System Of Differential Equations Research Articles

Tutorial: a beginner’s guide to building a representative model of dynamical systems using the adjoint method

AbstractBuilding a representative model of a complex dynamical system from empirical evidence remains a highly challenging problem. Classically, these models are described by systems of differential equations that depend on parameters that need to be optimized by comparison with data. In this tutorial, we introduce the most common multi-parameter estimation techniques, highlighting their successes and limitations. We demonstrate how to use the adjoint method, which allows efficient handling of large systems with many unknown parameters, and present prototypical examples across several fields of physics. Our primary objective is to provide a practical introduction to adjoint optimization, catering for a broad audience of scientists and engineers.

Modeling of the Efficiency of the Centrifugal Conical Disk Dispenser of Bulk Materials

Centrifugal disk dispensers are widely used in various tasks of dosing bulk, dispersed materials. The design of the disk depends on the physical and mechanical characteristics of the dosing medium. The work discusses the development of an analytical model of the movement of a material particle along a conical centrifugal disk depending on the kinematic characteristics of the dosing process and the characteristics of the dosing material, as well as experimental confirmation of the theoretical model, which is relevant for the calculation and design of working elements of this type. The obtained system of differential equations is solved using the Runge–Kutta numerical method. Experimental studies were carried out using the method of a planned factorial experiment. The experiment was conducted for three factors at three levels. The feedback criterion was the performance of a centrifugal conical disk dispenser for bulk materials. The disk cone angle was set at 10, 20, and 30°. The disk diameter was 130, 150, and 170 mm, the gap between the disk and the edge of the hopper neck was 6, 8, and 10 mm, and the rotational speed of the conical disk was 0.65, 1.02, and 1.39 rad/s. The dispensing rate of the dispenser ranged from 15 to 770 g/s, depending on the values of the experimental factors. For use in the regression equation of the natural values of the factors, a method of transforming the terms of the equation from coded values to natural ones is provided. The obtained experimental correlation dependencies were checked for reproducibility with Cochrane’s test, and the adequacy of the model was checked using Fisher’s test. The significance of the coefficients in the correlation equation was evaluated using the Student’s t-test. The difference between the experimental data and the results of the theoretical modeling does not exceed 5%. The obtained system of differential equations makes it possible to model the radial velocity of the ascent of bulk material from the conical rotating disk depending on the rotation frequency, disk diameter, and the height of the annular gap between the discharge throat of the hopper and the conical disk. The analytical model enables the modeling of the productivity of the conical dispenser for bulk materials for arbitrary parameters of rotation frequency, disk diameter, and the size of the annular gap between the discharge throat of the hopper and the conical disk.

Analytical Study of Nonlinear Systems of Higher-Order Difference Equations: Solutions, Stability, and Numerical Simulations

This paper aims to derive analytical expressions for solutions of fractional bidimensional systems of difference equations with higher-order terms under specific parametric conditions. Additionally, formulations of solutions for one-dimensional equations derived from these systems are explored. Furthermore, rigorous proof is provided for the local stability of the unique positive equilibrium point of the proposed systems. The theoretical findings are validated through numerical examples using MATLAB, facilitating graphical illustrations of the results.

Numerical scrutinization of heat transfer subject to physical quantities through bioconvective nanofluid flow via stretching permeable surfaces

Background: The mechanics of heat and mass transfer via nanofluid flow across many media are currently being discussed. “Nanofluids” are fluids that include highly heat-conductive nanoparticles, and they are essential for resolving engineering problems. Under the effects of activation energy, thermal radiation, and motile microorganisms, the process of heat and mass transfer through steady nanofluid flow crosses over stretched surfaces in this scenario.Methodology: For mathematical evaluation, the system of partial differential equations (PDEs) is used to describe this physical framework. By introducing suitable similarity variables with a set of boundary conditions, this mathematical system of PDEs has become a system of ordinary differential equations (ODEs). To obtain numerical results, the MATLAB built-in program “bvp4c” is used to solve the system of first-order equations.Results: In the findings and discussion section, the resulting outcomes are thoroughly examined and visually shown. The flow rate in these systems increases due to the erratic movement of microorganisms. The graphical representation shows the impacts of involving physical factors on the microorganism, thermal, concentration, and momentum profiles. Variations/changes in these profiles can be observed by adjusting the parametric values, as depicted in the graphs. Consequently, thermal transport is boosted by 25%. Additionally, the skin friction, Nusselt, Sherwood, and microbe density numbers are determined numerically. The findings demonstrate that increasing the magnetic field parameter causes the velocity profile to decrease, increasing the radiation parameter leads to an increase in temperature description, and increasing the Lewis number causes the microorganism profile’s transport rate to decrease.

Thermal inspection of hybrid nanofluid flows over a stretched cylinder at an oblique stagnation point with variable characteristics

AbstractThe enhanced thermal characteristics of hybrid nanofluids make them more versatile compared to conventional fluids. These improved thermal properties render hybrid nanomaterials highly practical for a wide range of applications, including solar systems, energy production, and cooling processes. In line with this perspective, the current study concentrates on evaluating the thermal performance of two unique hybrid nanofluid flows that impinge obliquely on a stretched cylinder. Two base fluids, FC‐77 and a binary mixture of water and ethylene glycol (50:50)%, have been considered, with the addition of nanoparticles such as single‐walled carbon nanotubes (SWCNTs) and multiwalled carbon nanotubes (MWCNTs). The said model's novelty is enhanced by the temperature dependent viscosity and thermal conductivity. Appropriate transformations are applied to derive a system of ordinary differential equations (ODEs), which are then solved numerically using the bvp4c method. A thorough examination is conducted on the physical phenomenon of pertinent parameters, accompanied by graphical representations. The results revealed that, for the FC‐77 coolant‐based hybrid nanofluid, mounting the particle volume fraction leads to a significant reduction in temperature distribution. Additionally, it is perceived that the presence of a variable viscosity parameter causes a reduction in the surface drag coefficient as well as the axial and tangential velocities. The validity of the proposed flow model is demonstrated by comparing the results with those from an earlier study.

Dynamic behavior of a third-order system of difference equations

Dynamic behavior of a third-order system of difference equations

Averaging of a normal system of ordinary differential equations of high frequency with a multipoint boundary value problem on a semi-axis

A multipoint boundary value problem for a nonlinear normal system of ordinary differential equations with a rapidly time-oscillating right-hand side is considered on a positive time semi-axis. For this problem, which depends on a large parameter (high oscillation frequency), a limiting (averaged) multipoint boundary value problem is constructed and a limiting transition in the Hölder space of bounded vector functions defined on the considered semi-axis is justified. Thus, for normal systems of differential equations in the case of a multipoint boundary value problem, the Krylov-Bogolyubov averaging method on the semi-axis is justified.

Mass and heat transfer in the boundary layer region of modified second-grade fluid flow over a linearly stretched sheet embedded in porous media

Here, we have improved a model to describe the flow variables, velocity, mass and heat transfer in the boundary layer region of modified second-grade fluid flow over a linearly stretched sheet in a porous media. The modified model is used to study the qualitative impact of buoyancy parameter, second-grade fluid parameter, magnetic parameters, porous parameter, power-law index, and chemical reaction parameter on the flow profiles, radial and axial velocities, temperature, and concentration. Starting with the steady-state governing equations of mass, momentum, heat, and concentration of the fluid flow, we obtained the boundary layer approximations of the flow near the linearly stretched sheet with the no-slip boundary condition. Similarity transformation has been used to convert the partial differential equations system into a nonlinear ordinary differential equations system. The radial velocities, temperature, and concentration profiles have been solved numerically, and the qualitative influence of the flow parameters on the flow variables has been simulated and graphically presented for comparison. We have observed that radial and axial velocities were increasing for the shear-thinning and shearthickening fluids with solutal Grashof number, thermal Grashof number and second-grade fluid parameters. In contrast, the porous and chemical reaction parameters slow down both fluids’ radial and axial velocities. The temperature and concentration increase with the porous and magnetic parameters. However, thermal and solutal Grashof and second-grade fluid parameters suppress temperature and concentration. In shear-thickening fluids, chemical reaction parameters enhance concentration but suppress in shear-thinning fluids.

Heat and mass transfer analysis of conducting non-Newtonian nanofluid flows over an elongating sheet with a non-uniform heat source

Nanofluids help in many fields to improve the performance of thermal systems by augmenting heat transfer rates through their thermophysical properties. The performance of the nanofluids with various base fluids may be different. Therefore, the study and comparison of behaviors of various nanofluids are useful in several applications such as fuel as a coolant in automobiles, and in medical and electronic equipment to reduce the thermal resistance. This research proposed a novel model to investigate the flow behavior of three different nanofluids over an elongating surface in the presence of a non-uniform heat source and thermal radiation effects. This investigation describes how the considered nanofluids behave in the presence of a transverse magnetic field, and other effects. The proposed governing boundary layer partial differential equations (PDEs) are reformed into a system of nonlinear ordinary differential equations (ODEs) by introducing the proper similarity transformation. The finalized equations are solved numerically with the help of the ND solve package in Mathematica software. We intended how the fluid flow and heat transfer are affected by non-dimensional controlling factors with the help of graphics. Further, the calculations and discussions are accompanied by the numerical values of the skin friction coefficient and heat and mass transfer rates. According to the current findings, the Maxwell nanofluid exhibits superior performance in velocity, and the Oldroyd-B nanofluid shows more concentration and less temperature. As a special case, the results of this investigation are compared with the existing results, and found a good agreement between the results.

Thermodynamic survey of a reactive MHD couple stress fluid through saturated porous substances with convective boundary conditions

This study explores the thermodynamic behavior of a reactive hydromagnetic liquid flowing through permeable materials, with convective cooling applied to the walls. This study holds practical significance in optimization of thermal management systems and it is crucial for enhancing the efficiency of controlling thermal runaway. The flow is modeled as a system of partial differential equations which are numerically solved. The modified Adomian Decomposition Method and Pade approximation technique are utilized in solving the equations. The results acquired for velocity and temperature distributions are thereby employed to estimate entropy generation rate with critical values over numerous effects of various boundaries over improvement of thermal runaway utilizing Pade approximation technique to show the significant impact of convective cooling term (Biot number) and other thermophysical parameters on the fluid flow. The outcomes show that fluid velocity continuously increases with rising values of inverse couple stress and fluid temperature increases Biot number.

Integration of a sine-Gordon type equation with an additional term in the class of periodic infinite-gap functions

In this paper, the inverse spectral problem method is used to integrate a nonlinear sine-Gordon type equation with an additional term in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin equations and the first trace formula satisfies sine-Gordon-type equations with an additional term.

Numerical Studies on the Classical Long Wave System Describing Shallow Water Waves With Dispersion

In this study, a reliable and efficient local discontinuous Galerkin scheme for numerically solving the classical long wave system is proposed. This scheme approximates the temporal derivatives using the explicit strong-stability-preserving higher-order Runge-Kutta method, while the space derivatives are approximated using the local discontinuous Galerkin method, yielding an ordinary differential equation system. Numerical examples for various test problems are presented to offer a comprehensive understanding of the accuracy and reliability of the proposed method. The results obtained, which confirm the sub-optimal order of accuracy, are displayed in multiple tables. Additionally, several two-dimensional and three-dimensional graphical depictions of the problem have been provided to illustrate the behavior of the solution.

Comparative study of Hiemenz flow in hybrid Carreau nanofluid (Fe3O4-Cu/kerosene-engine oil): nonlinear effects, Newtonian heating and viscous dissipation

ABSTRACT This study is important because it helps us understand how different factors affect the flow of a special liquid with nanoparticles, giving us useful insights for real-world applications in engineering and technology. This research endeavours to explore and contrast the distinctive characteristics of steady, 2-D magnetohydrodynamic (MHD) and stagnation point (Hiemenz) flow in Carreau hybrid nanofluid {Fe3O4-Cu/kerosene-engine Oil} over a nonlinear stretching-shrinking surface with its counterpart, Carreau nanofluid. Employing the laws of conservation encompassing mass, momentum, energy, and concentration, our methodology probes into the multifaceted effects induced by factors like the suction-injection, viscous dissipation, Newtonian heating, Hiemenz flow, thermophoresis, and Brownian motion. This exploration involves the transformation of governing partial differential equations (PDEs) into a system of ordinary differential equations (ODEs) through similarity transformations. Subsequently, the intricate non-linear ODEs are meticulously solved using the bvp4c technique, an adept MATLAB built-in function, facilitating a comprehensive understanding of the intricate dynamics within the system. In addition, the study considers the Newtonian heating effect and a numerical comparison is performed to verify the results. Using two different types of nanoparticles dispersed in kerosene-engine oil, it was demonstrated that increasing Weissenberg parameter values resulted in significant improvements in thermal conductivity and thermal velocity. A comparison of Carreau nanofluid thermal and concentration velocities with Carreau hybrid nanofluid showed that Carreau nanofluid values are lower (Weissenberg parameter). Higher Biot numbers are associated with reduced temperature velocity and elevated concentration velocity.

Exact Solutions of System of Fourth-Order Difference Equations

Exact Solutions of System of Fourth-Order Difference Equations

A model for predicting the phenology of Philaenus spumarius.

The design and implementation of Philaenus spumarius control strategies can take advantage of properly calibrated models describing and predicting the phenology of vector populations in agroecosystems. We developed a temperature-driven physiological-based model based on the system of Kolmogorov partial differential equations to predict the phenological dynamics of P. spumarius. The model considers the initial physiological age distribution of eggs, the diapause termination process, and the development rate functions of post-diapausing eggs and nymphal stages, estimated from data collected in laboratory experiments and field surveys in Italy. The temperature threshold and cumulative degree days for egg diapause termination were estimated as 6.5°C and 120 DD, respectively. Preimaginal development rate functions exhibited lower thresholds ranging between 2.1 and 5.0°C, optimal temperatures between 26.6 and 28.3°C, and upper threshold between 33.0 and 35°C. The model correctly simulates the emergence of the 3rd, 4th, and 5th nymphal instars, key stages to target monitoring actions and control measures against P. spumarius. Precision in simulating the phenology of the 1st and 2nd nymphal stages was less satisfactory. The model is a useful rational decision tool to support scheduling monitoring and control actions against the late and most important nymphal stages of P. spumarius.

Modelling dielectric behaviour and anomalous relaxation in chromonic liquid crystals

ABSTRACT We discuss the feasibility of applying a frequency-dependent diffusive model to study the dielectric response of complex fluids. The model considers the effect of free and bound particles by solving a non-exponential relaxation behaviour and anomalous diffusion phenomena into a coupled system of differential equations and boundary conditions. To illustrate the potentialities of the model in the data interpretation, we used the experimental results that provide the frequency spectra of a system formed by an aqueous solution of sunset yellow, a chromonic liquid crystal. Our model yields a considerably better adjustment of the experimental data compared to the simple Havriliak-Negami model and provides the relaxation frequency due to the director mode associated with the stack fluctuations about their short axis. Furthermore, we obtained parameters associated with the ion motion nature of the sample, including diffusion coefficient, Debye length and the ionic diffusive regime, which present a similar subdiffusive behaviour found in nematic thermotropic liquid crystals.

Stability analysis of a three-dimensional system of difference equations with quadratic terms

AbstractThis study is involved with a class of three-dimensional system of difference equations incorporating quadratic term, which naturally extends and improve several results in the literature. Firstly, we demonstrate the existence of fixed points, the boundedness, persistence and invariance of positive solution of the mentioned system. Later, for this system, we give the global asymptotic stability at fixed point and the rate of convergence result which play an important role in the discrete dynamical systems. And lastly, some numerical examples are given to validate the effectiveness and feasibility of the theoretical findings.

∂¯‐problem for a second‐order elliptic system in Clifford analysis

In the framework of Clifford analysis, we study a second‐order elliptic (generally nonstrongly elliptic) system of partial differential equations of the form: , where stands for the Dirac operator with respect to a structural set . The solutions of this system are known as ‐inframonogenic functions. Our main purpose is to describe necessary and sufficient conditions for the solvability of a ‐problem associated with the sandwich operator .

Significance of shape factor on magnetohydrodynamic buoyancy thin film flow of nanofluid over inclined sheet with slip condition: Irreversibility analysis

This work aims to examine the entropy production, heat transport, and dynamics of the unsteady thin film magnetohydrodynamic (MHD) flow of a nanofluid composed of alumina (Al2O3) and water. The fluid flow is seen to pass over an inclined sheet, taking into account the effects of buoyancy force, viscous dissipation, and joule heating. The system of partial differential equations (PDEs) is optimized under the boundary layer assumptions. Appropriate transformations are used to convert the governing partial differential equations (PDEs) and boundary conditions into dimensionless forms. Using MATLAB’s bvp4c code and a local non-similarity technique with up to second-degree truncation, we can obtain the findings of the enhanced model. The effect of multi-shape Al2O3 nanoparticles on flow, heat, and entropy-generating features is also investigated after the calculated results have been successfully aligned with published data. Mixed convection, nanoparticle volume percent, inclination angle, magnetic field intensity, mass suction, Eckert number, and Biot number are only a few of the governing parameters whose effects are graphically shown for selected values. As a result, the local Nusselt number and skin friction coefficient may be calculated. The skin friction and Nusselt number profiles exhibit a decreasing trend as the values of nanoparticle volume fraction ([Formula: see text]) magnetic [Formula: see text] and unsteadiness (A) increase toward mixed convection ([Formula: see text]). On the other hand, Nusselt number profile increases with increasing values of mass suction parameter [Formula: see text] The profiles of entropy generation and Bejan number show an upsurge as the values of the magnetic parameter [Formula: see text] and Brinkman number (Br) increase. Conversely, the entropy generation reduces with an increase in the temperature difference parameter [Formula: see text] and Bejan number increases.

Relative Entropy Gradient Sampler for Unnormalized Distribution

We propose a relative entropy gradient sampler (REGS) for sampling from unnormalized distributions. REGS is a particle method that seeks a sequence of simple nonlinear transforms iteratively pushing the initial samples from a reference distribution into the samples from an unnormalized target distribution. To determine the nonlinear transforms at each iteration, we consider the Wasserstein gradient flow of relative entropy. This gradient flow determines a path of probability distributions that interpolates the reference distribution and the target distribution. It is characterized by an ordinary differential equation (ODE) system with velocity fields depending on the density ratios of the density of evolving particles and the unnormalized target density. To sample with REGS, we need to estimate the density ratios and simulate the ODE system with particle evolution. We propose a novel nonparametric approach to estimating the logarithmic density ratio using neural networks. Extensive simulation studies on challenging multimodal 1D and 2D mixture distributions and Bayesian logistic regression on real datasets demonstrate that REGS has reasonable performance compared with popular samplers based on Wasserstein gradient flows.