System Of First-order Differential Equations Research Articles

Identification of the Lamé parameters of an inhomogeneous pipe based on the displacement field data

The present paper considers the direct and inverse problems on steady-state oscillations of functionally graded elastic pipe. The direct problem is reduced to solving a numerical set of canonical systems of first-order differential equations with variable coefficients. The influence of the variable Lamé parameters on the displacement functions and amplitude–frequency characteristics is estimated. The solution of the inverse problem on determination of the functions describing the change in the Lamé parameters is obtained numerically by inverting differential operators on the basis of the displacement field data gathered in a finite set of points at a fixed frequency. The results of computational experiments are discussed.

Thermal in-plane stability of concrete-filled steel tubular arches

This paper analytically and numerically investigates the pre-buckling response and in-plane stability boundaries of circular concrete-filled steel tubular (CFST) arches subjected to combined thermal and mechanical loading. The governing non-linear equations of equilibrium are obtained using energy methods and both elastic and inelastic material behaviour is considered. A novel mechanically derived non-discretisation numerical method is proposed for the pre-buckling analysis. The stress-strain relation of the confining steel tube is described using a bi-linear plasticity model, and an inelastic material model is adopted for the concrete core which considers the effects of confinement and transient thermal strain. The result is a system of first-order differential equations which can be numerically solved with known boundary conditions including fixed ends, pinned ends or crowned-pinned cases. Closed-form solutions are presented for the elastic anti-symmetric bifurcation loads, whilst the inelastic anti-symmetric buckling strength was studied using finite element (FE) analysis. The FE model is verified by comparison to the derived analytical and numerical models which show a high level of agreement. Additionally, a sensitivity analysis is conducted which explores the influence of the constitutive material law for the concrete core and contact model for the steel-concrete interface on critical buckling loads.

Dog Treat Ball: An Activity to Introduce Systems of Differential Equations

This paper describes an activity using a dog treat ball to introduce systems of first-order differential equations. Beads are placed in the first of two hemispherical chambers of a food-dispensing dog toy. As the ball is turned, students track the number of beads in the first chamber, the second chamber, and the exterior of the ball. Students develop a working assumption, write a system of differential equations to model the number of beads in each region, and use a spreadsheet to compare their experimental and model results.

ОБ УПРАВЛЯЕМОСТИ ТЕМПЕРАТУРНОГО РЕЖИМА ОТАПЛИВАЕМЫХ ЗДАНИЙ

The temperature of the internal air of different rooms of heated buildings often must be different. At the same time, the number of control actions applied in this case is rather limited – as a rule, they regulate either the temperature or the coolant flow rate at the individual heat point (IHP) of the building. Moreover, the coolant flow rate almost never changes due to the inevitability of hydraulic deregulation of the heating system, therefore, as a rule, only one control action is available to control the temperature regime of buildings – this is the temperature of the coolant at the IHP. In this regard, an issue quite naturally arises on the controllability of the temperature regime of buildings, that is, on the possibility of maintaining different temperatures in different rooms. To solve this problem, a mathematical description of the temperature conditions of various rooms in a building with a two-pipe heating system has been proposed. This description is a matrix system of first-order differential equations. The concepts of excess internal air temperature and excess coolant temperature are used. The conditions of complete controllability of the object were found. For this, the controllability matrix was calculated, it was shown that its determinant can be expressed in terms of the Vandermonde determinant. This circumstance greatly simplified the definition of the conditions of controllability. It was noted that for the complete controllability of the object, it is necessary that all its premises would differ either in heat-shielding and heat-inertial properties, or in the characteristics of the heating devices installed in them. In most practical cases, usually, there is no such distinction; therefore, the temperature regime of heated buildings, as a rule, is not fully controlled.

О КОЛЕБАНИЯХ НЕОДНОРОДНОГО ПЬЕЗОДИСКА

In the framework of the model of coupled electroelasticity of inhomogeneous bodies, the problem of steady-state oscillations of a thin piezodisc with inhomogeneous properties is considered, in particular, in the presence of radial polarization. The necessary simplifications are made within the framework of traditional hypotheses, the formulated boundary-value problem is reduced to a canonical system of first-order differential equations with respect to dimensionless components of radial displacement and radial stress with corresponding boundary conditions. The direct problem of oscillations of an inhomogeneous disk is solved numerically based on the shooting method by numerically analyzing auxiliary Cauchy problems. The analysis of the amplitude-frequency characteristics and resonance frequencies depending on various laws of variation of the inhomogeneous properties of the piezodisc is performed, which in the presented model are characterized by two functions, one of which characterizes the change in the elastic modulus, the second changes in the piezomodule. The inverse problem is formulated in the first statement, in which the laws of variation of the piezodisc heterogeneity (two functions) are restored from the values of the functions characterizing the radial displacement and stress, known in a finite set of points. The results of computational experiments on solving the inverse problem in the first formulation are presented, various aspects of reconstruction are discussed. The second formulation of the inverse problem is formulated to determine the piezoelectric characteristics of the disk, where a function that describes the laws of change in the elastic characteristics of the disk and the amplitude-frequency characteristic is considered known. To solve the inverse problem, in this formulation, the Fredholm integral equation of the first kind with a smooth kernel is formulated. The results of numerical experiments on solving the Fredholm integral equation of the first kind using the Tikhonov regularizing method are presented, various aspects of reconstruction are discussed.

Forced convection heat transfer of Casson fluid in non-Darcy porous media

Forced convection of non-Newtonian Casson fluid laminar boundary layer flow past an isothermal horizontal flat plate in non-Darcy porous media is studied using Darcy–Forchheimer–Brinkman model. Similarity variables are used to transform the boundary layer equations. The boundary layer equations are reduced into system of first-order differential equations using similarity method. Then, solved numerically using adaptive Runge–Kutta–Fehlberg scheme simultaneously with shooting technique. The effects of Casson parameter, porosity, first- and second-order porous resistances, and Prandtl number on the fluid flow and heat transfer are investigated in terms of the local skin friction and local heat transfer parameters. In addition, velocity and temperature boundary layer profiles are plotted for all considered parameters. It is found that the heat transfer could be enhanced by increasing the Casson parameter and the porous resistance terms. To the contrary, the increase in the porosity reduces heat transfer rates. Finally, the increase in the Prandtl number enhances the heat transfer rates.

Diagonal Block Method for Solving Two-Point Boundary Value Problems with Robin Boundary Conditions

This numerical study presents the diagonal block method of order four for solving the second-order boundary value problems (BVPs) with Robin boundary conditions at two-point concurrently using constant step size. The solution is obtained directly without reducing to a system of first-order differential equations using a combination of predictor-corrector mode via shooting technique. The shooting method was adapted with the Newton divided difference interpolation approach as the strategy of seeking for the new initial estimate. Five numerical examples are included to examine and illustrate the practical usefulness of the proposed method. Numerical tested problem is also highlighted on the diffusion of heat generated application that imposed the Robin boundary conditions. The present findings revealed that the proposed method gives an efficient performance in terms of accuracy, total function calls, and execution time as compared with the existing method.

Generalized conformal Hamiltonian dynamics and the pattern formation equations

We demonstrate the significance of the Jacobi last multiplier in Hamiltonian theory by explicitly constructing the Hamiltonians of certain well known first-order systems of differential equations arising in the activator–inhibitor (AI) systems. We investigate the generalized Hamiltonian dynamics of the AI systems of Turing pattern formation problems, and demonstrate that various subsystems of AI, depending on the choices of parameters, are described either by conformal or contact Hamiltonian dynamics or both. Both these dynamics are subclasses of another dynamics, known as Jacobi mechanics. Furthermore we show that for non Turing pattern formation, like the Gray–Scott model, may actually be described by generalized conformal Hamiltonian dynamics using two Hamiltonians. Finally, we construct a locally defined dissipative Hamiltonian generating function Hudon et al. (2008) of the original system. This generating function coincides with the “free energy” of the associated system if it is a pure conformal class. Examples of pattern formation equation are presented to illustrate the method.

On integrability of the geodesic deviation equation

The Jacobi equation for geodesic deviation describes finite size effects due to the gravitational tidal forces. In this paper we show how one can integrate the Jacobi equation in any spacetime admitting completely integrable geodesics. Namely, by linearizing the geodesic equation and its conserved charges, we arrive at the invariant Wronskians for the Jacobi system that are linear in the ‘deviation momenta’ and thus yield a system of first-order differential equations that can be integrated. The procedure is illustrated on an example of a rotating black hole spacetime described by the Kerr geometry and its higher-dimensional generalizations. A number of related topics, including the phase space formulation of the theory and the derivation of the covariant Hamiltonian for the Jacobi system are also discussed.

Existence and Multiplicity Results for Systems of First-Order Differential Equations via the Method of Solution-Regions

Abstract In this paper, we establish existence and multiplicity results for systems of first-order differential equations. To this end, we introduce the method of solution-regions. It generalizes the method of upper and lower solutions and the method of solution-tubes. Our results can also be seen as viability results since we obtain solutions remaining in suitable regions. We give conditions insuring the existence of at least three viable solutions of a system of first-order differential equations. Many examples are presented to show that a large variety of sets can be solution-regions.

Interacting holographic dark energy model in Brans–Dicke cosmology and coincidence problem

We study the dynamics of interacting holographic dark energy model in Brans–Dicke cosmology for the future event horizon and the Hubble horizon cut-offs. We determine the system of first-order differential equations for the future event horizon and Hubble horizon cut-offs and obtain the corresponding fixed points, attractors, repellers and saddle points. Finally, we investigate the cosmic coincidence problem in this model for the future event horizon and Hubble horizon cut-offs and find that for both cut-offs and for a variety of Brans–Dicke parameters, the coincidence problem is almost resolved.

Transient Thermal Response of a Guarded-Hot-Plate Apparatus for Operation Over an Extended Temperature Range.

A mathematical model is presented for a new-generation guarded-hot-plate apparatus to measure the thermal conductivity of insulation materials. This apparatus will be used to provide standard reference materials for greater ranges of temperature and pressure than have been previously available. The apparatus requires precise control of 16 interacting heated components to achieve the steady temperature and one-dimensional heat-transfer conditions specified in standardized test methods. Achieving these criteria requires deriving gain settings for the 16 proportional-integral-derivative (PID) controllers, comprising potentially 48 parameters. Traditional tuning procedures based on trial-and-error operation of the actual apparatus impose unacceptably lengthy test times and expense. A primary objective of the present investigation is to describe and confirm the incremental control algorithm for this application and determine satisfactory gain settings using a mathematical model that simulates in seconds test runs that would require days to complete using the apparatus. The first of two steps to achieve precise temperature control is to create and validate a model that accounts for heating rates in the various components and interactions with their surroundings. The next step is to simulate dynamic performance and control with the model and determine settings for the PID controllers. A key criterion in deriving the model is to account for effects that significantly impact thermal conductivity measurements while maintaining a tractable model that meets the simulation time constraint. The mathematical model presented here demonstrates how an intricate apparatus can be represented by many interconnected aggregated-capacity masses to depict overall thermal response for control simulations. The major assemblies are the hot plate with four subcomponents, two cold plates with three subcomponents each, and two edge guards with three subcomponents each. Using symmetry about the hot plate, the number of components in the simulation model is reduced to 12 or 15, depending on the mode of operation for the apparatus. Configurations of the main components with embedded heating elements were carefully designed earlier using detailed finite-element analyses to give essentially isothermal surfaces and one-dimensional heat flow through test specimens. It is not tractable, or perhaps justified, to extend these prior analyses to simulate the controlled transient responses of the apparatus. The earlier design criterion does, however, support the aggregated-capacity simplification implemented in the present thermal model. The governing equations follow from dynamic energy balances on components with controlled heating elements and additional intermediate ("floating") components. Thermal bridges comprise conduction paths, with and without surface convection and radiation, between components and fixed-temperature "heat sinks." An implicit finite-difference numerical method was used to solve the resulting system of first-order differential equations. The mathematical model was initially validated using measurement data from test runs where a step change in heating rate was applied to single elements in turn, and component temperatures were recorded up to a nearly steady condition. Thermocouples and standard platinum resistance thermometers were used to measure temperatures, and thermopiles were used to measure temperature differences. Next, extensive simulations were conducted with the mathematical model to estimate suitable gain settings for the various controllers. The criteria were tight temperature control after reaching set points and acceptable times to achieve quasi-steady-state operation. Comparisons between measurements and predicted temperatures for heated components are presented. The results show that the model incorporating the above simplifying approximations is satisfactory for components comprising the hot-plate and cold-plate assemblies. For the edge guards, however, the conventional aggregated-capacity criteria are not as fully satisfied because of their configuration. Temperature variations in the edge guards, fortunately, have a lesser effect on the accuracy of the thermal conductivity measurements. Therefore, the thermal response model is deemed satisfactory for simulating PID feedback to investigate "closed-loop" control of the apparatus, thus meeting the primary objective.

Automatic Reformulation of ODEs to Systems of First-Order Equations

Most numerical ODE solvers require problems to be written as systems of first-order differential equations. This normally requires the user to rewrite higher-order differential equations as coupled first-order systems. Here, we introduce the treeVar class, written in object-oriented Matlab, which is capable of algorithmically reformulating higher-order ODEs to equivalent systems of first-order equations. This allows users to specify problems using a more natural syntax and saves them from having to manually derive the first-order reformulation. The technique works by using operator overloading to build up syntax trees of expressions as mathematical programs are evaluated. It then applies a set of rules to the resulting trees to obtain the first-order reformulation, which is returned as another program. This technique has connections with algorithmic/automatic differentiation. We present how treeVar has been incorporated in Chebfun, greatly improving the ODE capabilities of the system.

The Homotopy Analysis Method for a Fourth-Order Initial Value Problems

In this paper, we apply the homotopy analysis method for solving the fourth-order initial value problems by reformulating them as an equivalent system of first-order differential equations. The analytical results of the differential equations have been obtained in terms of convergent series with easily computable components. Several examples are given to illustrate the efficiency and implementation of the homotopy analysis method.

A validated time-stepping analytical model for 3D transient vapor chamber transport

Advances in the computational performance of electronic devices have created a clear need for improved methods of passive thermal management. This has led to renewed interest in the use of vapor chambers as heat spreaders in applications ranging from mobile devices to high-performance-computing and power electronics systems. While there has been significant effort to develop vapor chambers for these applications, their designs have largely relied on steady-state analyses and performance prediction. In many applications, however, the heat load is inherently transient in nature. Heat spreader design must consider transient performance in response to these use-case scenarios. While detailed numerical models of transient vapor chamber operation have been developed, a transient modeling approach with low computational cost is needed for parametric study and quick assessment of vapor chamber performance in system-level models.In the current work, a low-cost, transient vapor chamber model is developed targeting the geometries and operating conditions typical of thermal management applications. The model considers mass, momentum, and energy transport in the vapor chamber wall, wick, and vapor core as well as phase change at the wick-vapor interface. The governing equations are simplified to a system of first-order differential equations based on a scaling analysis and assuming a functional form for the temperature profile along the thickness dimension. The errors in the temperature and pressure fields due to these simplifying assumptions are estimated for a wide range of operating conditions. These estimates indicate low errors in the model predictions over the range considered. For two example cases, the model predictions are compared to a finite-volume-based numerical model. Any deviation from the numerical model prediction is on the same order as the errors estimated based on the simplifying assumptions. The time-stepping analytical model is demonstrated to have a computational cost reduction of three to four orders of magnitude compared to the finite-volume based model.

Non-linear problems with non-monotonic blow-up solutions: Non-local transformations, test problems, exact solutions, and numerical integration

The method of non-local transformations is proposed for numerical integration of non-linear Cauchy problems having non-monotonic blow-up solutions. In such problems there exists a singular point whose position is unknown in advance (for this reason, the standard numerical methods for solving blow-up problems can lead to significant errors). In addition, the non-monotonic behavior of the solution excludes a possibility of applying the hodograph transformation and some other methods that are used for numerical investigation of simpler problems having monotone blow-up solutions. In this paper, the method is described for numerical integration of similar problems for non-linear nth-order ordinary differential equations xt(n)=g(t,x,xt′,…,xt(n−1)), based on the introduction of a new non-local independent variable ξ, which is related to the original variables t and x by the equation ξt′=g(t,x,xt′,…,xt(n−1),ξ), and the subsequent transformation of the original problem to the Cauchy problem for the corresponding system of first-order differential equations. With a suitable choice of the regularizing function g, the proposed method leads to problems whose solutions are presented in parametric form and do not have blowing-up singular points; therefore the transformed problems allow the application of the standard fixed-step numerical methods. A number of test problems with non-monotonic blow-up is constructed for differential equations of the first, second, third, and fourth orders, which have exact solutions expressed in elementary functions. The numerical integration of test problems has shown high efficiency of methods based on non-local transformations of a special type (and the practical inapplicability of the arc-length transformation for numerical solving blow-up problems with ODEs of high order). In addition to problems for single ODEs, Cauchy problems for systems of coupled equations also are considered. Some recommendations are given on a suitable choice of a variable step in direct adaptive numerical methods in which the transformations of equations are not used. It is important to note that the method of non-local transformations can also be used to numerically integrate other problems with large solution gradients (including problems of boundary-layer type).

A numerical study of the axisymmetric rotational stagnation point flow impinging radially a permeable stretching/shrinking surface in a nanofluid

PurposeThe purpose of this study is to analyze numerically the steady axisymmetric rotational stagnation point flow impinging on a radially permeable stretching/shrinking sheet in a nanofluid.Design/methodology/approachSimilarity transformation is used to convert the system of partial differential equations into a system of ordinary (similarity) differential equations. This system is then reduced to a system of first-order differential equations and solved numerically using the bvp4c function in MATLAB software.FindingsDual solutions exist when the surface is stretched, as well as when the surface is shrunk. For these solutions, a stability analysis is carried out revealing that the first solution (upper branch) is stable and physically realizable, while the second solution (lower branch) is unstable and therefore not physically realizable.Originality/valueThe present results are original and new for the study of fluid flow and heat transfer over a stretching/shrinking surface, as they successfully extend the problem considered by Weidman (2016) to the case of nanofluids.

The effect of vertical throughflow on the boundary layer flow of a nanofluid past a stretching/shrinking sheet

PurposeThe purpose of this paper is to study the effects of vertical throughflow on the boundary layer flow and heat transfer of a nanofluid driven by a permeable stretching/shrinking surface.Design/methodology/approachSimilarity transformation is used to convert the system of boundary layer equations into a system of ordinary differential equations. The system of governing similarity equations is then reduced to a system of first-order differential equations and solved numerically using the bvp4c function in Matlab software. The generated numerical results are presented graphically and discussed based on some governing parameters.FindingsIt is found that dual solutions exist in both cases of stretching and shrinking sheet situations. Stability analysis is performed to determine which solution is stable and valid physically.Originality/valueDual solutions are found for positive and negative values of the moving parameter. A stability analysis has also been performed to show that the first (upper branch) solutions are stable and physically realizable, while the second (lower branch) solutions are not stable and, therefore, not physically possible.

An Analytic Technique for the Solutions of Nonlinear Oscillators with Damping Using the Abel Equation

Using the Chiellini condition for integrability we derive explicit solutions for a generalized system of Riccati equations x+αx2n+1x+x4n+3 = 0 by reduction to the first-order Abel equation assuming the parameter α ≥ 2 2(n+1). The technique, which was proposed by Harko et al, involves use of an auxiliary system of first-order differential equations sharing a common solution with the Abel equation. In the process analytical proofs of some of the conjectures made earlier on the basis of numerical investigations in [1] is provided.

Sequences of maps and convergence properties of first-order difference equations with variable coefficients

We consider sequences of continuous functions on that converge to a continuous monotone limit function. We prove a convergence theorem for the iterative map This result is used to solve an open problem in the field concerning the convergence properties of first-order rational difference equations that are linear in numerator and denominator with variable coefficients. We also establish convergence properties for a certain class of systems of first-order difference equations.