Higher Order Ordinary Differential Equations Research Articles

Chaotic systems based on higher-order oscillatory equations

This paper discusses the design process toward new lumped chaotic systems that originates in higher-order ordinary differential equations commonly used as description of ideal oscillators. In investigated third-order case, two chaotic oscillators were constructed. These systems are dual in the sense of vector field geometry local to fixed points. The existence of robust chaos was proved by both standard routines of numerical analysis and practical measurement. For the fourth-order oscillatory equation, the concept based on interaction between superinductor and supercapacitor was examined in detail. Since both “superelements” are active, the nonlinearity essential to the evolution of chaos is fully passive. It is demonstrated that complex motion is robust and does not represent long transient behavior or numerical artefact. The existence of chaos was verified using standard quantifiers of the flow, such as the largest Lyapunov exponents, recurrence plots, approximate entropy and sensitivity calculation. A good final agreement between theoretical assumptions and practical results will be concluded, on a visual comparison basis.

Entropy generation in a chemically reactive magnetohydrodynamic unsteady micropolar nanofluid flow with activation energy over an inclined stretching sheet: A Buongiorno model approach

The goal of this research is to inspect the heat and mass transfer trends and entropy generation in a time-reliant stagnation point stream of a micropolar fluid across an inclined stretched surface. For this objective, a chemically reactive, electrically conducting fluid exposed to an orthogonal magnetic field is studied. The flow governing equations are modelled using Buongiorno model and are reformed to a system of higher order ordinary differential equations by administering appropriate similarity transformations. This system is quantitatively examined by employing the fourth-order Runge-Kutta scheme with shooting approach. The effects of thermal radiation, magnetic field, uniform heat source/sink, Brownian motion, thermophoresis, activation energy, and binary chemical reaction are studied on velocity, microrotation, temperature, and concentration profiles. It is observed that magnetic field and Brownian motion elevate the flow temperature. Increased activation energy spikes the fluid concentration while increase in binary chemical reaction reduces the particle concentration. Later, impact of various parameters on skin friction coefficient and heat and mass transfer rates are tabularised. Increasing values of thermophoretic diffusion parameter, Brownian diffusion parameter, and chemical reaction parameter improve the rate of mass transfer. Unsteadiness parameter triggers the skin friction coefficient 53.2% when the parameter value was increased from 0.7 to 1.0. Viscous dissipation and thermal radiation increase the rate of entropy generation. A comparison of skin friction coefficient with previous studies demonstrates a strong agreement.

Optimization of heat transfer properties in micropolar ternary (Al2O3 + TiO2 + SiO2/water) nanofluid with porosity and magnetic field effects

Decades of study provide substantial evidence supporting that nanofluids possess improved heat transmission capabilities. Optimal qualities of nanoliquids may be achieved by manipulating the volume concentration of the nanoparticles. Nevertheless, this method has constraints due to the balance between negative and positive viscosity. A recent advancement has been produced in creating and thoroughly examining a particular form of hydraulic fluid that incorporates three solid nanoparticles scattered inside an existing fluid. This improvement aims to tackle this limitation. This study investigates the characteristics of a micro-rotational incompressible micropolar ternary hybrid nanofluid in a porous media. It considers the influence of magnetohydrodynamics, Darcy-Forchheimer, and linear thermal radiations on a stretched surface. The nanofluid consists mostly of water, which is blended with titanium oxide, aluminum oxide, and silicon oxide nanoparticles. Using dimensionless similarity transformations, the collection of higher-order non-dimensional partial differential equations is converted into higher-order ordinary differential equations. The BVP4C technique in MATLAB software is used to solve these equations and assess the graphical and tabular outcomes. The graphical analysis examines the influence of various physical characteristics. The heat transfer of a ternary hybrid nanofluid is significantly affected by many factors, including the Forchheimer coefficient, Prandtl number, thermal radiation parameter, and magnetic field parameters, as seen. The research demonstrates that an increase in the magnetic field and Darcy-Forchheimer parameters reduces linear momentum. Conversely, increased porosity, magnetic field, and radiation parameters increase the temperature profile.

Mixed convective flow of engine oil-based non-Newtonian tri-hybrid nanofluid across a porous rotating disk

Purpose Flow induced by rotating disks is of great practical importance in several engineering applications such as rotating heat exchangers, turbine disks, pumps and many more. The present research has been freshly displayed regarding the implementation of an engine oil-based Casson tri-hybrid nanofluid across a rotating disk in mass and heat transferal developments. The purpose of this study is to contemplate the attributes of the flowing tri-hybrid nanofluid by incorporating porosity effects and magnetization and velocity slip effects, viscous dissipation, radiating flux, temperature slip, chemical reaction and activation energy. Design/methodology/approach The articulated fluid flow is described by a set of partial differential equations which are converted into one set of higher-order ordinary differential equations (ODEs) by using convenient conversions. The numerical solution of this transformed set of ODEs has been spearheaded by using the effectual bvp4c scheme. Findings The acquired results show that the heat transmission rate for the Casson tri-hybrid nanofluid is intensified by, respectively, 9.54% and 11.93% when compared to the Casson hybrid nanofluid and Casson nanofluid. Also, the mass transmission rate for the Casson tri-hybrid nanofluid is augmented by 1.09% and 2.14%, respectively, when compared to the Casson hybrid nanofluid and Casson nanofluid. Originality/value The current investigation presents an educative response on how the flow profiles vary with changes in the inevitable flow parameters. As per authors’ knowledge, no such scrutinization has been carried out previously; therefore, our results are novel and unique.

A hybrid kernel-based meshless method for numerical approximation of multidimensional Fisher’s equation

We propose and analyze a meshless method of lines by considering some hybrid radial kernels. These hybrid kernels are constructed by linearly combining infinite smooth radial functions to piecewise smooth radial functions; which are then used for spatial approximation on trial spaces spanned by translates of positive definite radial functions. After spatial approximation, a high-order ODE solver is invoked for efficient and stable time-integration of the resultant semi-discrete system of ordinary differential equations (ODEs). Unlike the mesh-based method of lines, the proposed method works for arbitrary scattered data points and is equally effective for problems over non-rectangular domains. The proposed method is tested on one-, two- and three-dimensional reaction–diffusion Fisher equation for its numerical stability, accuracy, and efficiency against the contemporary meshless and mesh-based methods. The economical computational cost, improved accuracy, eigenvalues stability, and well-conditioning of system matrices are observed against RBF collocation and RBF-PS methods.

New Numerical Model for the Direct Solution of Higher Order Ordinary Differential Equations

New Numerical Model for the Direct Solution of Higher Order Ordinary Differential Equations

A New Method for Finding Lie Point Symmetries of First-Order Ordinary Differential Equations

The traditional algorithm for finding Lie point symmetries of ordinary differential equations (ODEs) faces challenges when applied to first-order ODEs. This stems from the fact that for first-order ODEs, unlike higher-order ODEs, the determining equation lacks derivatives, rendering it impossible to decompose into simpler PDEs to be solved for the infinitesimals. Consequently, a common technique for determining Lie point symmetries of first-order ODEs involves making speculative assumptions about the form of the infinitesimal generator. In this study, we propose a novel and more efficient approach for finding Lie point symmetries of first-order ODEs and systems of first-order ODEs. Our method leverages the inherent connection between first-order ODEs and their corresponding second-order counterparts derived through total differentiation. By exploiting this connection, we develop a systematic algorithm for determining Lie point symmetries of a wide range of first-order ODEs. We present the algorithm and provide illustrative examples to demonstrate its effectiveness.

Analysis of Forchheimer Effect for Double Diffusive Convection With Dusty Fluids and MHD

An attempt has been made to analyze the effect of second order resistance for a steady, dusty fluid considering magnetohydrodynamic (MHD) and also the characteristics of fluid like permeability, porosity, solutal diffusivity and thermal conductivity being varied. Here the basic equations are coupled, non-linear Partial Differential Equations (PDEs), which are changed by similarity transformations to higher order Ordinary Differential Equations (ODE). After being transformed the higher order ODE that where obtained are resolved numerically. Shooting technique is employed here and the values are tabulated for various pertinent parameter variations. The effects of the inertia, concentration and interaction, mixed convection, magnetic and many other parameters are discussed and plotted graphs for velocity, concentration and temperature. The magnetic force enhances velocity. It was found that the present study correlates with the existence results.

A fast and efficient computation method for reflective diffraction simulations

We present a new computation method for simulating reflection high-energy electron diffraction and the total-reflection high-energy positron diffraction experiments. The two experiments are used commonly for the structural analysis of material surface. The present paper improves the conventional numerical method, the multi-slice method, for faster computation, since the present method avoids the matrix-eigenvalue solver for the computation of matrix exponentials and can adopt higher-order ordinary differential equation solvers. Moreover, we propose a high-performance implementation based on multi-thread parallelization and cache-reusable subroutines. In our tests, this new method performs up to 2,000 times faster than the conventional method without compromising accuracy.

Computational analysis of melting radiative heat transfer for solar Riga trough collectors of Jeffrey hybrid-nanofluid flow: A new stochastic approach

Current investigation deals with the melting heat transfer for the Jeffrey hybrid-nanofluid flow in parabolic trough solar collectors through Darcy Forchheimer porous media over a variable thick vertical elongation Riga surface under the effect of solar radiation. The impacts of viscous dissipation and Joule heating are also investigated. Equations governing the Jeffrey’s hybrid nanofluid flow are higher-order non-linear partial differential equations (PDEs). These governing PDEs are transformed into the non-linear ordinary differential equations (ODEs) by introducing appropriate similarity transformations and dimensionless parameters. Runga Kutta’s fourth-order numerical scheme is implemented with the shooting technique to solve coupled higher-order ODEs. Results for velocity profile, temperature profile, drag coefficient and Nusselt number are discussed for various influential parameters. Artificial neural networking is also performed to predict Nusselt numbers in different cases and scenarios. The Artificial Neural Network gives the desired outputs with the highest possible accuracy. It is observed that the temperature profile rises with increase in Hartmann number, porosity parameter, and Forchheimer number. However thermal profile worsens with escalates in the exponential index. Radiant energy of the Sun is a renewable energy source available in considerable amounts in our living environment. A parabolic trough solar collector is an efficient solar collector that stores the concentrated incoming radiant energy from the Sun to fulfill the requirement of high temperatures in thermal energy storage systems. It has various applications in solar-powered appliances for cooking and air conditioning, solar-powered systems for treating wastewater, photovoltaic lighting, solar-powered cars, aircraft, etc.

MHD nanofluid flow through Darcy medium with thermal radiation and heat source

In this analysis, we have considered heat transmission in two-dimensional steady laminar nanofluid flow past a wedge. Magnetohydrodynamic (MHD), Brownian motion, viscous dissipation and thermophoresis effects are considered over the porous surface. Similarity transformations have been used to change the governing partial differential equations (PDEs) into nonlinear higher-order ordinary differential equations (ODEs). Governing ODEs with boundary conditions are then converted to the system of first-order initial value problem. After that the modeled system is solved numerically by RK4 technique. Impact of the magnetic number, Eckert number, Prandtl number, Lewis number, Brownian motion, thermophoresis and permeability parameters on the flow domain is analyzed graphically as well as in tabular form. It is noted that magnitude of Nusselt number for the flow regime increases with the increase of nondimensional parameter [Formula: see text] while opposite behavior is observed in case of R.

Thermal characteristics for the flow of Williamson hybrid nanofluid with distinct shape factors

AbstractThe main purpose of the present article is to investigate the flow of 2‐D, incompressible, steady, hydro magnetic Williamson hybrid nanofluid with three distinct shape factors namely spherical, cylindrical, and platelet shapes under the influence of thermal radiation and viscous dissipation effects on the flow. The aim of the current work is to investigate the thermal conduction capacity of three different shaped nanoparticles by comparison. We have modelled copper and molybdenum disulfide nanoparticles suspension in Williamson fluid blood as a conventional real fluid passing through a horizontal stretching cylinder in this case. A set of non‐linear PDEs are used to conceivably formulate the problem's physical model. The transformation from these modelled PDEs to ODEs is accomplished through the use of appropriate similarity variables. To address the problem, the RK method of order four is used in conjunction with shooting system in order to get first order ordinary equations from non‐linear higher order ordinary differential equations. To run the code for numerical results, computational Matlab bvp4c solver is used and graphs are depicted to explain the impact of various embedded physical quantities on the momentum and energy profiles accompanying the rates of shear stress and heat transfer for the considered Williamson hybrid nanofluid. The use of spherical shaped nanoparticles is thought to improve the thermal conductivity rate of the flowing fluid more than cylinder and platelet shaped nanoparticles. The skin friction coefficient is enhancing for larger values of magnetic parameter and curvature parameter but Weissenberg number has a negative trend. The rate of cooling is high for greater values of magnetic parameter, Williamson fluid parameter, heat generation parameter, thermal conduction parameter, viscous dissipation parameter and thermal radiation parameter.

Heat and mass transfer by stirring nanofluids with the presence of renewable solar rays, Joule heating, and entropy procreation

Renewable solar radiation is the foremost energy source because of its accessibility, natural replication, and sustainability in an environmentally safe manner. Here, researchers intended to inspect the heat and mass transfer via nanofluid transported on an inclined permeable expanded sheet in the presence of solar thermal radiation without any barrier. Mainly, the formation of non-recovery energy called entropy and Joule heating are also weighed. The guiding non-linear partial differential equations were transformed into systems of non-linear higher-order ordinary differential equations by felicitous similarity transformation. They are solved by the prevalent technique called the Homotopy Analysis Method, which is executed by the BVPh2.0 package in Mathematica 12.1 software. Comparisons with preceding published articles confirm the method's validity and accent its admirable uniformity. Afterward, the magnetic field interaction delays the mobility of nanofluid while increasing the magnitude of local skin friction and temperature distribution. By intensifying the thermal radiation parameter and Eckert number, the temperature and entropy production escalated. Furthermore, the heat transfer by convective surpasses that of conductive owing to the particles' Brownian motion. Thermophoresis established surplus tiny-particles concentration. Heat transfer from solar radiation in moving nanofluids has been applicable for cooking, heating water, and producing electricity.

WKB analysis of the linear problem for modified affine Toda field equations

We study the WKB analysis of the solutions to the linear problem for a modified affine Toda field equation, which is equivalent to the higher-order ordinary differential equation (ODE) studied in the ODE/IM correspondence. After gauge transformation, we diagonalize the flat connection of the linear problem to reduce the latter to a set of independent first-order linear differential equations. We explicitly perform this procedure for classical affine Lie algebras with lower ranks. In particular, we study the WKB solutions of the {D}_r^{(1)} - and {D}_{r+1}^{(2)} -type linear problems, which correspond to the higher-order ODEs with the pseudo-differential operator. The diagonalized connection is obtained from the Riccati equations of the adjoint linear problem and related to the conserved currents of the integrable hierarchy constructed by Drinfeld and Sokolov up to total derivatives.

Mathematical analysis of MHD hybrid nanofluid flow with variable viscosity and slip conditions over a stretching surface

There are number of real-world uses for hybrid nanofluids. Studies have shown that hybrid nanofluid is better at transferring heat than nanofluid with only one type of nanoparticle. It can be used in new ways, like in microfluidics, dynamic sealing, heat dissipation, and so on. The use of hybrid nanofluids in MHD flow over a stretching surface with varying viscosity can enhance heat transfer rates. This has various applications in industries such as, cooling systems in electronics, thermal management in aerospace, and energy system. But the properties of boundary layer flow and rate of heat transfer for hybrid nanofluids could be studied more in a space with more dimensions and with other physical assumptions around the flow of fluids. The current study's major purpose is to investigate the flow of a three-dimensional hybrid nanofluid over a stretched sheet as its viscosity changes. Furthermore, the effects of the Smoluchowski temperature and Maxwell velocity slip boundary conditions are considered. Hybrid nanofluid was created by dispersing copper (Cu) and alumina (Al2O3) nanoparticles in a base fluid (H2O). Similarity variables are employed to transform a set of nonlinear partial differential equations (PDEs) that govern fluid flow and heat transfer into a system of higher-order ordinary differential equations (ODEs). The RKF method in Mathematica solves the higher-order nonlinear ODEs. Graphical analyses and in-depth discussions examine how leading variables affect velocity and temperature profiles. The expressions for skin friction and local Nusselt number are explicitly formulated, and graphs illustrate the variations of these two quantities across different parameter values. When velocity slip parameter goes up, velocity profile f′ηgoes down, and the temperature profile θ(η) also goes down when the temperature slip parameter goes up. As the velocity slip parameter goes up, skin friction goes down, while the Nusselt number goes up as the temperature slip parameter goes up. When results of this study are compared to some of results that have already been published, they are found to be very interesting as heat transfer increases due to the impacts of physical parameters.

Analysis of Kerosene oil conveying silver and Manganese zinc ferrite nanoparticles with hybrid Nanofluid: Effects of increasing the Lorentz Force, Suction, and volume fraction

The current study aims to explore the magnetic field on a spinning disk with the hybrid nanofluid flow and Cattaneo-Christov heat theory in the existence of nonlinear thermal radiation incorporating Ag and MnZnFe2O4 nanoparticles. Because silver may increase the thermal characteristics of the base material, it has a wide variety of industrial, pharmaceutical, power generation, and heating and cooling applications. The thermal properties of a hybrid nanofluid were to be found by exploring the aspect of nanomaterials on heat transfer and fluid flow. The principal partial differential equations are converted into ordinary differential equations using appropriate similarity treatments. With the aid of the shooting strategy, higher-order ordinary differential equations are converted to first-order ordinary differential equations. To present the numerical data and graphical results of the flow parameters, the built-in solver Bvp4c in the computational tool MATLAB is used. Several plots are also used to investigate the upshot of physical parameters on the graphically depicted profiles, such as the suction and injection parameter, magnetic parameter, Prandtl number, temperature ratio parameter, thermal radiation parameter, thermal relaxation parameter, and volume friction nanoparticles. For the magnetic parameter, both velocity profiles dropped, but the heat profile increased. When the temperature ratio and heat source-sink parameters were increased while the Prandtl number was decreased, the heat field increased.

Photo-thermoelastic wave propagation with changing thermal conductivity on excited pulsed laser nanoscale microelongated semiconductor material

Indroduction: The novel model of a non-local elastic semiconductor material that is microelongated is created. The process of photothermal transfer is responsible for the stimulation of the material. Photo-thermoelastic theories are used when the thermal conductivity is changed for the non-local medium. Under the influence of laser pulses, the effective framework describes the nanoscale microelongation instance as well as the interference between the photo-thermoelastic propagation waves in the non-local medium. It is possible to think of thermal conductivity as a linear function of temperature when electronic and thermoelastic deformation mechanisms are described. Two-dimensional deformation (2D) is used to extract the main fields, which obtained in non-dimension.Methods: Harmonic wave analysis, which is described by the normal mode, has been used to convert the basic equations into non-homogeneous higher-order ordinary differential equations. Applying a small subset of the possible non-local semiconductor surface conditions leads to exhaustive solutions.Result and Discussion: The outcomes of numerical simulations for silicon (Si) are graphically shown. There are comparisons made and explanations given for the investigated physical factors like their thermal conductivity, laser pulses and microelongation parameters.

APPROXIMATE SOLUTIONS TO BOUNDARY VALUE PROBLEMS OF HIGHER-ORDER BY THE MODIFIED ADOMAIN DECOMPOSITION METHOD

This paper propose a new modified differential operator for solving third-order boundary value problems into higherorder ordinary differential equation. We found the differential operator for new two inverse operators which can be applied for solving equations at more than one type in different conditions. This paper is given five non-linear examples from a high-order, we got effective and easily approximate solutions to the exact solution.

Responses, optimization and prediction of energy harvesters under galloping and base excitations

In this paper, the complexification-averaging method is firstly used to derive the theoretical solutions and analyze the dynamical performance of the energy harvesters under galloping and base excitations. Based on the distributed parameter model of the energy harvesters, the model equations can be turned into a set of high-order ordinary differential equations by solving the ordinary differential equations with coupled terms, then the equation of the amplitude frequency and the analytical relationship to determine the displacement amplitude and the voltage amplitude are derived. In addition, the influence of parameters such as the wind speed, the electromechanical coupling parameter and the base excitation frequency on the nonlinear behaviors of the energy harvesters are explored. It is found that there exists very good consistency between the theoretical results and the numerical results. The response properties are verified by means of the phase portraits and time histories. Additionally, according to the implicit function derivative theorem, the critical condition for identifying the unstable region is obtained. And with the different system parameters, the critical unstable boundary can be identified. In order to improve the efficiency of energy harvesting, the parameters of the energy harvesters are optimized based on the purpose of maximum voltage amplitude. Finally, the results are predicted by means of the multilayer feedforward neural network, and the excellent prediction results can be found.

Elliptic and multiple-valued solutions of some higher order ordinary differential equations

<abstract><p>In the present paper, by the complex method, the meromorphic solutions of the higher order ordinary differential equation $ w^{(5)}+aw^{''}+bw^2-cw+d = 0 $ are investigated, where $ a, b, c, d $ are constant complex numbers, and $ b \neq0 $. Furthermore, by Theorem 1.1, we built elliptic and multiple-valued solutions for the higher order ordinary differential equations $ u^{(6)}-u^{(5)}+u'^2-2u'u+u^2+2u'-2u+1 = 0 $ and $ u^{(6)}-u^{(5)}+au^{'''}-au''+bu'^2-2bu'u+bu^2-cu'+cu+d = 0 $. At the end, we give some new meromorphic solutions for two higher-order KdV-like equations.</p></abstract>