System Of First Order ODEs Research Articles

Application of Lie symmetry to find an analytical solution for reactor point kinetics equation with prompt jump approximation and power feedback

The reactor Point Kinetics Equations (PKE) are simpler zero-dimensional approximation to space dependent dynamical models of nuclear reactor core, that are accurate enough to describe transients in small to medium size fast reactor cores. Even these simplified equations can be solved only by numerical methods, except in a very few restrictive cases, where they are amenable to analytical solution. Symmetry methods using Lie's point symmetry have been shown to be a systematic and powerful tool to solve any given ordinary or partial differential equation. An approximation of the PKE, known as Prompt Jump Approximation (PJA) converts the coupled system of first order ODEs with one delayed-neutron precursor group and power feedback, into a single first order nonlinear ordinary differential equation. In this study, we demonstrate an application of Lie symmetry method for solving the point kinetics equation under PJA. The analytical solution obtained is compared with benchmark numerical solution of PKE with PJA.

Optimizing entropy in mixed convective MHD dissipative nanofluid with cross-diffusion and nonlinear velocity slip

Entropy optimization is crucial for enhancing fluid dynamics and heat transfer, leading to improved efficiency, energy conservation, and environmental sustainability. Recognizing its significance, we examine the influence of radiation, Joule heating, heat sources, mixed convection, and nonlinear slip on the entropy-optimized flow of a dissipative nanofluid over a curved surface. To seamlessly integrate within the MATLAB bvp4c framework, we convert the set of multi-order ODEs, derived from fundamental PDEs via a similarity transformation approach, into a system of first-order ODEs. Graphs and analytical discussions are used to evaluate the parameters of the problem. The velocity within the boundary layer, along with entropy, experiences reduction due to the enhancement in both first- and second-order slip velocity parameters, as well as curvature parameters. The temperature plot improves with the increase in parameters such as the Soret number and Dufour number, as well as radiation. Conversely, the temperature plot reduces with the rise in suction values. The study indicates that a higher Brinkman number is associated with a lower Bejan number, while the first- and second-order velocity slip parameters have a contrasting effect, leading to an increase in the Bejan number.

Magnetic williamson hybrid nanofluid flow around an inclined stretching cylinder with joule heating in a porous medium

The study of an inclined stretching cylinder in a porous medium holds significant implications for understanding complex fluid dynamics and heat transfer phenomena, mainly when influenced by a hybrid nanofluid containing Cu and Al2O3, Joule heating, and a magnetic field in the presence of Williamson fluid. The nonlinear ordinary differential equations are derived by applying pertinent similarity transformations to partial differential equations. Subsequently, these non-dimensional ordinary differential equations are transformed into a system of first-order ODEs and numerically solved using Maplesoft's symbolic computation software MAPLE 2023. The influence of governing parameters on dimensionless velocity, temperature, concentration, skin friction, heat, and mass transfer rates is investigated. This investigation is crucial for various practical applications, including geothermal reservoirs, enhanced oil recovery, and environmental processes involving porous media. It is demonstrated that Biot and Eckert numbers and nanofluid parameters contribute to increased surface temperature, while the Lewis number leads to a decrease in dimensionless concentration. Moreover, the curvature parameter is associated with an increase in dimensionless concentration, and skin friction demonstrates a direct relationship with the magnetic parameter while inversely correlating with the cylinder's curvature.These results contribute to the scientific understanding of these complex interactions and provide valuable insights for engineering applications, such as designing efficient heat exchangers, optimizing cooling systems, and advancing technologies involving fluid flow and magnetic fields.

Significance of irregular heat source and Arrhenius energy on electro-magnetohydrodynamic hybrid nanofluid flow over a rotating stretchable disk with nonlinear radiation

For its applications in nuclear reactors, food processing, chemical engineering, water emulsions and thermal power generating systems, the significance of irregular heat source and Arrhenius energy on electro-magnetohydrodynamic hybrid nanofluid flow over a rotating stretchable disk with nonlinear radiation have been investigated. The flow problem has been modeled utilizing the modified Buongiorno model and the thermophysical characteristics of water-based Cu − F e 3 O 4 hybrid nanoliquid. Effects like passive control of nanoparticles, hydrodynamic slip and convective boundary conditions are also heeded to boost the realistic nature of this work. Further, engineering quantities like moment coefficient and pumping efficiency of the disk are also elucidated which boosts the novelty of this research work. The modeled governing equations are transmuted into a system of first-order ODEs, with the help of apposite similarity transformations, which are then numerically resolved using the finite-difference based bvp5c algorithm. It is noticed that per unit increase in the electric parameter decreases the skin friction coefficient by 41.75% and increases the heat transfer rate by 15.31%. It is also observed that the entrainment velocity is directly proportional to the changes in electric field parameter and is inversely proportional to the changes in volume fraction of copper and magnetite nanoparticles.

Response surface optimization for the electromagnetohydrodynamic Cu-polyvinyl alcohol/water Jeffrey nanofluid flow with an exponential heat source

This article examines the sensitivity analysis of Jeffrey fluid flow in the presence of copper nanoparticles with polyvinyl alcohol water serving as the base fluid under the influence of source terms like Joule heating, viscous dissipation, exponential heat source, and Arrhenius activation energy via a non-linear vertically elongating surface of non-uniform thickness. Governing PDEs are first converted into ODEs using a suitable similarity transformation. The resulting coupled ODEs are converted into a system of first-order ODEs and then solved using the shooting iteration technique with the RK-4th order method. The remarkable impacts of numerous parameters like Deborah number, Hartmann number, electric field parameter, thermal and solutal Grashof number, Prandtl and Eckert numbers, exponential heat source parameter, Lewis number, chemical reaction parameter, and activation energy parameter associated with the flow properties are discussed graphically. The effects of various physical parameters and the dependence of the response factors on the input parameter are explored using an experimental design and sensitivity analysis based on Response Surface Methodology. Response Surface Methodology is utilized to determine the output response variables of model dependencies, such as the drag coefficient and Nusselt number on the independent input parameters, namely the electric field parameter and Deborah numbers for the drag coefficient, and electric field parameter, thermophoresis diffusion parameter, and exponential heat source parameter for the Nusselt number. Local Nusselt number enhances with increment in the electric field parameter and thermophoresis diffusion parameter values, whereas it decelerates with the rise in exponential heat source parameter. The non-Newtonian nanofluids have proven their great potential in thermal transportation which has a diverse range of applications in thermal energy technologies such as lubrication, natural gas networks, spray processes, and cooling of nuclear reactors, spinning of fibers, cryopreservation, drawing of copper wires, nano-drug delivery, electronics, and many more.

Transport phenomena in Darcy-Forchheimer flow over a rotating disk with magnetic field and multiple slip effects: modified Buongiorno nanofluid model

The current study focuses on investigating the significance of magnetic field and multiple slip constraints on the water-based graphene Darcy-Forchheimer nanofluid flow over a rotating disk. For a realistic approach, the modified Buongiorno nanofluid model that incorporates the combination of effective thermophysical properties, thermophoretic diffusion, and Brownian diffusion has been utilized. Engineering quantities like moment coefficient and pumping efficiency of the disk are also elucidated. The transmutation of the mathematically modeled nonlinear equations into a system of first-order ODEs are achieved through Von Kármán’s similarity transformations and are then resolved using the shooting technique along with the Runge–Kutta-Fehlberg algorithm. The highest heat transfer rate is experienced for smaller values of magnetic field parameter, Forchheimer number, porosity parameter, and thermal slip constraint. An increment rate of 88.14355% in the azimuthal drag coefficient and a decrement rate of 39.849732% in the radial drag coefficient are noted when the values of hydrodynamic slip constraint are augmented. Further, per unit increase in the volume fraction of graphene nanoparticles descends the moment coefficient and the entrainment velocity by 456.25501% and 12.48875%, respectively. The tidings of this numerical simulation have applications in spin coating, centrifugal filtration, medical equipment, gas turbine rotors, and thermal power generating systems.

Entropy Generation and Thermal Radiation Analysis of EMHD Jeffrey Nanofluid Flow: Applications in Solar Energy.

This article examines the effects of entropy generation, heat transmission, and mass transfer on the flow of Jeffrey fluid under the influence of solar radiation in the presence of copper nanoparticles and gyrotactic microorganisms, with polyvinyl alcohol-water serving as the base fluid. The impact of source terms such as Joule heating, viscous dissipation, and the exponential heat source is analyzed via a nonlinear elongating surface of nonuniform thickness. The development of an efficient numerical model describing the flow and thermal characteristics of a parabolic trough solar collector (PTSC) installed on a solar plate is underway as the use of solar plates in various devices continues to increase. Governing PDEs are first converted into ODEs using a suitable similarity transformation. The resulting higher-order coupled ODEs are converted into a system of first-order ODEs and then solved using the RK 4th-order method with shooting technique. The remarkable impacts of pertinent parameters such as Deborah number, magnetic field parameter, electric field parameter, Grashof number, solutal Grashof number, Prandtl number, Eckert number, exponential heat source parameter, Lewis number, chemical reaction parameter, bioconvection Lewis number, and Peclet number associated with the flow properties are discussed graphically. The increase in the radiation parameter and volume fraction of the nanoparticles enhances the temperature profile. The Bejan number and entropy generation rate increase with the rise in diffusion parameter and bioconvection diffusion parameter. The novelty of the present work is analyzing the entropy generation and solar radiation effects in the presence of motile gyrotactic microorganisms and copper nanoparticles with polyvinyl alcohol-water as the base fluid under the influence of the source terms, such as viscous dissipation, Ohmic heating, exponential heat source, and chemical reaction of the electromagnetohydrodynamic (EMHD) Jeffrey fluid flow. The non-Newtonian nanofluids have proven their great potential for heat transfer processes, which have various applications in cooling microchips, solar energy systems, and thermal energy technologies.

Significance of nanoparticles' shape and thermo-hydrodynamic slip constraints on MHD alumina-water nanoliquid flows over a rotating heated disk: The passive control approach

This work aims at the numerical exploration of the hydromagnetic alumina –water nanoliquid flow due to a rotating rigid disk. The nanoliquid flow considering different nanoparticle shapes (namely sphere, platelet, cylinder, and brick) and the thermo-hydrodynamic slip constraints have been modeled utilizing the two-phase modified Buongiorno model (MBM). MBM modifies the conventional two-phase Buongiorno model by additionally considering the volume fraction-dependent nanoparticle properties. The present study is relevant in both engineering and industrial fields like aerodynamics, spin coating, extrusion process, etc. Von Kármán's similarity transformations are exercised in the transmutation of the mathematically modeled equations into a system of first-order ODEs and treated numerically using the generalized differential quadrature method. The consequence of pertinent parameters on the physical quantities and the flow profiles are explained with the aid of tables and graphs. It is observed that the nanoliquid temperature increases with the hydrodynamic slip parameter and decreases with the thermal slip parameter. Further, it is noted that the highest drag is exhibited by platelet-shaped alumina nanoparticles.

Numerical study for entropy generation and melting heat in flow of modified hybrid nanomaterial (Ag + MWCNTs + SWCNTs + Water)

The main theme behind this work is to model and analyze melting heat in the squeezed flow of a modified hybrid nanomaterial (Ag + MWCNTs + SWCNTs + Water). Entropy generation in the presence of viscous dissipation is investigated. Modeling is performed for basefluid (Water), nanofluid (SWCNTs + Water), hybrid nanofluid (MWCNTs + SWCNTs + Water), and modified hybrid nanomaterial (Ag + MWCNTs + SWCNTs + Water). Entropy production rate, velocity, Bejan number, skin friction coefficient, temperature, and Nusselt number are evaluated graphically under sundry variables. Non-linear coupled equations (ODEs) are constructed from the expressions (PDEs (continuity, momentum, and energy)) by adopting suitable transformations. The obtained coupled systems are then solved numerically by the bvp4c technique (shooting method with RK-4 algorithm). To execute this methodology, first we have converted our governed non-linear ODEs into a system of first-order ODEs. It is due to the fact that bvp4c works on the basis of the Runge–Kutta method, which is applicable to first-order ODEs. The velocity of the fluid increases with an increment in squeezing and melting parameters while it is reduced via nanoparticle concentration for silver. Fluid temperature reduces with melting while it boosts with squeezing number. Both melting and squeezing number boost skin friction. Nusselt number increases with both melting and squeezing number. Entropy is minimized via the squeezing number. The Bejan number is higher for both melting and squeezing number. It is concluded that performance of modified hybrid nanomaterial is best when compared with nanomaterial and hybrid nanomaterial.

Approximation of 3D convection diffusion equation using DQM based on modified cubic trigonometric B-splines

This paper presents an approximate solution of 3D convection diffusion equation (CDE) using DQM based on modified cubic trigonometric B-spline (CTB) basis functions. The DQM based on CTB basis functions is used to integrate the derivatives of space variables which transformed the CDE into the system of first order ODEs. The resultant system of ODEs is solved using SSPRK (5,4) method. The solutions are approximated numerically and also presented graphically. The accuracy and efficiency of the method are validated by comparing the solutions with existing numerical solutions. The stability analysis of the method is also carried out.

Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem

We investigate a singularly perturbed, non-convex variational problem arising in material science with a combination of geometrical and numerical methods. Our starting point is a work by Stefan Müller, where it is proven that the solutions of the variational problem are periodic and exhibit a complicated multi-scale structure. In order to get more insight into the rich solution structure, we transform the corresponding Euler-Lagrange equation into a Hamiltonian system of first order ODEs and then use geometric singular perturbation theory to study its periodic solutions. Based on the geometric analysis we construct an initial periodic orbit to start numerical continuation of periodic orbits with respect to the key parameters. This allows us to observe the influence of the parameters on the behavior of the orbits and to study their interplay in the minimization process. Our results confirm previous analytical results such as the asymptotics of the period of minimizers predicted by Müller. Furthermore, we find several new structures in the entire space of admissible periodic orbits.

The Ocean and Climate Change: Stommel's Conceptual Model

The ocean plays a major role in our climate system and in climate change. In this article we present a conceptual model of the Atlantic Meridional Overturning Circulation (AMOC), an important component of the ocean's global energy transport circulation that has, in recent times, been weakening anomalously. Introduced by Henry Stommel, the model results in a two-dimensional system of first order ODEs, which we explore via Mathematica. The model exhibits two stable regimes, one having an orientation aligned with today's AMOC, and the other corresponding to a reversal of the AMOC. This material is appropriate for a junior-level mathematical modeling or applied dynamical systems course.

Numerical study of magnetohydrodynamics and thermal radiation on Williamson nanofluid flow over a stretching cylinder with variable thermal conductivity

The effect of nanofluid and thermal radiation on MHD Williamson fluid over a stretching cylinder in the presence of convective boundary conditions is analyzed. The variable thermal conductivity is considered during the formulation of energy equation. Nonlinear ordinary differential equations are attained by proposing relevant similarity transformations on partial differential equations. These non-dimensional ordinary differential equations are then converted to the system of first order ODEs and solved numerically by shooting method. Results are further compared by using bvp4c, a MATLAB built-in function. The results obtained from MATLAB codes are compared with previously published results of the same nature in limiting case. Numerical results of skin-friction coefficient, local Nusselt and Sherwood numbers are systematized in the form of tables. The effect of prominent parameters on velocity, temperature and concentration is displayed by using graphs. It is observed that velocity profile diminishes for the higher values of Weissenberg number but a reverse relation is noticed for the temperature profile.

Cubic trigonometric B-spline differential quadrature method for numerical treatment of Fisher’s reaction-diffusion equations

This paper concerns through the numerical treatment of Fisher’s reaction-diffusion equation by using a hybrid numerical method. In this method, the combination of cubic trigonometric B-spline (CTB) base functions and differential quadrature method is used. This reduces the problem to a system of first order ODEs which is solved by “an optimal five stage and fourth-order strong stability preserving Runge-Kutta (SSP-RK54)” scheme. Four examples are considered to compare the present results with exact solutions and the results obtained by existing methods. It is found that the present method is not only quite easy to implement, but also it gives better results than the ones already existing in the literature.

Mixed-mode oscillations and chaotic solutions of jerk (Newtonian) equations

We analyze jerk equations (third-order ODEs) resulting from an underlying prototypical model of mixed-mode oscillations and propose their circuit realizations in this paper. The scalar ODEs and their corresponding circuit realizations are obtained from a system of first-order ODEs with one nonlinearity (third-degree polynomial). One of the jerk equations is Newtonian as it is obtained by computing the time-derivative of the second Newton’s law x″−F/m=0 for a constant mass m and specially designed nonlinear force function F(x,x′,τ). The second jerk equation is non-Newtonian. The two circuits are op-amp RC circuits with interesting dynamical properties, including the mixed-mode and chaotic oscillations. The mixed-mode oscillations follow the rules of Farey arithmetic and the circuits’ dynamics is of a fractal nature.

The Fourier transform of tubular densities

We consider the Fourier transform of tubular volume densities, with arbitrary axial geometry and (possibly) twisted internal structure. This density can be used to represent, among others, magnetic flux or the electron density of biopolymer molecules. We consider tubes of both finite radii and unrestricted radius. When there is overlap of the tube structure the net density is calculated using the super-position principle. The Fourier transform of this density is composed of two expressions, one for which the radius of the tube is less than the curvature of the axis and one for which the radius is greater (which must have density overlap). This expression can accommodate an asymmetric density distribution and a tube structure which has non-uniform twisting. In addition we give several simpler expressions for isotropic densities, densities of finite radius, densities which decay at a rate sufficient to minimize local overlap and finally individual surfaces of the tube manifold. These simplified cases can often be expressed as arclength integrals and can be evaluated using a system of first-order ODEs.

Reduction of systems of first-order differential equations via Λ-symmetries

The notion of λ-symmetries, originally introduced by C. Muriel and J.L. Romero, is extended to the case of systems of first-order ODEs (and of dynamical systems in particular). It is shown that the existence of a symmetry of this type produces a reduction of the differential equations, restricting the presence of the variables involved in the problem. The results are compared with the case of standard (i.e. exact) Lie-point symmetries and are also illustrated by some examples.

Numerical Solution of Boundary Value Problems for Selfadjoint Differential Equations of 2nth Order

The paper is devoted to solving boundary value problems for self-adjoint linear differential equations of 2nth order in the case that the corresponding differential operator is self-adjoint and positive semidefinite. The method proposed consists in transforming the original problem to solving several initial value problems for certain systems of first order ODEs. Even if this approach may be used for quite general linear boundary value problems, the new algorithms described here exploit the special properties of the boundary value problems treated in the paper. As a consequence, we obtain algorithms that are much more effective than similar ones used in the general case. Moreover, it is shown that the algorithms studied here are numerically stable.

Computing eigenvalues of periodic sturm-liouville problems using shooting technique and direct integration method

This paper deals with computing eigenvalues of periodic Sturm-Liouville (SL) problem by direct integration (DI) method using shooting technique without reducing to the system of first order ODEs. Floquet theory is applied to find a non-trivial solution of SL problems and the eigenvalues are approximated by the application of shooting techniques. Computational advantages are presented comparing the results obtained by the DI method with that of reducing to the system of first order ODEs.

Using nonconvergence of iteration to partition ODEs

Given is a variation of partitioning ODEs when using the multistep method. Essentially the nonconvergence property is made use of for partitioning. The criteria used are explained by using a test system of two linear equations and generalisation is made to a general system of first-order ODEs. Finally, some numerical results are given that include systems greater than two.