variable Hamiltonian simulation on D+1 qumodes can be used. This very simple methodology does not require one to discretise partial differential equations first, and it is not only applicable to linear partial differential equations but also to some nonlinear partial differential equations and systems of nonlinear ordinary differential equations. We show some examples using this method, including the Liouville equation, heat equation, Fokker-Planck equation, Black-Scholes equations, wave equation and Maxwell's equations. We also devise new protocols for linear partial differential equations with random coefficients, important in uncertainty quantification, where it is clear how the analog or continuous-variable framework is most natural. This also raises the possibility that some partial differential equations may be simulated directly on analog quantum systems by using Hamiltonians natural for those quantum systems.

Analytical solution to boundary layer flow and convective heat transfer for low Prandtl number fluids under the magnetic field effect over a flat plate

AbstractThe present study aims to quantify the flow field, flow velocity, and heat transfer features over a horizontal flat plate under the influence of an applied magnetic field, with a particular emphasis on low Prandtl number fluids. Nonlinear partial differential expressions can be incorporated into the ordinary differential framework with the use of appropriate transformations. This research utilizes the variational iteration method (VIM) to approximate solutions for the system of nonlinear differential equations that define the problem. The objective is to demonstrate superior flexibility and broader application of the VIM in addressing heat transfer issues, compared to alternative approaches. The results obtained from the VIM are compared with numerical solutions, revealing a significant level of accuracy in the approximation. The numerical findings strongly suggest that the VIM is effective in providing precise numerical solutions for nonlinear differential equations. The analysis includes an examination of the flow field, velocity, and temperature distribution across various parameters. The study found that improving temperature patterns, velocity distribution, and flow dynamics were all positively impacted by increasing the Prandtl numbers. As a result, this leads to the thickness of the boundary layer to decrease and improves heat transfer at the moving surface. Thus, the convection process becomes more efficient. When the strength of the magnetic field is increased, the velocity of the fluid decreases. This observation aligns with expectations since the magnetic field hampers the natural flow of convection. Notably, the convection process can be precisely controlled by carefully applying magnetic force.

Analytical Properties and Numerical Preservation of an Age-Group Sir Model: Application to the Diffusion of Information

Analytical Properties and Numerical Preservation of an Age-Group Sir Model: Application to the Diffusion of Information

Analyzing epidemics and designing policies with a modified SIR model

This research paper aims to study the impact of COVID-19 on the economy, where we have extended the standard epidemiological model, the Susceptible–Infected–Recovered (SIR) model. In the economy’s dynamics, we have incorporated aggregate demand and aggregate supply and further set up a system of three-dimensional nonlinear differential equations, which is more likely the Labor–Infected–Quarantine (LIQ) model, an extension of the SIR model. Here, we have derived the steady-state condition and analyzed the model’s stability. In addition, we have done a comparative static analysis of parameters with some policy implications. We aim to achieve the twin objective, i.e., controlling the pandemic and reviving the economy under the set of the policy mix.

Opers on the projective line, Wronskian relations, and the Bethe Ansatz

It is well-known that the spectra of the Gaudin model may be described in terms of solutions of the Bethe Ansatz equations. A conceptual explanation for the appearance of the Bethe Ansatz equations is provided by appropriate G-opers: G-connections on the projective line with extra structure. In fact, solutions of the Bethe Ansatz equations are parameterized by an enhanced version of opers called Miura opers; here, the opers appearing have only regular singularities. Moreover, this geometric approach to the spectra of the Gaudin model provides a well-known example of the geometric Langlands correspondence. Feigin, Frenkel, Rybnikov, and Toledano Laredo have introduced an inhomogeneous version of the Gaudin model; this model incorporates an additional twist factor, which is an element of the Lie algebra of G. They exhibited the Bethe Ansatz equations for this model and gave an interpretation of the spectra in terms of opers with an irregular singularity. In this paper, we consider a new geometric approach to the study of the spectra of the inhomogeneous Gaudin model in terms of a further enhancement of opers called twisted Miura-Plücker opers. This approach involves a certain system of nonlinear differential equations called the qq-system, which were previously studied in [20] in the context of the Bethe Ansatz. We show that there is a close relationship between solutions of the inhomogeneous Bethe Ansatz equations and polynomial solutions of the qq-system and use this fact to construct a bijection between the set of solutions of the inhomogeneous Bethe Ansatz equations and the set of nondegenerate twisted Miura-Plücker opers. We further prove that as long as certain combinatorial conditions are satisfied, nondegenerate twisted Miura-Plücker opers are in fact Miura opers.

Bioconvective flow of thermally radiative Casson nanofluid with aerobic microorganisms over a stretching surface

A mathematical model is proposed to analyze the flow characteristics of a thermally radiative, chemically reacting Casson nanofluid incorporating aerobic microorganisms over a stretching sheet. Nonlinear dimensionless ordinary differential equations are formulated through suitable similarity transformations. The resulting coupled system of nonlinear ordinary differential equations is numerically solved using the finite difference method with the bvp4c routine in MATLAB. The verification of the numerical solution’s accuracy is conducted by comparing it with previously established results under specific conditions pertinent to the present investigation. Skin friction coefficients, local Nusselt number, and Sherwood number are computed for various parameters and their implications for engineering applications are discussed. The investigation also explores the impact of different dimensionless parameters on velocity, temperature, mass concentration, motile microorganism density, and oxygen concentration profiles for both Casson and Newtonian nanofluids, offering a comprehensive analysis. The study highlights that suspending non-Newtonian nanofluids with fast-moving aerobic microorganisms and nanoparticles results in enhanced heat transfer rates, thus presenting potential benefits for practical applications such as the manufacturing of biofilms.

A well-balanced second-order finite volume approximation for a coupled system of granular flow

A well-balanced second-order finite volume scheme is proposed and analyzed for a 2×2 system of non-linear partial differential equations which describes the dynamics of growing sandpiles created by a vertical source on a flat, bounded rectangular table in multiple dimensions. To derive a second-order scheme, we combine a MUSCL type spatial reconstruction with strong stability preserving Runge-Kutta time stepping method. The resulting scheme is ensured to be well-balanced through a modified limiting approach that allows the scheme to revert to a well-balanced first-order scheme if it fails to capture or preserve the discrete steady state solution during its time evolution. Further, the scheme maintains the second-order accuracy away from the steady state. The well-balance property of the scheme is proven analytically in one dimension and demonstrated numerically in two dimensions. Additionally, numerical experiments reveal that the second-order scheme reduces finite time oscillations and provides better resolutions of the physical properties of the state variables, as compared to the existing first-order schemes of the literature.

Application of generalized Haar wavelet technique on simultaneous delay differential equations

In this paper, the Haar wavelet technique is applied to a system of simultaneous linear and nonlinear proportional delay differential equations. By implementing this method, the delay differential equations are transformed into a series of linear and nonlinear algebraic equations, respectively. These equations are then solved using numerical code constructed in MATLAB. The obtained approximate solutions are compared with exact solutions, demonstrating the accuracy and efficacy of the Haar wavelet method. Using the fixed point theorem, the existence and uniqueness of the solutions is established. The reliability and efficiency of the proposed technique are illustrated through two numerical examples. Additionally, a comprehensive error analysis is presented and the convergence result is discussed, providing a thorough examination of the robustness of the method. This study not only shows that the Haar wavelet technique is a powerful tool for solving delay differential equations but also sets the groundwork for future research in this area.

Fractional view analysis of coupled Whitham- Broer-Kaup and Jaulent-Miodek equations

In this study, I systematically investigate the fractional Whitham-Broer-Kaup (WBK) system and the fractional Coupled Jaulent-Miodek (CJM) equation under the Caputo fractional calculus. The nonlinear fractional differential equation systems are investigated via the Aboodh transform iteration method and the Aboodh residual power series method, thus offering a thorough analytical investigation. The dynamics of the fractional WBK system are accomplished using the Aboodh transform iteration method, and the Aboodh residual power series method is employed to study the behavior of the fractional CJM equation. Using established solutions, we completely analyze their dynamics employing both symbolic computations and numerical simulations. Consequently, novel solutions are identified, and the behavior of these fractional systems in the Caputo operator sense are clarified. The results obtained show good agreement of convergence of the numerical and analytical solutions, highlighting the effectiveness of the schemes adopted for unraveling the complex dynamics of fractional nonlinear systems.

Determining the parameters of the functioning for a nonlinear ballistic system in a real external environment

Purpose. Development of an approximate nonlinear model for the solution of the external ballistics problem with determination of the nonlinear system parameters and development of a methodology for calculating the aerodynamic characteristics of the cargo, located on the external suspension of unmanned aerial vehicle (UAV) in order to increase the efficiency of their delivery to the specified landing target by means of asymptotic approach with given parameters of the studied system and external load. Methodology. The development of an effective model was carried out using analytical and numerical research algorithms based on a nonlinear system of differential equations in a general form with time-varying coefficients. In order to obtain a solution to the nonlinear problem of external ballistics in a two-dimensional formulation, the assumption of a significant influence of the projection of the velocity function on the ordinate axis in relation to the component on the abscissa axis is introduced. The problem is reduced to the solution of a related system of differential equations with variable coefficients along the corresponding coordinates using the asymptotic approach for a small parameter of the coefficient of frontal aerodynamic resistance. Applied mathematical analysis and modeling have been used for the problem formulation considering studied environmental parameters. Findings. Analytical dependences of the nonlinear problem of ballistics and application of finite-element analysis (FEA) with respect to the cargo motion from the UAV in the presence of the initial speed and wind load in the plane of motion are proposed. It is shown that the obtained analytical solution is correlated with the direct numerical calculation of the basic differential equation with respect to the ordinate axis. Originality. A mathematical nonlinear model of the dynamic process is proposed, assuming the prevailing influence of the speed function along the ordinate axis compared to the function along the abscissa axis. To obtain an approximate analytical solution of the basic nonlinear system of differential equations with variable coefficients the asymptotic perturbation method is applied. The dependence for the axial displacement function is presented considering actual time-flight parameter. Practical value. The obtained analytical dependencies for estimating the time and distance reaching the target with the initial speed of movement and the presence of wind load can be used in applied problems of mathematical physics and engineering calculations of functional dependencies and control of the cargo delivery process and target reaching from an UAV. The obtained analytical results and the solution algorithm can be integrated into applied problems of mathematical physics and engineering calculations, particularly the development of ballistic system control algorithms.

Convex hull property for elliptic and parabolic systems of PDE

We study the convex hull property for systems of partial differential equations. This is a generalization of the maximum principle for a single equation. We show that the convex hull property holds for a class of elliptic and parabolic systems of non-linear partial differential equations. In particular, this includes the case of the parabolic p-Laplace system. The coupling conditions for coefficients are demonstrated to be optimal by means of respective counterexamples.

Thermal attributes of sodium alginate (Na.C6H7O6) based binary and ternary hybrid nanofluids under activation energy and induced magnetic field environment

Ternary hybrid nanofluids possess several potential applications in the modern science beyond just the heat transfer efficiency. A continuous research and advancements on binary and ternary hybrid nanofluids might lead towards new developments in many technological and scientific zones. Ongoing research interprets and distinguishes the novel attributes of the sodium alginate (Na.C6H7O6) based unary, binary and ternary nanofluids through a porous medium under induced magnetic field environment. The nano-composition of single-wall carbon nanotubes (SWCNT) with silver (Ag) and molybdenum dioxide (MoS2) gives rise to the binary (SWCNT-Ag/Na.C6H7O6) and ternary hybrid (SWCNT-Ag-MoS2/Na.C6H7O6) nanofluids respectively. A complex and highly nonlinear system of differential equations, obtained via the boundary layer approximations, is tackled numerically. An algorithmic approach, using MATLAB software, is incorporated to determine the iterative solutions. The efficiency as well as validity of the developed algorithm is confirmed by equating the results with the previous ones. The results evidently portray the fact that the magnetic Prandtl number and magnetic interaction parameter not only dissuade the induced magnetic field but also resist the fluids’ speed. With the larger values of activation energy parameter, higher will be the concentration of fluid.

Numerical Scheme by Legendre-Galerkin for Characterization of Micropolar-Maxwell Unsteady Fluid Model

In this dissertation, Micropolar-Maxwell unsteady fluid flow prior an expanding sheet is asymptotically investigated using an efficient Legendre-Galerkin technique in the existence of magnetic field. By employing fitting similarity mapping to the controlling partial differential equations, a system of nonlinear ordinary differential equations is obtained. In comparison to the prior studies, the suggested asymptotic outcomes are guaranteed. The impacts of elasticity/material parameters on velocity/micro-rotation profiles are shown and explored with the help of tables and figures to enhance the mechanical properties of the sheet.

Effects of Variable Thermal Conductivity on MHD Fluid Flow on Rotating Vertical Cone in the Presence of Darcy Forchheimer, Soret and Dufour Effects through a Porous Medium

This research investigated the effects of Variable Thermal Conductivity on MHD fluid on a Rotating Vertical Cone in the Presence of Darcy Forchheimer, Soret and Dufour effects through a porous medium. The system of nonlinear differential equations were transformed into first order differential equations and solved using shooting technique in Matlab package Bvp4c.Furthermore, the effects of various parameters over the main physical quantities are shown graphically and also extensively discussed. The result shows that Thermal conductivity increases Tangential Velocity, Temperature and decreases Normal Velocity. Dufour increases Tangential Velocity and decreases Normal Velocity. Radiation parameter R enhances Tangential Velocity and Temperature.

Chelyshkov wavelet-based numerical technique for the solution of Darcy–Forchheimer flow of Erying–Powell radiated dusty fluid over a stretching sheet with Cattaneo–Christov heat flux

This article presents the Chelyshkov wavelet-based numerical technique (CWBNT) for the solution of the system of non-linear differential equations arising in the study of Darcy–Forchheimer flow of Erying–Powell radiated dusty fluid over a stretching sheet with Cattaneo–Christov heat flux. Initially, the proposed technique is implemented on the test problem having exact solution. The obtained results are highly accurate in comparison with the solutions obtained by shooting technique based Runge–Kutta Fehlberg 4th–5th order method. The proposed technique is applied to the governing equations of the considered dusty fluid problem. The results obtained for some fixed values of parameters resemble the previously published results. The comparisons validate the method. The flow behavior of dusty fluid is analyzed and the effect of various factors on the velocity and thermal attribute of the dust and fluid phase are studied. It is inferred that the Erying–Powell fluid parameters ε and α have entirely different effects on temperature, velocity profiles, and skin-friction coefficient. The local inertia parameter decreases the velocity and thermal relaxation parameter decreases the temperature. Enhancement in temperature ratio and radiation parameters escalates the temperature and Nusselt number. The proposed technique is simple, reliable, efficient tool to solve the differential equations arising in dusty fluid flow problems.

Outlier-resistant non-fragile observer-based fuzzy boundary control for interconnected nonlinear PDE systems

This paper investigates an observer-based boundary controller design for interconnected nonlinear partial differential equation (PDE) systems. First, the Takagi–Sugeno (T–S) fuzzy model is adopted to accurately describe the target systems. Then, boundary measurements are employed to reduce the number of sensors. Next, considering the phenomenon of abnormal interference that may lead to measurement outliers and observer parameters’ uncertainties, an outlier-resistant non-fragile observer expressed by a saturation function is designed to guarantee the desired control objectives. Moreover, the boundary control approach is employed to trade-off the cost of system design and system performance. Furthermore, utilizing the membership function-dependent Lyapunov functions and free-weight matrixes, sufficient conditions ensuring the closed-loop systems’ exponential stability are obtained while decreasing the conservativeness of the system stability analysis. Finally, the proposed method’s feasibility and effectiveness are validated by an example.

Numerical simulations of third-grade fluid flow with exothermic reactions in porous media-saturated vertical microchannels

The aim of this article is to analyze the electro-osmotic flow behavior of a third-grade (TG) incompressible fluid flowing in a vertical microchannel saturated with porous material. The electro-osmotic flow is regarded as an essential part of many scientific and engineering applications used in microfluidic devices. The flow is subjected to joule heating, exothermic reactions, and asymmetrical convective cooling. The viscosity which is assumed temperature dependent via a Naham-type law and the convective heat transfer rates governing the asymmetric convective cooling at the microchannel walls are modeled via Newton’s law of cooling. The governing system of nonlinear partial differential equations is solved computationally using an efficient and reliable finite difference method. The spatial and temporal convergence of the numerical scheme is showcased graphically. Our study findings reveal that the combination of exothermic reactions and Joule heating significantly influences the electroosmotic flow. Higher values of variable viscosity parameter (α), Brinkman number (Br), and Buoyancy parameter (Gr) result an enhancement in the flow variables. Conversely, a reverse trend is observed when augmenting the Prandtl number (Pr), non-Newtonian parameter (γ), and porous media parameters (Ω2). Numerical results for temperature and velocity are also discussed qualitatively and validated with the data previously published.

Nonlinear thermal radiation analysis for bioconvection peristaltic transport of fourth-grade nanofluid

Nanoparticles mixed with base liquids result in the formation of nanofluids, known for their exceptional thermal conductivity and their ability to enhance the performance of electrical and mechanical devices. Therefore, this study investigates the peristaltic activity of fourth-grade nanomaterial by taking the consideration of magnetic aspects. Analysis is carried out in the presence of motile gyrotactic microorganisms, Ohmic heating, and viscous dissipation. Thermal transport incorporated effects of nonlinear thermal radiation along with Brownian motion and thermophoresis. Thermal and mass convective conditions are imposed on elastic symmetric channels. The assumptions of low Reynolds number and long wavelength approximations have led to the simplification of the nonlinear partial differential equations system into ordinary differential equations. The governing system is solved numerically by NDSolve technique via Mathematica. Graphs are plotted and examined for velocity, thermal field, density of motile gyrotactic microorganisms, and heat transfer rate.

Improving the rheological behavior of magnetiz couple stress Buongiorno's nanofluid through resilient wavy channel with curvature effect: Nonlinear analysis

In biological systems, fluids with complex rheological behavior often encounter complex microenvironments. Couple stress fluids can be applied to model the flow of biological fluids in micro-vessels, capillaries, and other intricate structures. Couple stress fluids can be employed to model drug transport in microfluidic devices and capillaries, providing insights into the behavior of pharmaceuticals at small scales. Studying the couple stress fluids is crucial, particularly when considering the impact of particle viscosity in the existence of induced magnetic fields and nanoparticles that might enhance the characteristic motion. So, we studied the moderate and magnetic Reynold numbers and curvature effects of magnetized couple stress Buongiorno's nanofluid model through resilient wavy channel. The governing equations of this model are formed in non-dimensional form without any approximation, i.e., in the companionship of moderate Reynolds number and curvature effect, which aren't studied in this form. Using the Adomian decomposition approach, it is possible to obtain a solution of nonlinear system of partial differential equations analytically. The semi-analytical results are obtained and represented graphically for characteristic motion and bio-thermal properties for temperature and concentration, as well as magnetic features. Physically, by rising the magnetic Reynolds number, the magnetic permeability raises, so the ability of the couple stress fluid to form an internal magnetic field within itself with the impact of an applied magnetic field increases, which increases the magnetic force.

Dynamics of Open States and Promoter Functioning in the appY_red and appY_green Genetic Constructions Based on the pPF1 Plasmid.

Although the role of dynamic factors in DNA function still remains unclear, research in this direction is a rapidly developing area of molecular biology. In this work, the genetic constructions appY_red and appY_green, based on the plasmid pPF1 and containing a fragment of Escherichia coli (E. coli) DNA with predicted promoter-like regions, are considered complex dynamic systems in which local sites of double helix unwinding, called open states, can arise and propagate. The purpose of the article is to show the existence of a connection between the dynamics of open states and the functioning of predicted promoters. We experimentally verified the functionality of the predicted promoters using a reporter vector. Using a reverse transcription reaction, transcription start sites were identified indicating the presence of two divergent promoters, one on each strand. In mathematical studies, a dynamic model was used that described open states as one-soliton solutions (kinks) of a system of nonlinear partial differential equations, and the influence of the torque Mτ on the dynamics of kinks was taken into account. Fluorescence analysis of colonies of E. coli cells transformed with plasmid constructions showed that one of the two promoters is stronger than the other and that the strength of the promoters depends on the orientation of the fragment under study in the plasmid. On the other hand, using mathematical modeling, the energy profiles of genetic constructions were calculated and the kink trajectories were constructed. In addition, by studying the effect of torsion moment in model studies, we found threshold torque values at which the behavior of kinks changes dramatically: from oscillatory to translational motion. The minimum values of torsion moment required to initiate the transcription process were also assessed. A comparative analysis of the results of experimental data and model calculations showed a good correlation between the preferred starting points and the direction of transcription, which in turn confirmed the existence of a relationship between the dynamics of open states and the functioning of promoters.