System Of ODEs Research Articles

Within-chain parallelization-Giving Stan Jet Fuel for population modeling in pharmacometrics.

Stan is a powerful probabilistic programming language designed mainly for Bayesian data analysis. Torsten is a collection of Stan functions that handles the events (e.g., dosing events) and solves the ODE systems that are frequently present in pharmacometric models. To perform a Bayesian data analysis, most models in pharmacometrics require Markov Chain Monte Carlo (MCMC) methods to sample from the posterior distribution. However, MCMC is computationally expensive and can be time-consuming, enough so that people will often forgo Bayesian methods for a more traditional approach. This paper shows how to speed up the sampling process in Stan by within-chain parallelization through both multi-threading using Stan's reduce_sum() function and multi-processing using Torsten's group ODE solver. Both methods show substantial reductions in the time necessary to sufficiently sample from the posterior distribution compared with a basic approach with no within-chain parallelization.

A Between-Host Cholera Mathematical Model Incorporating Temperature Dependence

This paper establishes a between-host cholera model with temperature dependent parameter. This is done using system of ODEs to analyse the effect of temperature change on cholera disease. The model analysis reveals that when R0 < 1, the disease free equilibrium point is locally and globally asymptotically. It is also noticed that if R0 > 1, the endemic equilibrium point is locally and globally asymptotically stable. The sensitivity analysis of model parameters shows that R0 ​depends intensively on infection rate of pathogen α1​ normalized with temperature. An increase in infection rate of pathogen α1​ that is dependent on temperature by 10% would increase R0​ by 10% and decreasing it by 10% reduces R0 by 10%; hence, increasing the temperature of the environment where the pathogen lives would help reduce the rate of infection of the pathogen, thus reducing the reproduction number R0​. We conducted numerical simulation of the model in response to temperature changes, and the results indicate that Vibrio cholerae pathogens multiply faster at 23°C but between 23°C < T ≤ 43°C the pathogen multiplication is hindered, therefore, at 23°C, more pathogens are active to cause infection compared to high temperatures.

Hunter Self-Similar Implosion Profiles for the Gravitational Euler–Poisson System

Our result is a construction of infinitely many radial self-similar implosion profiles for the gravitational Euler–Poisson system. The problem can be expressed as solving a system of non-autonomous non-linear ODEs. The first rigorous existence result for a non-trivial solution to these ODEs is due to Guo et al. (Commun Math Phys 386(3):1551–1601, 2021), in which they construct a solution found numerically by Larson (Mon Not R Astron Soc 145(3):271–295, 1969) and Penston (Mon Not R Astron Soc 144(4):425–448, 1969) independently. The solutions we construct belong to a different regime and correspond to a strict subset of the family of profiles discovered numerically by Hunter (Astrophys J 218:834, 1977). Our proof adapts a technique developed by Collot et al. (Mem Am Math Soc 260(1255):v+97, 2019), in which they study blowup for a family of energy-supercritical focusing semilinear heat equations. In our case, the quasilinearity presents complications, most severely near the sonic point where the system degenerates.

Mathematical insights into the influence of delay and external recruitment on coral-macroalgae system

As anthropogenic pressures on coral reef ecosystems continue to increase due to global warming and mass coral bleaching events, there is growing interest in developing conservation strategies to restore degraded coral reefs. One such approach is the transplantation of coral larvae onto degraded reefs. In this paper, we use a delayed coral-macroalgae model to explore the effects of external recruitment of coral. By analyzing the local and global stability of macroalgae-free equilibrium and coexistence equilibrium in the system, we find that sustained external recruitment of coral will favor coral competition with macroalgae. In addition, we choose delay as a bifurcation parameter and demonstrate that Hopf bifurcation may occur at a critical delay near the coexistence equilibrium. Interestingly, the delay may cause the globally asymptotically stable equilibrium in the ODE system to become unstable, resulting in the appearance of periodic solutions. Furthermore, we analyze population dynamics using optimal control theory and determine the effect of minimum external recruitment on the population dynamics. In the numerical simulation section, parameters of coral-macroalgae dynamics are estimated by using a 12-year (2005–2017) benthic cover dataset of coral reefs in the Gulf of Mannar, southeastern India. The theoretical results are validated and supported by numerical simulations.

Analytical solutions of higher-dimensional coupled system of nonlinear time-fractional diffusion-convection-wave equations

This paper discusses how to generalize the invariant subspace method for deriving the analytical solutions of the multi-component [Formula: see text]-dimensional coupled nonlinear time-fractional PDEs (NTFPDEs) in the sense of Caputo fractional-order derivative for the first time. Specifically, we describe how to systematically find different invariant product linear spaces with various dimensions for the considered system. Also, we observe that the obtained invariant product linear spaces help to reduce the multi-component [Formula: see text]-dimensional coupled NTFPDEs into a system of fractional-order ODEs, which can then be solved using the well-known analytical methods. More precisely, we illustrate the effectiveness and importance of this developed method for obtaining a long list of invariant product linear spaces for the multi-component [Formula: see text]-dimensional coupled nonlinear time-fractional diffusion-convection-wave equations. In addition, we have shown how to find different kinds of generalized separable analytical solutions for a multi-component [Formula: see text]-dimensional coupled nonlinear time-fractional diffusion-convection-wave equations along with the initial and boundary conditions using invariant product linear spaces obtained. Finally, we provide appropriate graphical representations of some of the derived generalized separable analytical solutions with various fractional-order values.

Innovative binary nanofluid approach with copper (Cu - EO) and Magnetite ([formula omitted] - EO) for enhanced thermal performance

In this study, we present an innovative and detailed analysis of the motion of a binary nanofluid, specifically copper (Cu - EO) and Magnetite (Fe3O4 - EO), in a porous medium under solar radiation. Our research is driven by the potential to significantly enhance the thermal performance of a Parabolic Trough Solar Collector (PTSC) using a binary nanofluid. The boundary conditions, partial differential restrictions, and the governing PDEs are transformed into the ODEs system with appropriate similarity transformations. The finite element method (FEM) and semi-analytical technique memad Akbari-Ganji's Method (AGM) are used to solve the approximate solutions of ODEs. The findings of evaluating Copper (Cu - EO) and Magnetite (Fe3O4 - EO) were the two different nanofluids based on engine oil that were presented. Furthermore, the model's overall entropy variations are enhanced using Reynolds. Moreover, our study contributes to understanding fluidity and viscosity alterations in such systems. We use the Brinkman number to observe these alterations. We reveal that the thermal efficiency of (Cu - EO) is lower than (Fe3O4 - EO) by a range of 0.005–0.6. This indicates a substantial enhancement in the thermal performance of the PTSC.

A system of ODEs for representing trends of CGM signals

Diabetes Mellitus is a metabolic disorder which may result in severe and potentially fatal complications if not well-treated and monitored. In this study, a quantitative analysis of the data collected using CGM (Continuous Glucose Monitoring) devices from eight subjects with type 2 diabetes in good metabolic control at the University Polyclinic Agostino Gemelli, Catholic University of the Sacred Heart, was carried out. In particular, a system of ordinary differential equations whose state variables are affected by a sequence of stochastic perturbations was proposed and used to extract more informative inferences from the patients’ data. For this work, Matlab and R programs were used to find the most appropriate values of the parameters (according to the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC)) for each patient. Fitting was carried out by Particle Swarm Optimization to minimize the ordinary least squares error between the observed CGM data and the data from the ODE model. Goodness of fit tests were made in order to assess which probability distribution was best suitable for representing the waiting times computed from the model parameters. Finally, both parametric and non-parametric density estimation of the frequency histograms associated with the variability of the glucose elimination rate from blood were conducted and their representative parameters assessed from the data. The results show that the chosen models succeed in capturing most of the glucose fluctuations for almost every patient.

Modeling of KSHV/HHV-8 and HIV-1 Co-Dynamics in Vivo

Human immunodeficiency virus kind 1 (HIV-1) compromises the immune system by infecting and damaging CD4+ T cells. Infection can progress to the ultimate stage, acquired immune deficiency syndrome (AIDS), if HIV-1 therapy is not received. People living with HIV/AIDS are more vulnerable to infections that they otherwise wouldn’t develop. Opportunistic infections or malignancies are the terms used to describe them. Kaposi sarcoma (KS) is an AIDSrelated malignancy caused by Kaposi’s sarcoma-associated herpesvirus (KSHV) (also known as human herpesvirus 8 (HHV-8)). HIV-1 and KSHV co-infection cases has been shown in several studies. Using a system of ODEs, we develop a new mathematical model to study the co-dynamics of HIV-1 and KSHV in vivo. The model includes interactions between healthy CD4+ T cells, HIV-1-infected CD4+ T cells, HIV-1 particles, healthy B cells, KSHV-infected B cells, and KSHV particles. By analyzing the boundedness and nonnegativity of the solutions, we prove the mathematical well-posedness and biological compatibility of the model. The existence and stability of the model’s steady states are established by four threshold values that we identify. We prove that steady states are globally asymptotically stable by using Lyapunov’s method and LaSalle’s invariance principle. Numerical simulations are used to display the results. For both basic reproduction ratios of HIV-1 mono-infection (R1) and KSHV mono-infection (R2), sensitivity analysis is carried out. A comparison between HIV-1 or KSHV mono-infections and co-infections with HIV-1 and KSHV is given. Empirical evidence indicates that co-infection results in higher KSHV and HIV-1 concentrations compared to mono-infection cases. This result is in line with a number of findings found in the literature.

Investigation of a new similarity solution for the (2+1)-dimensional Schwarzian Korteweg–de Vries equation

We study the (2+1)-dimensional Schwarzian Korteweg–de Vries equation (SKdV). The explored solutions describe new Lump soliton colliding with visible soliton, an interaction between multi-soliton waves with one soliton, multi peaks of waves moving in a curved path, two hyperbolic waves moving together without interaction and some of periodic waves. We examine the commutative product between multi unknown Lie infinitesimals for the (2+1)-dimensional (SKdV) equation, and this study result in some new Lie vectors. The commutative product generates a system of nonlinear ODEs which had been solved manually. Through two stages of Lie symmetry reduction, SKdV equation is reduced to non-solvable nonlinear ODEs using various combinations of optimal Lie vectors. Using the Riccati–Bernoulli sub-ODE and Integration methods, we investigate new analytical solutions for these ODEs. Back substituting for the original variables generates new solutions for SKdV. Some selected solutions are illustrated through three-dimensional plots.

New insights of heat transfer in pistons and nozzles flow of graphene-transformer oil nanofluid: A differential transform method

This work aims to study heat transfer analysis in a squeezing flow problem of nanofluid. The flow between pistons and nozzles in an automobile engine is one industrial application where the knowledge of the squeezing flow plays a crucial role. The applied magnetic field has a longitudinal inclined angle that ranges from zero to ninety degrees. In addition, viscous dissipation, chemical reaction, and suction injection are taken into consideration. Graphene nanoparticles with transformer oil are considered for Tiwari-Das Model. The transformed system of nonlinear ODEs is solved using the differential transform method. The numerical findings of the skin friction coefficient, the Nusselt and the Sherwood numbers are assessed with previously available works for accuracy. The graphical findings for relevant parameters of all profiles are analyzed. It becomes apparent that the velocity and heat transfer in compressing flows are significantly influenced by the angle of the inclination of the magnetic field. The temperature of the nanofluid and the rate of the heat transfer are both improved in the existence of the viscous-dissipation and thermophoresis. The findings show that the temperature rises with increased Prandtl and Eckert numbers. Furthermore, when Brownian movements of nanoparticles take place, the rate of mass transfer decreases.

Comparative numerical study between MHD Forchheimer nano and hybrid nanofluid flows over stretching sheet under aligned magnetic field in the presence of radiation absorption

ABSTRACT This study investigates the mass and heat transfer properties of Ag-H2O nanofluid MHD Forchheimer flows and Ag-MoS2/H2O hybrid nanofluid over two-dimensional linear stretching surfaces in aligned magnetic fields with absorption of radiation. The fluid experiences rotational motion around the vertical axis at a consistent angular speed. The system of nonlinear ODEs is derived from the set of nonlinear PDEs by use of similarity transformations. The BVP-5C shooting approach in MATLAB is then employed to solve the associated system of ODEs. As the Forchheimer number, porosity parameter, rotation parameter and aligned magnetic field exhibit an ascent, the velocity profile along the x-axis and y-axis experiences a decrement. Simultaneously, an augmentation in the magnetic parameter, thermal radiation and rotation parameter induces a heightened temperature profile. Additionally, greater values of the Forchheimer number, rotation parameter and magnetic parameters correspond to an increase in concentration profile. Furthermore, the results indicated an increased temperature profile in the Ag-MoS2/H2O hybrid nanofluid compared to that in the Ag-H2O nanofluid for magnetic, rotation and radiation parameters. Also, the concentration profile of magnetic and rotation parameters along with the Forchheimer number were noted to be higher in the Ag-MoS2/H2O hybrid nanofluid compared to that in the Ag-H2O nanofluid.

A GPU-accelerated Lagrangian method for solving the Liouville equation in random differential equation systems

This work presents and analyzes a numerical approach to efficiently solve the Liouville equation in the context of random ODEs using GPGPUs. Our method combines wavelet compression-based adaptive mesh refinement, Lagrangian particle methods, and radial basis function interpolation to create a versatile algorithm applicable in multiple dimensions. We discuss the advantages and limitations of this algorithm. To demonstrate its effectiveness, we compute the probability density function for various 2D and 3D random ODE systems with applications in physics and epidemiology.

Bifurcation and Stability of a Haptotaxis Mathematical Model for Complex MAP

This paper studies the bifurcation and stability of a parabolic–ordinary-parabolic system to a model with three reactive species: molecular Anchors [Formula: see text], Nanoparticulates [Formula: see text] and Matrix constituents [Formula: see text] ([Formula: see text]). Global asymptotic stability is studied for both the ODE system and the reaction–diffusion system. The results show that the positive equilibrium is globally asymptotically stable in both the ODE and the corresponding reaction–diffusion systems. Furthermore, by selecting the haptotaxis coefficient as a bifurcation factor, we analyze the existence, direction, and stability of the Hopf bifurcation. The results also reveal that haptotaxis is crucial in determining the stability and bifurcation behavior of the model, suggesting that it has a destabilizing effect and can undergo Hopf bifurcation. Moreover, numerical simulations are presented to verify and illustrate the theoretical results.

Integrability and renormalizability for the fully anisotropic SU(2) principal chiral field and its deformations

For the class of 1 + 1 dimensional field theories referred to as the non-linear sigma models, there is known to be a deep connection between classical integrability and one-loop renormalizability. In this work, the phenomenon is reviewed on the example of the so-called fully anisotropic SU(2) Principal Chiral Field (PCF). Along the way, we discover a new classically integrable four parameter family of sigma models, which is obtained from the fully anisotropic SU(2) PCF by means of the Poisson-Lie deformation. The theory turns out to be one-loop renormalizable and the system of ODEs describing the flow of the four couplings is derived. Also provided are explicit analytical expressions for the full set of functionally independent first integrals (renormalization group invariants).

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.

Theoretical assessment of chemically reactive bioconvective flow of hybrid nanofluid by a curved stretchable surface with heat generation

ABSTRACT Bioconvection in hybrid nanofluids refers to the phenomenon where biological microorganisms such as algae or bacteria exhibit collective movement or pattern formation in a suspension containing nanoparticles. This phenomenon has significance in various fields, including biology, nanotechnology, and engineering. The current investigation emphasizes the bioconvective flow of a water (H2O)−ethylene glycol (C2H6O2)-based hybrid nanofluid flow by a curved stretched sheet. Copper (Cu) and silver (Ag) nanoparticles are suspended in the base fluid. The thermal field is analyzed in the presence of heat generation, dissipation, Joule heating, and the impact of thermal radiation. Binary reactions associated with Arrhenius energy are accounted in the modeling of mass concentration. The phenomenon of bioconvection is considered to regulate the random movement of tiny solid particles within the flow. Boundary layer constraints are implemented to eliminate ineffective terms from modeling. The transformation procedure is adopted to obtain the flow governing system of ODEs. The built-in code of Mathematica (NDSolve) is implemented to obtain the graphical and numerical results. The results show that the velocity field decreases with increasing porosity variable and Hartmann number. An opposite impression of the Eckert number and Schmidt variable on the thermal field is noticed. Bioconvection Peclet and Lewis numbers have a direct relationship with motile density.

Mathematical Modelling of the Thermosolutal Effect on Blood Flow through a Micro-Channel in the Presence of a Magnetic Field

Substances such as mass concentrations in the blood cause circulation challenges. Blood is a substance made up of 55% plasma and 45% hemoglobin, which flows through the blood vessels and is aided by the heart in pumping to all organs and other tissues and cells in the human body. Mathematical modelling plays an important role in understanding the blood circulation problem in the human body, and it helps transform the real-life cardiovascular problem into a system of mathematical equations. In this research, we transformed the real-life cardiovascular problem of the heat effect on mass concentration and blood flow through blood vessels into a system of partial differential equations representing blood momentum, energy, and mass concentration equations after modifying the Navier-Stoke equation. The models are scaled from dimensional to dimensionless using specific scaling parameters. The scaled dimensionless equations were further perturbed and reduced to a system of ordinary differential equations, and the system of ODEs was solved using the Laplace method, where mathematical models representing blood velocity profiles, temperature profiles, and mass concentration profiles were obtained. A numerical simulation was performed using Wolfram Mathematica, version 12, where the effect of the pertinent parameters on the flow profiles was investigated and the results were presented graphically. The variation of the pertinent parameters such as the Grashof number, solutal Grashof number, Prandtl number, Soret number, Schmidt number, Darcy number, and magnetic field affect the flow profile such that the increase in Grashof number and solutal Grashof number causes the blood velocity to increase substantially at the centre of the vessel before decreasing to zero as the wall boundary layer increases, while the increase in Prandtl number and Soret number increases the blood temperature and velocity.

Irreversibility analysis of Eyring Powell nanofluid flow in curved porous channel

AbstractThe aim of this study is to minimize entropy in the MHD Eyring‐Powell fluid through a semi‐porous curved channel. The flow phenomena are examined under the consideration of joule heating, viscous heating, thermophoresis, and Brownian motion. The motivation of this research is to minimize entropy production in curved porous channel because thermal systems become more efficient as a result of decreased energy consumption, operational expenses, and experimental costs. This approach is useful in designing industrial equipment that requires efficient thermal management, such as fuel cells, chemical processing, and advanced refrigeration systems. The coupled boundary layer equations of the problem are highly nonlinear PDEs, which are transformed into systems of coupled ODEs by using similarity variables. The solution of a coupled system of ODEs is obtained numerically via Bvp4c. The effect of several physical parameters for entropy analysis, Bejan number, concentration, and velocity/temperature are illustrated and analyzed using graphs. Furthermore, the computational outcomes of physical quantities, for example, heat and mass transfer rate are also presented. Results indicated that the increasing value of the Reynolds number increases the entropy generation rate, while the reverse tendency is noticed for the Eyring‐Powell parameter. The rising values of the Brownian parameter increase the Bejan number after its decrease, while reverse behavior is observed for the thermophoresis parameter.

Exploring the influence of morphology on magnetized Ree–Eyring tri‐hybrid nanofluid flow between orthogonally moving coaxial disks using artificial neural networks with Levenberg–Marquardt scheme

AbstractThe present study presents an analysis of Ree–Eyring tri‐hybrid nanofluid flow between two expanding/contracting disks with permeable walls by applying the computing power of Levenberg–Marquardt supervised neural networks (LM‐SNNs). The effects of thermal radiation, Brownian motion, and thermophoresis were also thoroughly examined. The results are presented for tri‐hybrid nanofluid with SWCNT and MWCNT and Fe2O3 and H2O base fluid. The coupled non‐linear PDE system is transformed into a system of ODE associated with convective boundary conditions by applying the appropriate transformations. This is then accomplished numerically by using the finite difference‐based BVP‐4c MATLAB code that implements the three‐stage Lobatto IIIA formula. The results are novel and have been validated with LM‐SNNs outcomes. It has been observed that both numerical outcomes and LM‐SNNs produce equivalent results, and both approaches exhibit a drop in the velocity profile for the magnetic field near the lower plate and a rise near the upper plate. The skin friction against the Prandtl number increases, whereas the Nusselt number decreases at the upper disc. Compared to BVP‐4c numerical approaches, the given LM‐SNNs model is more dependable, efficient, and time‐saving because it requires less work and produces results quickly.

Singular Perturbation Analysis for a Coupled KdV–ODE System

Singular Perturbation Analysis for a Coupled KdV–ODE System