System Of Nonlinear Differential Equations Research Articles

Hamiltonian formalism for Bose excitations in a plasma with a non-Abelian interaction I: Plasmon – hard particle scattering

Hamiltonian theory for collective longitudinally polarized gluon excitations (plasmons) interacting with classical high-energy test color-charged particle propagating through a high-temperature gluon plasma is developed. A generalization of the Lie-Poisson bracket to the case of a continuous medium involving bosonic normal field variable ak⁎a and a non-Abelian color charge Qa is performed and the corresponding Hamilton equations are presented. The canonical transformations including simultaneously both bosonic degrees of freedom of the soft collective excitations and degree of freedom of hard test particle connecting with its color charge in the hot gluon plasma are written out. A complete system of the canonicity conditions for these transformations is derived. The notion of the plasmon number density Nkaa1′, which is a nontrivial matrix in the color space, is introduced. An explicit form of the effective fourth-order Hamiltonian describing the elastic scattering of a plasmon off a hard color particle is found and the self-consistent system of Boltzmann-type kinetic equations taking into account the time evolution of the mean value of the color charge of the hard particle is obtained. On the basis of these equations, a model problem of the interaction of two infinitely narrow wave packets is considered. A system of nonlinear first-order ordinary differential equations defining the dynamics of the interaction of the colorless Nkl and color Wkl components of the plasmon number density is derived. The problem of determining the third- and fourth-order coefficient functions entering into the canonical transformations of the original bosonic variable ak⁎a and color charge Qa is discussed. With the help of the coefficient functions obtained, a complete effective amplitude of the elastic scattering of plasmon off hard test particle is written out.

Modeling heat and mass transfer in granular flows between vertical parallel plates

Gravity-driven granular flows between vertical parallel plates at elevated temperatures are integral in the development of the next generation of particle-based heat exchangers for concentrated solar power. Efficient modeling methodologies that accurately captures the heat transfer mechanisms of temperature-dependent granular flows are essential to effectively design and evaluate these heat exchangers. Transient, pseudo-one-dimensional heat and mass transfer models were developed for inlet flow temperatures between 473 and 1073 K. The mass transfer models for the granular flows were resolved using discrete element method simulation, providing detailed temporal and spatial distributions of flow parameters in good agreement with experimental measurements. The heat transfer model employed a one-dimensional, two-phase, continuum approach, and a system of non-linear differential equations was solved via finite difference method. The particle-to-wall contact heat transfer was determined with an effective thermal conductance model. Radiative heat transfer between particles and walls was also considered. The predicted particle and wall temperatures for two different channel widths of 6.4 and 2 mm were well-correlated to previously obtained experimental measurements with Pearson’s correlation coefficients greater than 0.91. A significant disparity in particle temperature of up to 72 K was observed when radiative heat transfer was omitted, and a maximum temperature difference of 11 K was observed when temperature-dependent granular flow properties were not considered. The model revealed that radiative heat transfer potentially contributed to > 30 % of heat losses in wider channels at high inlet temperatures, and conduction was the dominated heat transfer mechanisms in narrower channels (∼70 %) due to increased particle-to-wall contact. This accurate model, leveraging discrete element method in a streamlined approach, advances the understanding and modeling of heat transfer in granular flows at elevated temperatures, enabling more efficient and reliable solar thermal energy storage and transport.

Impacts of activation energy and electroosmosis on peristaltic motion of micropolar Newtonian nanofluid inside a microchannel

This study investigates the impact of electroosmosis on the peristaltic flow of unsteady micropolar nanofluid with heat transfer. The findings could enhance the design of peristaltic pumps, potentially improving drug delivery systems, simulations of blood flow in medical devices, and cancer treatments. The fluid under investigation adheres to a micropolar model and flows through a microchannel that exhibits peristalsis along its walls. Moreover, the system is subjected to various external effects, including a uniform magnetic field, the electroosmotic phenomenon, heat absorption, and a chemical reaction with activation energy. Consequently, the problem is mathematically modulated by a system of nonlinear partial differential equations governing the velocity, temperature, and nanoparticle concentration. By employing wave transformation, these governing equations are reduced to ordinary differential equations (ODEs). The reduced equations were solved both analytically, using the homotopy perturbation method, and numerically, using the Runge–Kutta–Merson method. A comparison was made between the solutions, which were found to be closely aligned. Furthermore, a series of figures were employed to provide visual representation and discussion of the implications of the physical properties. The calculations reveal that the electroosmotic flow (EOF) enhances the axial flow of the micropolar fluid along the direction of the applied electric field. It is also observed that the increase in the activation energy (which indicates a low reaction rate) increases the concentration profile whereas the increase in the reaction rate parameter reduces the concentration profile. Additionally, the spin velocity of the particles is diminished by either an increase in the magnetic parameter or the coupling parameter.

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

Abstract This paper analyzes an age-group susceptible-infected-recovered (SIR) model. Theoretical results concerning the conservation of the total population, the positivity of the analytical solution, and the final size of the epidemic are derived. Since the model is a nonlinear system of ordinary differential equations (ODEs), a numerical approximation is considered, based on Standard and non-Standard Finite Difference methods, and on a Modified Patankar-Runge–Kutta (MPRK) method. The numerical preservation of the qualitative properties of the analytical solution is studied. The obtained results are applied to the diffusion of information in social networks, and the effectiveness of the different numerical approaches is shown through several numerical tests on real data.

Modeling of micropolar hybrid nanofluids flow across porous medium between permeable parallel plates with nonlinear thermal radiative in MHD

The study pertains on the dynamics and thermal distribution of magnetohydrodynamics (MHD) micropolar hybrid nanofluids flowing between two parallel plates channel. It explores the enhancement of heat transport processes by blending the base fluid with nanoparticles at varying concentrations. A system of nonlinear partial differential equations is developed as governing equations for momenta and heat energy. To facilitate numerical solutions for fluid temperature and velocities, this formulation is translated into ordinary differential form using the necessary similarity transformation. By utilizing Matlab code for the Runge–Kutta method and shooting approach, we evaluate the results. It has been observed that an increase in the concentration of hybrid nanoparticles enhance the heat transmission. The salient findings of this study reveal alteration in the velocity profiles for both nanofluid (SWCNT/water) and hybrid nanofluids (SWCNT + MWCNT/water) flow. Notably, with a substantial input of magnetic field strength (M) and porosity parameters (P0), the velocity is increased near the walls but it decelerates toward the center of channel. Additionally, the temperature rises for hybrid nanofluids. Furthermore, an increase in the θ w temperature ratio parameter leads to an enhanced thermal distribution.

Analysis of fluid flow and heat transfer in CNT-infused spiraling disk

This study investigates the fluid flow and heat transfer characteristics influenced by the thermophysical properties of carbon nanotube suspension in engine oil over a stretching-spiraling disk. The surface comprises Homan stagnation point flow impinging normal to a rotating, radially stretching disk, which gives velocity a form of logarithmic spiral. Similarity ansatz transformed coupled nonlinear differential equation system is derived and solved numerically using MATLAB’s bvp4c collocation method. Numerical and asymptotic solutions are also obtained against the controlled parameters via graphical representations and tabular data. Findings indicate that the magnetic field enhances temperature distribution while reducing the velocity field. At the same time, stretching increases radial velocity but decreases azimuthal velocity and temperature profiles, regardless of carbon nanotube type. The asymptotic values of skin friction decline with an increase in the spiralisng angle and in the absence of a magnetic field for both SWCNTs and MWCNTs cases.

Combined effects of rotating and thermal radiation on heat and mass transfer of MHD flow in converging and diverging channels

ABSTRACT Current study deals with influence of rotating and thermal radiative fluxing on fluid’s thermal, hydrodynamic, and concentration behavior flowing between two nonparallel walls in existence of uniform magnetized force. The modeling is based on nonlinear partial differential equations (PDEs) such as continuity, momentum, temperature, and concentricity equations. These equations are converted into a system of nonlinear ordinary differential equations (ODEs) utilizing similarity transformations. Analytical solution has been constructed using the Differential Transform Method (DTM). The present results are compared to results obtained by the HAM package and by the numerical procedure (RKF-45) for validation. The impacts of active variables like the rotational factor, Hartmann, Schmidt numbers, and thermal radiation parameter, are investigated on the rapidity, temperature, and concentricity profiles. Moreover, frictional force factor, Nusselt, and Sherwood quantities. It is found that speed and concentricity of the fluid flow increase with the growth of Hartmann number, whereas a reverse behaviour is observed for thermal distribution. Results obtained also reveal an improvement in frictional force factor, Nusselt and Sherwood numbers with the enhancement of Reα parameter. It is found that an increase in the rotational parameter Ro and the radiation factor Rd leads to a decrease in the Nusselt number.

Analytical solutions of virus propagation model in blockchain networks

The main goal of this paper is to find analytical solutions of a system of nonlinear ordinary differential equations arising in the virus propagation in blockchain networks. The presented method reduces the problem to an Abel differential equation of the first kind and solve it directly.

Numerical analysis of the heat transfer application on a convective tangent hyperbolic nanofluid flow over a porous stretching cylinder with stratification effects

The nanofluids has attractive applications in the industrials and engineering field because nanofluids offer the improvement in the thermal efficiencies of flow, which is very fruitful in the energy sector. This study emphases on the numerical exploration of heat and mass transfer efficiencies on a mixed convective flow of tangent hyperbolic nanofluid under the consideration of gyrotactic microorganisms. To analyze the magnetized boundary layer flow, the stratification boundary conditions are considered on the surface of the stretching cylinder. The thermal analysis involves the important features such as activation energy, nonuniform heat source/sink, and nonlinear thermal radiation. The flow model is altered into system of nonlinear ordinary differential equations by utilization of suitable similarity variables and Bvp4c solver on the MATLAB to find numerical solutions. The graphical and tabulated investigation is presented for various developing parameters thoroughly. The results indicate that by the increment of mixed convection parameters the fluid velocity accelerate consequently, whereas raising values of the Weissenberg number reduce the fluid velocity. TThe greater values of the Prandtl number lead to a decrease in fluid temperature, while the concentration of nanoparticles rises with the increase in activation energy parameters. Additionally, the heat transfer rate reduces due to stronger values of thermophoresis parameter, although reverse tendency is observed for the greater values of Brownian motion parameter.

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.

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.

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.

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.

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.

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

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

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.