Nonlinear Coupled Differential Equations Research Articles

Numerical study of boundary stresses on Jeffery-Hamel flow subject to Soret/Dufour effects

We have mathematically modeled the problem of convective Jeffery-Hamel flow in the presence of thermo-diffusion and diffusion-thermo effects. The flow is considered to be the steady, incompressible and unidirectional by assuming the velocity only in radial direction. We have also modeled the boundary conditions representing the boundary stresses and assumed that these stresses apply only radially. Then we analyzed the effects of these stresses on the problem. The boundary stresses are defined through traction boundary conditions which are of Robin type. The flow is assumed in the domain [[Formula: see text]] and the constant flow rate condition is applied instead of taking symmetry of the channel. The governing equations are nonlinear coupled partial differential equations. They become a system of nonlinear coupled ordinary differential equations by adopting suitable transformations. Therefore, they are handled numerically. For this purpose, we have developed a scheme using shooting method to solve nonlinear coupled differential equations representing the flow problem and shown the solution procedure via a flow chart. In order to obtain the numerical results, we have taken different values of parameters. Since we have considered the problem of Newtonian fluid, therefore, we considered the values [Formula: see text] for air, carbon disulfide and water respectively at 25°C. Whereas the values of Schmidt number are considered as [Formula: see text] for hydrogen, water vapor, and ammonia respectively at 25°C. The numerical solution is validated by comparing the results with those published in literature. The numerical results of present work show that there occurs no-slip for wedge angle of [Formula: see text] by fixing the other parameters whereas slip is achieved in some other situations. It is noticed that the flow reversal occurs for convergent as well as divergent flow. All boundary layers are affected greatly due to presence of the boundary stresses and the hydrodynamic boundary layer that disappears in the absence of boundary stresses. The freedom in constraint of channel symmetry leads to the occurrence of symmetric as well as asymmetric flow. An interesting theoretical result of opposite behaviors of temperature and concentration profiles for convergent and divergent flows is observed in case of variation of all of the governing parameters. It is also noticed that the simultaneous increase of Soret number and decrease of Dufour number results in occurrence of opposite behavior of temperature and concentration profiles for divergent flow. However, it results in similar behavior of these profiles. The results show that the boundary layer completely disappears in the absence of boundary stresses.

Qualitative Investigation of Nonlinear Fractional Coupled Pantograph Impulsive Differential Equations

Qualitative Investigation of Nonlinear Fractional Coupled Pantograph Impulsive Differential Equations

Numerical solution of density-driven groundwater flows using a generalized finite difference method defined by an unweighted least-squares problem

Density-driven groundwater flows are described by nonlinear coupled differential equations. Due to its importance in engineering and earth science, several linearizations and semi-linearization schemes for approximating their solution have been proposed. Among the more efficient are the combinations of Newtonian iterations for the spatially discretized system obtained by either scalar homotopy methods, fictitious time methods, or meshless generalized finite difference method, with several implicit methods for the time integration. However, when these methods are used, some parameters need to be determined, in some cases, even manually. To overcome this problem, this paper presents a novel generalized finite differences scheme combined with an adaptive step-size method, which can be applied for solving the governing equations of interest on non-rectangular structured and unstructured grids. The proposed method is tested on the Henry and the Elder problems to verify the accuracy and the stability of the proposed numerical scheme.

Effect of Asymmetry in the Modulation Parameters on Self-Focusing of Asymmetric Finite Airy-Gaussian Laser Beam in Collisionless Plasma

The self-focusing/defocusing of asymmetric finite Airy-Gaussian (AiG) laser beam has been investigated by employing standard Wentzel–Kramers–Brillouin (WKB) and paraxial-ray approximations in a nonrelativistic regime for underdense plasma. The second-order non-linear coupled differential equations have been solved numerically by using the fourth-order Range-Kutta method. The effect of asymmetry in the modulation parameters on the self-focusing/defocusing of the asymmetric finite AiG laser beam in collisionless plasma has been studied. It is observed that the self-focusing/ defocusing of asymmetric finite AiG laser beam is strongly connected with the initial values of the laser and plasma parameters such as modulation parameters and plasma frequency.

Belousov–Zhabotinsky reaction: an open-source approach

The time-dependent change in concentration of reactants in the oscillatory Belousov–Zhabotinsky reaction defined by non-linear ordinary coupled differential equations has been derived by applying normalized non-dimensional computational methods and presented graphically. The accurate normalisation procedure eschews complex mathematical steps; however, choosing the appropriate references to define the dimensionless variables necessitates a deep physical understanding of the problem. The dimensionless groups are then derived to provide the functional dependences of the unknowns of interest. Finally, to cross-check all the mentioned dependences, a computational simulation has been carried out using domain-specific to general-purpose programming languages. Although there are countless programming languages present to carry out computer simulations, a flexible, portable, and open-source language like python proves fruitful.

Combined Effect of Spatio-Temporal Dynamics of Laser Pulse on Electron Acceleration in Relativistic Plasma

This article presents the spatial and temporal dynamics of high-power Gaussian laser pulse in relativistic plasma. At higher intensities, the self-focusing and self-compression of laser pulse take place due to the variation in the mass of plasma electrons moving with speed comparable to the speed of light. The two non-linear coupled differential equations for spatial and temporal pulsewidth parameters have been obtained using the moment theory approach. These equations have been solved numerically using the Runga-Kutta method. The variation in the spatial and temporal pulsewidth parameters with the distance of propagation is then used to calculate the energy gained by the electrons, which are trapped by the excited electron plasma wave during laser-plasma interaction. It is observed that spatio-temporal dynamics play an important role in electron acceleration in relativistic plasma.

Existence and U-H Stability Results for Nonlinear Coupled Fractional Differential Equations with Boundary Conditions Involving Riemann–Liouville and Erdélyi–Kober Integrals

The purpose of this article is to discuss the existence, uniqueness, and Ulam–Hyers stability of solutions to a coupled system of fractional differential equations with Erdélyi–Kober and Riemann–Liouville integral boundary conditions. The Banach fixed point theorem is used to prove the uniqueness of solutions, while the Leray–Schauder alternative is used to prove the existence of solutions. Furthermore, we conclude that the solution to the discussed problem is Hyers–Ulam stable. The results are illustrated with examples.

Fractional analysis of thin-film flow in the presence of thermal conductivity and variable viscosity

This article contains a fractional analysis of the thin-film steady flow of magneto-hydrodynamic fluid with variable viscosity and thermal conductivity by permeable stretched surface. The obtained nonlinear coupled fractional differential equations are solved through fractional calculus and the homotopy perturbation method, and the validity of obtained solutions is confirmed by finding residual errors. The impact of various parameters on the velocity and temperature profiles is studied graphically in fractional space. Some key observations related to these effects are witnessed. It is observed that the temperature fractional parameter has a negligible effect on the temperature profile. Also, it is seen that the fluid fractional parameter has an inverse relation with velocity and temperature profiles. The investigation also discloses that radiation and magnetic parameters have direct association while Prandtl number has an inverse association with temperature.

On system of nonlinear coupled differential equations and inclusions involving Caputo-type sequential derivatives of fractional order

We investigate a new class of boundary value problems of a nonlinear coupled system of sequential fractional differential equations and inclusions involving Caputo fractional derivatives and boundary conditions. We use standard fixed-point theory tools to deduce sufficient criteria for the existence and uniqueness of solutions to the problems at hand. Examples are discussed to illustrate the validity of the proposed results.

Numerical experiment to examine activation energy and bi-convection Carreau nanofluid flow on an upper paraboloid porous surface: Application in solar energy

The current article examines the physical features of magneto-hydrodynamics (MHD) Carreau nanofluid bi-convection flow past an upper paraboloid porous surface. The presence of activation energy and chemical reaction are also examined. The notion of the Carreau nanofluid was established with recent parametric quantities to accomplish the momentum governing equation.The momentum equation is furthermore reformed by applying Darcy law. The governing partial differential equations are reconstructed into ordinary differential equations by making use of similarity transforms. The reformed non-linear coupled differential equations are solved by implementing the spectral local linearization technique (SLLM). The SLLM technique is forthright to grow and employing since this algorithm relies upon a smoothly Uni-variant linearizing of non-linear functions. The numerical interpretation of SLLM is admirable cause it establishes a system of equations, which consecutively resolve by functioning the outcomes through the first equation into the subsequent equation. To advance and strengthen the converged solutions for the SLLM algorithm, the SOR method is also implemented. The numerical outcomes of temperature, velocity, and concentration profiles, even though the skin friction, Sherwood number, and Nusselt number are adorned via graphical curves for all the arising parametric quantities and analyzed in detail.

Dual Solutions of EMHD Nanofluid at Stretching Sheet with Mixed Convection Slip Boundary Condition

In this paper stagnation boundary layer flow of nanofluid with mixed convection heat and mass transfer with Electrical Magneto Hydrodynamic (EMHD) effects over a second-order momentum slip boundary condition have been mathematically analysed. The governing equations are transformed by similarity variable and the problem becomes coupled third-order nonlinear coupled differential equations. We use fourth-order Runge Kuta method and shooting technique to find the solution. The effect of second-order momentum slip condition with linear thermal slip condition has determined. Variation of all nano energy conversion parameters depends on different factors has shown graphically. Some of the parameters possesses dual solution at different values of second-order velocity slip parameter (β1&β2).

Numerical and Statistical Analysis of Dissipative and Heat Absorbing Graphene Maxwell Nanofluid Flow Over a Stretching Sheet

This paper is basically devoted to carry out an investigation regarding the unsteady flow of dissipative and heat absorbing hydromagnetic graphene Maxwell nanofluid over a linearly stretched sheet taking momentum and thermal slip conditions into account. Ethylene glycol is selected as a base fluid while graphene particles are considered as nanoparticles. The highly nonlinear mathematical model of the problem is converted into a set of nonlinear coupled differential equations by means of fitting similarity variables. Further, Runge-Kutta Fehlberg algorithms along with the shooting scheme are instigated to analyse the numerical solution. The variations in graphene Maxwell nanofluid velocity and temperature owing to different physical parameters have been demonstrated via numerous graphs whereas Nusselt number and skin friction coefficients are illustrated in numeric data form and are reported in different tables. In addition, a statistical method is implemented for multiple quadratic regression estimation analysis on the numerical figures of wall velocity gradient and local Nusselt number to establish the connection among heat transfer rate and physical parameters. Our numerical findings reveal that the magnetic field, unsteadiness, inclination angle of magnetic field and porosity parameters boost the graphene Maxwell nanofluid velocity while Maxwell parameter has a reversal impact on it. The regression analysis confers that Nusselt number is more prone to heat absorption parameter as compared to Eckert number. Finally, the numerical findings are compared with those of earlier published articles under restricted conditions to validate the numerical solution. The comparison of numerical findings shows an excellent conformity among the results.

Semi Analytic Solution of Hodgkin-Huxley Model by Homotopy Perturbation Method

Hodgkin-Huxley model is a system of four non-linear coupled differential equations which describes and explains the threshold and action potential by a stimulus arising in a single neuron. The solution and analysis of Hodgkin-Huxley equations is a formidable task because of the coupling between non-linear differential equations, lots of unknowns and their dependence on many physical parameters. Although this model has been solved by numerical methods, finding an analytic solution is interesting due to the challenges that the continuum model offers. In this paper, first order semi analytic solution of this model, in space-clamped situation, is derived by Homotopy Perturbation Method. We applied this technique in piece wise manner due to the strong and complex coupling between the variables in the model. Without this modification, finding an accurate analytic solution is impossible for this neural model. Results show that computed analytic solution has excellent agreement with higher order numerical solution. Robustness of the computed analytic solution in different physical scenarios is examined. Further, this analytic solution can describe many key properties such as the threshold potential, the action potential and the refractory period. MATLAB software is used to simulate the solution.

Approximate Analytical Solutions for the Euler Equation for Second-Row Homonuclear Dimers.

This work presents a new method how to obtain approximate analytical solutions for the Euler equation for second-row homonuclear dimers. In contrast to the well-known Kohn-Sham method where a system of N nonlinear coupled differential equations must be solved iteratively, orbital-free density functional theory allows to access the minimizing electron density directly via the Euler equation. For simplified models, here, an atom-centered monopole expansion with one free parameter, solutions of the electron density can be obtained analytically by solving the Euler equation at the bond critical point. The procedure is exemplarily carried out for N2, C2, and B2, yielding bound molecules with an internuclear distance of 2.01, 2.43, and 3.07 bohr, respectively.

The LISE package: Solvers for static and time-dependent superfluid local density approximation equations in three dimensions

Nuclear implementation of the density functional theory (DFT) is at present the only microscopic framework applicable to the whole nuclear landscape. The extension of DFT to superfluid systems in the spirit of the Kohn-Sham approach, the superfluid local density approximation (SLDA) and its extension to time-dependent situations, time-dependent superfluid local density approximation (TDSLDA), have been extensively used to describe various static and dynamical problems in nuclear physics, neutron star crust, and cold atom systems. In this paper, we present the codes that solve the static and time-dependent SLDA equations in three-dimensional coordinate space without any symmetry restriction. These codes are fully parallelized with the message passing interface (MPI) library and the time-dependent code takes advantage of graphic processing units (GPU) for accelerating execution. The dynamic code has checkpoint/restart capabilities and for initial conditions one can use any generalized Slater determinant type of wave function. By generating the appropriate initial quasi-particle wave-functions in a static calculation only, the time-dependent code can describe a large number of physical problems: nuclear fission, collisions of heavy ions, the interaction of quantized vortices with nuclei in the nuclear star crust, excitation of superfluid fermion systems by time dependent external fields, quantum shock waves, domain wall generation and propagation, the dynamics of the Anderson-Bogoliubov-Higgs mode, dynamics of fragmented condensates, vortex rings dynamics, generation and dynamics of quantized vortices, their crossing and recombinations and the incipient phases of quantum turbulence. Program summaryProgram title: LISECPC library link to program files:https://doi.org/10.17632/mwr7rsxpjw.1Licensing provisions: BSD 3-ClauseProgramming language: C, CUDANature of problem: The description of nuclear fission and nuclear reactions within the mean field approximation in real time within the extension of the density functional theory to superfluid systems is an extremely computationally demanding problem, which requires the solutions of a very large system of nonlinear coupled complex partial differential equations in 3+1 coordinates. Similar problems also appear in the case of cold atoms and in the dynamics of the neutron star crust, which have been tackled within the same framework with the same codes, using appropriate initial quasi-particle wave functions and/or an appropriate time-dependent external potential.Solution method: The evolution equations are discretized on a 3-dimensional spatial lattice and propagated in time. Spatial derivatives are evaluated using the fast Fourier transform technique. The propagation in time is performed using a predictor-modifier-corrector algorithm due to Adams-Bashforth-Milne, which requires only two evaluations of the right hand side of the equations per time step. The numerical accuracy of the time integration is ∼O(Δt)6. This method has a low truncation error, excellent numerical stability, and low roundoff errors.Additional comments including restrictions and unusual features: The code has been implemented on a variety of supercomputers (Jaguar, Titan, Piz Daint, Tsubame, Summit, Sierra) and demonstrates excellent scaling properties. The strong scaling capabilities can be significantly affected if a very large number of GPUs is used, when the communication time between processes overtakes the computation time as the dominant run time cost.

Effect of nonlinear radiation on flow and heat transfer of dusty fluid over a stretching cylinder with Cattaneo–Christov heat flux

The heat transfer through the thermal radiation is very significant in many industrial processes, biomedical engineering, technological devices and thermal therapeutic process. These comprise nuclear power plants, propulsion of rockets, missiles, satellites and space vehicles, photochemical reactors, solar collector performance, heat exchangers. The boundary layer flow and the heat transport of dusty liquid over a stretching cylinder are discussed. The impact of nonlinear thermal radiation is explored. Cattaneo–Christov heat flux model is considered to formulate the energy equation. The resulting equations of the physical system are converted into a system of nonlinear ordinary coupled differential equations with the help of suitable similarity transformations and numerically solved by using Runge–Kutta–Fehlberg’s method along with shooting technique. The influence of nondimensional parameters is analyzed and interpreted graphically. The heat transfer rate and local coefficient of skin friction are also calculated for diverse nondimensional constraints and plotted.

Simultaneous effects of Brownian motion and thermophoretic force on Eyring–Powell fluid through porous geometry

Abstract The current study addresses the influences of Brownian motion and thermophoretic force on non-Newtonian fluid flow. Eyring–Powell fluid serves as the base fluid for heat and mass transfer through a porous channel. Buongiorno model for nanofluid is incorporated into the convection–diffusion equation to investigate the random motion of tiny spherical particles. Additional contributions of viscous dissipation and thermal radiation have also been applied by formulating two different types of flows. A system of nonlinear coupled differential equations is solved with the help of the “regular perturbation method”. For the limiting case, a numerical solution is obtained to validate the computational results with existing literature and it is found to be in complete agreement. Eventually, it is inferred that the heat transfer rate dominates in nanofluid flow due to the moving plate, while the mass transfer is more prominent in generalized Couette nanoflow of Eyring–Powell fluid.

Magnetized Dissipative Soret Effect on Chemically Reactive Maxwell Fluid over a Stretching Sheet with Joule Heating

The present research paper deals with the study of heat and mass transfer characteristics of steady viscous incompressible two-dimensional Maxwell fluid flow past a stretching sheet under the influence of magnetic field and the Soret effect. A well-known non-Newtonian Maxwell fluid flow model is used to differentiate it from the Newtonian fluids. The present physical problem gives the set of highly nonlinear-coupled partial differential equations that are not amenable to any of the direct techniques. The resultant nonlinear system of partial differential equations is reduced to a set of nonlinear ordinary differential equations by using suitable similarity transformations. Due to the inadequacy of analytical techniques, a bvp4c MATLAB function is used to solve the developed nonlinear system of equations. The simulated results are shown for various values of physical parameters in the flow regime. Additionally, the numerical values of skin-friction coefficient, heat, and mass transfer rates are calculated and tabularized. From the present investigation, it is observed that the normal and axial velocity profiles decreased for the enhancing values of the magnetic parameter. Increasing the Prandtl and Schmidt numbers reduces the temperature and concentration profiles in the flow region, respectively. Increasing the Maxwell fluid parameter decreases the velocity profile and magnifies the temperature field. Additionally, increasing the Soret number increases the concentration profile in the flow regime. Comparison of current similarity solutions with available results indicates the accuracy and guarantee of the present numerical results and the used method.

Entropy generation and heat transfer analysis for ferrofluid flow between two rotating disks with variable conductivity

Present model analyze the flow and heat transfer of water-based carbon nanotubes (CNTs) [Formula: see text] ferrofluid flow between two radially stretchable rotating disks in the presence of a uniform magnetic field. A study for entropy generation analysis is carried out to measure the irreversibility of the system. Using similarity transformation, the governing equations in the model are transformed into a set of nonlinear coupled differential equations in non-dimensional form. The nonlinear coupled differential equations are solved numerically through the finite element method. Variable viscosity, variable thermal conductivity, thermal radiation, and volume concentration have a crucial role in heat transfer enhancement. The results for the entropy generation rate, velocity distributions, and temperature distribution are graphically presented in the presence of physical and geometrical parameters of the flow. Increasing the values of ferromagnetic interaction number, Reynolds number, and temperature-dependent viscosity enhances the skin friction coefficients on the surface and wall of the lower disk. The local heat transfer rate near the lower disk is reduced in the presence of Harman number, Reynolds number, and Prandtl number. The ferrohydrodynamic flow between two rotating disks might be useful to optimize the use of hybrid nanofluid for liquid seals in rotating machinery.

Water-Based Fe3O4 Ferrofluid Flow Between Two Rotating Disks with Variable Viscosity and Variable Thermal Conductivity

The impact of variable viscosity and variable conductivity is presented on the water-based Fe3O4 nanofluid flow between two parallel stretchable rotating disks. The stretching and rotating rates of both disks play an essential role in velocity distribution and heat transfer study. The similarity transformation is utilized to customize the momentum and energy equations into nonlinear-coupled differential equations with some influential dimensionless physical parameters. The nonlinear-coupled differential equations of the transformed system are solved numerically through the finite element method using COMSOL Multiphysics. The outcomes for velocity and temperature distributions are shown in the presence of considered physical parameters in the flow. Enhancing the stretching of disks, rotation speed, and variable thermal conductivity enhances the temperature of nanofluid. Permeability parameter, ferromagnetic interaction numbers, and Reynolds numbers also have a major role in the velocity and temperature distributions. Strengthening the rotation number, ferromagnetic interaction number and Reynolds number amplifies the friction on the surface of the disk and shrinks the local heat transfer rate. The rheological behavior of ferrofluid and the role of variable thermal conductivity in the present study might be useful to select the best possible nanofluid for medical purposes.