Ordinary Differential Equations Research Articles

Mathematical model for heat transfer during underground coal gasification process

Purpose. To develop a mathematical model of the “coal-gas” medium heat transfer during underground coal gasification to predict the combustion face advance velocity and the duration of gasification column mining. Methodology. To detect the temperature fields in the coal seam and gas, depending on the displacement length of the phase transition boundary, a boundary-value problem of mathematical physics has been developed. To solve this boundary-value problem, Boltzmann transformations, as well as methods for solving ordinary differential equations, are used. Newton-Raphson method, which has quadratic convergence, is used to find the roots of the transcendental equation. Findings. Tendencies in the formation of mathematical models when studying temperature fields around an underground gasifier have been analyzed, with highlighting their disadvantages. A mathematical model of heat transfer during underground gasification has been developed, taking into account the phase transition boundaries of the “coal-gas” medium. A computational experiment was conducted to determine the temperature of the phase transition boundary based on the length of the gasification column and the duration of the process. Originality. A mathematical model for heat transfer during coal gasification in the form of a boundary-value problem of mathematical physics, which consists of parabolic heat-transfer equations, the Stefan condition at the phase transition boundary, and the Dirichlet boundary conditions, has been constructed. As a result of solving the boundary-value problem, a self-similar solution has been obtained for the distribution of the coal seam and gas temperature fields, as well as the position of the phase transition boundary depending on the gasification duration and on the medium density, thermal conductivity coefficients, specific heat capacity of gas and coal, specific calorific value and temperature of coal combustion, initial coal temperature and constant temperature of the gasification process. The conducted analysis of numerical calculations provides a deeper understanding of the dynamics of underground coal gasification process and makes necessary corrections to achieve the maximum process efficiency. Practical value. A methodology for determining the displacement length of the phase transition boundary of the “coal-gas” medium has been developed, taking into account the change in the combustion face temperature along the gasification zone length depending on the duration of this process. Application of the methodology makes it possible to predict the time of mining the gasified coal column for drawing up a calendar plan for mining operations.

Conjugate Impacts of Linear Variable Thermal Conductivity and Thermal Radiation on Nonlinear Electrically Conducting Fluid

This study investigates the conjugate impacts of variable thermal conductivity and thermal radiation on nonlinear heat conducting fluid moving vertically through parallel channels. The dimensional governing equations were reduced to dimensionless partial differential equations and consequently transformed to ordinary differential equations. The resulting ordinary differential equations were solved using the homotopy perturbation method. The study aimed at discovering the possible effects of thermal radiation parameter , thermal conductivity parameter , temperature difference parameter and Magnetic parameter through the help of line graphs. It was found that, Velocity and temperature distributions were increasing functions of , while Velocity and temperature distributions are decreasing functions of

Construction and analysis of symmetric Sprott B multi-attractors with electric implementation

Abstract In chaos theory, a number of systems and models which apparently contain simple ordinary differential equations (ODEs) turn out to show a dynamic characterized by complicated behaviors and complex trajectories. One of such systems is the Sprott B model. We construct some set of multi-attractors based on the Sprott B model where additional parameters and operators are considered. After summarizing important preliminaries relevant to simple chaotic differential systems, the model is firstly solved analytically and numerically, then graphical simulations are provided. The later show coexistence and evolution of two chaotic attractors in a symmetrical representation. Lastly, similar results and expected outcomes are recovered via an electrical circuit implementation, realized using the Field Programmable Gate Array (FPGA) board, the Digital-to-Analog Converter (DAC) and the Rigol Oscilloscope. They also show progressing sets of coexisting multi-attractors.

Adaptive time step selection for spectral deferred correction

AbstractSpectral Deferred Correction (SDC) is an iterative method for the numerical solution of ordinary differential equations. It works by refining the numerical solution for an initial value problem by approximately solving differential equations for the error, and can be interpreted as a preconditioned fixed-point iteration for solving the fully implicit collocation problem. We adopt techniques from embedded Runge-Kutta Methods (RKM) to SDC in order to provide a mechanism for adaptive time step size selection and thus increase computational efficiency of SDC. We propose two SDC-specific estimates of the local error that are generic and do not rely on problem specific quantities. We demonstrate a gain in efficiency over standard SDC with fixed step size and compare efficiency favorably against state-of-the-art adaptive RKM.

Mathematical modeling predicts that endemics by generalist insects are eradicated if nearly all plants produce constitutive defense

Plants with constitutive defense chemicals exist widely in nature. The phenomenon is backed by abundant data from plant chemical ecology. Sufficient data are also available to conclude that plant defenses act as deterrent and repellent to attacking herbivores, particularly deleterious generalist insects. In the wild, generalist species are usually not endemic, meaning they are not restricted to certain plant species in a region. Therefore, our objective is to inspect theoretically whether evolution of chemical defenses in all plant species eradicate an endemic by any generalist species. The objective is addressed by developing deterministic ordinary differential equations under the following conditions: Plants without constitutive defenses are susceptible to oviposition by generalist insects, while they become defended against generalists by storing chemical defenses. From the models, we explicitly obtain that a generalist-free stable state is only possible if the vast majority of all plant individuals have chemical defenses. The model also allows one to predict the highest possible percentage of undefended plant individuals, which may be considered as free-riders.

Exploring nanoparticle dynamics in binary chemical reactions within magnetized porous media: a computational analysis

Artificial Neural Networks are incredibly efficient at handling complicated and nonlinear mathematical problems, making them very useful for tackling these challenges. Artificial neural networks offer a special computational architecture that is extremely valuable in disciplines like biotechnology, biological computing, and computational fluid dynamics. The present work investigates the applicability of back-propagation artificial neural networks in conjunction with the Levenberg-Marquardt algorithm for evaluating heat transmission in hybrid nanofluids. This work focuses on the computational analysis of a MgO + GO/EG hybrid nanofluid’s steady mixed convection flow over an exponentially stretched sheet, considering multiple slip boundary conditions, thermal conductivity, heat generation, and thermal radiation. A nonlinear system of ordinary differential equations is produced from the basic associated partial differential system by performing the proper exponential similarities modifications. For generating benchmark datasets, the resulting ordinary differential equations are processed employing the bvp4c method. Considering benchmark datasets set aside for training (70%), testing (15%), and validation (15%), the Levenberg-Marquardt algorithm, which employs back-propagation in artificial neural networks, is implemented. The accuracy of the suggested strategy for handling nonlinear problems is verified utilizing mean squared error, error histograms, and regression analysis, which are all used to evaluate the methodology. Outstanding agreement is seen when ANN outputs are compared to numerical results. The flow properties, including temperature, velocity, and concentration profiles, are shown graphically and numerically. For practical purposes, it is therefore essential to analyze the flow and heat transfer in hybrid nanofluids over exponentially extending and shrinking surfaces under mixed convection and heat source scenarios. Hybrid nanofluid problems have a wide range of practical and industrial applications, such as medication delivery, manufacturing, microelectronics, nuclear plant cooling, and marine engineering.

Shape factor and thermal radiation effects on convective MHD flow of Casson blood ternary hybrid nano‐fluid through a stretched rotating rigid disk

AbstractThe mathematical modeling of blood flow with the incorporation of ternary hybrid nanoparticles (i.e., titania, silica, and alumina nanoparticles) is the focus of this work. The flow of blood is simulated using the Casson fluid model. The impacts of ternary hybrid nanoparticles, shape factor, and Geometry of solid nanoparticles are visualized and investigated. The momentum and thermal characteristics of a flowing liquid are determined by considering magnetization, porosity, and nonlinearized radiating thermal flux. The basic partial differential equations resulting from mathematical modeling are turned into non‐linear ordinary differential equations using suitable velocity transforms. The derived nonlinear equations are then numerically calculated using the 4th‐5th order Runge‐Kutta‐Fehlberg method with the shooting technique and analytically solved by the Adomian decomposition method (ADM). The effects of the factors involved on the dimensionless profiles produced are fully addressed. It is found that the Skin friction factor and heat transfer rate in the radial direction upsurge with the augment of both rotation parameter and nanoparticles volume fraction; however, the Skin friction factor drops in the azimuthal direction. Also, results obtained reveal an enhancement in the local Nusselt number with the upsurge in the magnitude of radiation parameter, Rd, solid nanoparticles concentration, , shape factor value, s, and disk temperature, . For validation, the outcomes of this inquiry were compared to the outcomes of the HAM‐based Mathematica software. In addition, the acquired analytical DRA data are compared to numerical RKF45 values and those given in the literature.

Efficient probabilistic inference in biochemical networks

Biochemical networks are usually modeled by ordinary differential equations that describe the time evolution of the concentrations of the interacting (biochemical) species for specific initial concentrations and certain values of the interaction rates. They consist, in general, of many variables and parameters. Parameter estimation of such systems using standard optimization algorithms is computationally expensive since a large number of numerical simulations must be performed for numerous values of initial conditions and parameters, making these approaches either inefficient or inaccurate. In this paper, we explain how biochemical networks can be approximated by dynamic Bayesian networks, a class of discrete probabilistic models. This type of approximation enables the use of Bayesian inference for performing tasks such as parameter estimation. We explain how to make this type of approximations and the subsequent parameter estimation task as accurate and computationally efficient as possible. Our approach is applied on concrete examples such as the EGF-NGF cellular signaling pathway.

Entropy generation in radiative motion of tangent hyperbolic nanofluid in the presence of gyrotactic microorganisms and activation energy

In this work, entropy generation is optimized through the application of the second law of thermodynamics. The slip mechanisms, Brownian diffusions, and thermophoresis are elaborated using the tangent hyperbolic nanomaterial model. Magnetohydrodynamic (MHD) fluid is taken into consideration. To characterize the impact of activation energy, a unique model involving the binary chemical reaction is deployed. The effects of mixed convection that is nonlinear in nature, bioconvection, and Joule effect are all taken into consideration. The key partial differential equations (PDEs) are reduced into ordinary differential equations (ODEs) by utilizing appropriate similarity transformations and then solved numerically with the help of a built-in ‘bvp4c’ technique of MATLAB software. Varied flow parameters’ impacts on the nanoparticle volume concentration, entropy number, microorganism concentration, temperature, and velocity fields are analyzed using graphs. Various flow variables are taken into consideration to calculate the total rate of entropy generation. The obtained results show that concentration irreversibility, Joule effect irreversibility, viscous dissipation, and heat irreversibility all influence the entropy. The numerical outcomes were observed by fixing the physical parameters as 0.1<α<4.0, 0.1<M<1.2, 0.1<Nr<2.2, 0.1<Le<2.2, 0.1<Nb<0.4, 0.1<Nt<1.0, 2.0<⁡Pr⁡<5.0, and 0.1<Lb<2.0, as well as their impact on the momentum, thermal, concentration, and microorganism density profiles. From results, an increasing estimate of the variable representing chemical reaction indicates a decline in the concentration. The higher the chemical reaction variable, Hartmann number, and Weissenberg number, the higher the entropy number, while the Bejan number has a contrary behavior. Subsequently, all the outcomes are plotted in graphs and discussed in detail, when subjected to the involving physical quantities.

A Chebyshev Generated Block Method for Directly Solving Nonlinear and Ill-posed Fourth-Order ODEs

This study presents a block method induced by Chebyshev polynomials of first kind for directly solving initial value problems of fourth-order ordinary differential equations without reducing the problems to a system of first-order differential equations. The method was developed by applying interpolation and collocation procedures to a Chebyshev approximate polynomial. The unknown parameters were obtained using the Gaussian elimination method, then substituted into the approximate solution to get the continuous scheme and evaluated at the selected point to give the discrete scheme. The method was zero stable, consistent, and convergent, p-stable as shown by the region of absolute stability and has an order of seven. The accuracy and usability of the developed method were tested by applying it to solve six numerical examples. The method was found to be efficient as it gives minimal error. The numerical results of the method were compared with other works cited in the literature and found to be better as it gives minor errors.

Analytical simulation of Darcy-Forchheimer nanofluid flow over a curved expanding permeable surface

Abstract This research paper presents an analytical simulation of Darcy-Forchheimer flow over a porous curve stretching surface. In fluid dynamics, the Darcy-Forchheimer model combines Forchheimer adjustment and high-velocity effects with Darcy's formula for porous media flow: two nanofluid particles, molybdenum disulfide, and graphene oxide, form nanofluid with the base fluid blood. The governing partial differential equations for momentum and energy are converted into a nonlinear ordinary differential equations system by applying the appropriate similarity transformations. The homotopy analysis method (HAM) is used to solve the transform equations analytically. The impact of essential factors includes the Forchheimer parameter, porosity parameter, slip parameter, Eckert number, nanoparticle volume friction, magnetic field parameter, and curvature parameter. The results have applications in the design of sophisticated cooling systems, where effective thermal control is essential.

A mathematical model of glomerular fibrosis in diabetic kidney disease to predict therapeutic efficacy

BackgroundGlomerular fibrosis is a tissue damage that occurs within the kidneys of chronic and diabetic kidney disease patients. Effective treatments are lacking, and the mechanism of glomerular damage reversal is poorly understood.MethodsA mathematical model suitable for hypothesis-driven systems pharmacology of glomerular fibrosis in diabetes was developed from a previous model of interstitial fibrosis. The adapted model consists of a system of ordinary differential equations that models the complex disease etiology and progression of glomerular fibrosis in diabetes.ResultsWithin the scope of the mechanism incorporated, advanced glycation end products (AGE)—matrix proteins that are modified due to high blood glucose—were identified as major contributors to the delay in the recovery from glomerular fibrosis after glucose control. The model predicted that inhibition of AGE production is not an effective approach for accelerating the recovery from glomerular fibrosis. Further, the model predicted that treatment breaking down accumulated AGE is the most productive at reversing glomerular fibrosis. The use of the model led to the identification that glucose control and aminoguanidine are ineffective treatments for reversing advanced glomerular fibrosis because they do not remove accumulated AGE. Additionally, using the model, a potential explanation was generated for the lack of efficacy of alagebrium in treating advanced glomerular fibrosis, which is due to the inability of alagebrium to reduce TGF-β.ImpactUsing the mathematical model, a mechanistic understanding of disease etiology and complexity of glomerular fibrosis in diabetes was illuminated, and then hypothesis-driven explanations for the lack of efficacy of different pharmacological agents for treating glomerular fibrosis were provided. This understanding can enable the development of more efficacious therapeutics for treating kidney damage in diabetes.

The Role of Two Models of Thermal Radiation and Variable Properties on Thermal Explosion Critical and Transition Conditions of the Optically Thin Droplets

ABSTRACT Determining the criticality and disappearance of optically thin droplets (gas phase) by thermal explosions is crucial for maintaining thermal stability in various industrial and nonindustrial applications. The governing nonlinear ordinary differential equations of this problem make use of two distinct approximation models for thermal radiation loss, which were completed by Cogley et al. and Sohrab et al. Two methods of solving the problem – the Semenov method and the inflection point method – are used to identify the circumstances of criticality and disappearance of criticality (transition conditions). The governing equations utilize the use of the temperature-dependent Arrhenius frequency factor of the reaction and the thermal conductivity of the gas. The analytical solution of the equations yields the transitional values of the distinctive parameters at various initial conditions. Numerical methods are used to determine the equation’s exact solution. At specific initial conditions, the impact of θo, θa, γSh,F, and αSh,F on the temperature-time histories of ignition (supercritical region), non-ignition (subcritical region), and the critical boundary between them is derived. Notable effects on the ignition, non-ignition, and critical temperatures and times include the changing frequency of the reaction, the gas thermal conductivity, and two approximation models of radiation heat loss. The two radiation heat loss approximation models of the analytical and numerical findings are compared.

A mathematical model describing the tip-stalk regulation in angiogenesis

Angiogenesis is the process of new blood vessel growth from existing vessels, involving extensive cell signaling. Under normal conditions, new vessels are robust and organized, with a balance among angiogenesis factors. In abnormal conditions, such as tumor development, vessels are stunted and tangled due to an imbalance of these factors. Pathological angiogenesis stimulates rapid vessel growth to feed the oxygen and nutriente starved tumor. Inhibiting angiogenesis can cause side effects like hypertension, thrombosis, and fatigue. To better understand this process, significant effort has gone into studying signaling pathways, contributing to drug development for diseases like cancer. This study presents a mathematical model describing angiogenesis on a microscopic scale, comparing its results with experimental data on vascular network topology. The model, implemented in MatLab®, uses ordinary differential equations to represent cell behavior. Results show that altering VEGF (Vascular Endothelial Growth Factor) disrupts system balance, impacting angiogenesis and possibly explaining differences in network topology seen experimentally.

Free evolution in the Ginzburg-Landau equation and other complex diffusion equations

Abstract New ordinary differential equations (ODEs) for the evolution of spectral components are derived from the complex Ginzburg-Landau equation (CGLe) for one-dimensional spatial domains without boundaries (free evolution) and with one fixed boundary (semi-free evolution). For such evolution, a complex or imaginary diffusion term creates a tendency for waves to lengthen. This requires a novel ansatz and auxiliary condition that treat wavenumbers as time-varying. The ansatz consists of a discrete spatial Fourier transform modified with a time-dependent wavenumber for the peak spectral component. The wavenumbers of the other components are fixed relative to this wavenumber. The new auxiliary condition is the terminal condition for complex diffusion (after wavenumbers evolve to zero, they remain at zero). The derived free and semi-free ODEs are solved along characteristic lines located symmetrically about a fixed spatial point. Waves lengthen with time away from this point in both directions. Laboratory experiments on the formation of channel sandbars, theoretically described by the CGLe, show two regions whose evolutionary behaviour is qualitatively predicted by the free and semi-free evolution equations. This analysis applies to other time-dependent partial differential equations with complex or imaginary diffusion terms. New freely evolving solutions are derived for the complex heat equation and Schrödinger equation (linear and nonlinear).

On the solutions of the Lane-Emden model via non-dyadic wavelet scheme to unravel astrophysical phenomena

The research focuses on exploring the characteristics of the Lane Emden equation, which is a second-order ordinary differential equation that has its roots in astrophysics. To approximate this equation, a technique that employs non-dyadic Haar wavelets in conjunction with quasi-linearization is utilized. By utilizing non-dyadic wavelets, the ordinary differential equation can be simplified into a system of algebraic equations. Error estimates are provided to assess the accuracy of the produced data. A comparison between the existing solutions and the numerical results obtained using the suggested approach is performed to showcase its efficiency and advantages. The non-dyadic Haar wavelet approach is found to be a rich structure for numerous solutions that span a wide range of physical parameter.

Benchmarking methods for computing local sensitivities in ordinary differential equation models at dynamic and steady states.

Estimating parameters of dynamic models from experimental data is a challenging, and often computationally-demanding task. It requires a large number of model simulations and objective function gradient computations, if gradient-based optimization is used. In many cases, steady-state computation is a part of model simulation, either due to steady-state data or an assumption that the system is at steady state at the initial time point. Various methods are available for steady-state and gradient computation. Yet, the most efficient pair of methods (one for steady states, one for gradients) for a particular model is often not clear. In order to facilitate the selection of methods, we explore six method pairs for computing the steady state and sensitivities at steady state using six real-world problems. The method pairs involve numerical integration or Newton's method to compute the steady-state, and-for both forward and adjoint sensitivity analysis-numerical integration or a tailored method to compute the sensitivities at steady-state. Our evaluation shows that all method pairs provide accurate steady-state and gradient values, and that the two method pairs that combine numerical integration for the steady-state with a tailored method for the sensitivities at steady-state were the most robust, and amongst the most computationally-efficient. We also observed that while Newton's method for steady-state computation yields a substantial speedup compared to numerical integration, it may lead to a large number of simulation failures. Overall, our study provides a concise overview across current methods for computing sensitivities at steady state. While our study shows that there is no universally-best method pair, it also provides guidance to modelers in choosing the right methods for a problem at hand.

Parametric identification of coefficients for a model of fatigue stiffness degradation of a composite material

The problem of finding the fatigue characteristics of a composite material based on test results is considered. The results of endurance tests of unidirectional polymer composite materials with different initial stiffness, breaking stress and working cycle stress were used as the initial data. As a mathematical model of stiffness degradation, a nonlinear ordinary differential equation with five unknown parameters is used, reflecting characteristic changes in the properties of the material. It is required to find such parameter values that the solution of the differential equation should describe the available test results with sufficient accuracy. The solution procedure is reduced to the problem of optimizing the objective function, the value of which characterizes the achieved accuracy. As optimization methods, a method simulating the behavior of a flock of moths and a method of sequential reduction of the search set were used. A step-by-step algorithm for finding unknown model parameters is proposed, and numerical results of processing input data containing information on changing the elasticity modulus of the composite material in the course of applying load cycles are presented.

A Six-Point y-Function Hybrid Block Method for Direct Solution of Third Order Ordinary Differential Equations

In this article, a new continuous numerical method was developed for the numerical integration of general third-order initial value problems (IVPs) of ordinary differential equations (ODEs). The method was derived by adopting interpolation and collocation approach using a combination of power series and exponential functions as the basis function for the approximate solution to the ODEs. The approximate solution was interpolated at both nodal and off-nodal points while the differential systems from the approximate solution was collocated at all nodal points to obtain the set of methods with given step-numbers. This derived approach ensured that the hybrid points are obtained at the y-values of the methods. This class of hybrid linear multi-step method satisfied the conditions for usability and accuracy of any given method. These conditions were confirmed by testing the consistency, stability and convergence of the developed method. The method was applied to solve directly linear and non-linear third order ODEs without reduction to system of first-order ordinary differential equations. Results from the computation compared favorably with some existing numerical methods in literature with comparable properties to confirm the superiority of the newly developed method.

Study of MHD Casson Nanofluid Flow Over an Inclined Stretching Sheet in Porous Media with Thermal Slip, Chemical Reaction and Radiation Effects

In the current study, the behaviour of Casson nanofluid subjected to magnetohydrodynamic (MHD) flow across an inclined stretched sheet within a porous medium has been investigated numerically. The governing equations are transformed into ordinary differential equations with the corresponding boundary conditions by employing similarity transformations. The solutions of essential equations are achieved by using the 4th-order Runge-Kutta method combined with the shooting technique. The novelty and innovative contribution are showcased through illustrative graphs that scrutinize the effect of factors that impact the velocity, temperature, and concentration profiles within assorted flow scenarios. The primary focal points of the study encompass examining variations in magnetic field strength, angle of inclination, and suction intensity that affect the fluid's velocity moderation, while improved porosity and radiation parameters lead to a rise in fluid temperature. Higher Biot numbers correlate with an increase in fluid temperature. The implications of positive coefficients of heat transfer are crucial across various fields to ensure efficient thermal management ensuring optimal performance and longevity. The numerical data presented is aligned with earlier published results for comparison. Furthermore, the variations in skin friction, Nusselt, and Sherwood numbers driven by different parameters are displayed in tables to highlight significant modifications.