System Of First-order Differential Equations Research Articles

Research on Lateral Load Bearing Characteristics of Deepwater Drilling Conductor Suction Pile

The vast reserves of natural gas hydrates in offshore areas present significant challenges to development. Surface well construction technology is crucial for the extraction of deepwater natural gas hydrates. To ensure the safety of the subsea wellhead during the drilling process for deepwater natural gas hydrates, a novel conductor suction pile device has been designed, comprising a combination of suction piles and surface conductors. And research has been conducted to investigate the lateral stability characteristics of the conductor suction pile. Drawing upon the pile foundation load-bearing theory and the equilibrium of the differential element, a theoretical analysis model and corresponding governing equations of the conductor suction pile system are established. A solution for a multi-point boundary value problem by simplifying the conductor suction pile system into a two-end free beam is proposed. The governing equations are then converted into a first-order differential equation system, and the four-stage Lobatto IIIa collocation method program for the multi-point boundary value problem is developed and resolved using MATLAB 2023a. Furthermore, a case study of a well in the South China Sea elucidates the effects of wellhead load and seabed soil properties on the lateral load-bearing capacity of the conductor suction pile system, verifying the collocation method’s validity against the results from the finite difference method. After conducting a comparative analysis of the lateral load-bearing performance between conductor suction piles and traditional surface conductors, it is observed that conductor suction piles exhibit lower horizontal displacement and bending moments compared to surface conductors. Therefore, conductor suction piles demonstrate a substantial safety margin. The research findings provide a theoretical basis for the lateral stability of conductor suction piles during deepwater natural gas hydrate drilling. This offers a safe and efficient method for surface well construction in the extraction of natural gas hydrates.

New nonlinear models of dynamic longitudinal deformation of a viscoelastic rod

In this paper we study a general model of dynamic longitudinal deformation of a viscoelastic rod with a nonlinear dependence of the stress on the strain and on the strain rate. This model is given by a nonlinear differential equation with partial derivatives of the third order. A group classification of this model which depends on two arbitrary functions is carried out. Six basic models are obtained, for which their main Lie groups of transformations are found. A model that admits the widest Lie group of transformations is studied in detail. For it, formula for the production of new solutions is obtained and all invariant submodels are found. For all its invariant submodels, the invariant solutions defining these submodels are found either in explicit form, or their search reduces to solving the systems of first-order differential equations. For these systems, boundary value problems that have a physical meaning are studied. Conditions are obtained that ensure the existence and uniqueness of solutions to these boundary value problems. This allows one to correctly solve these boundary value problems numerically. Boundary value problems for some specific values of the parameters included in them are solved numerically. The research carried out is especially relevant in rocket engineering, aircraft engineering, shipbuilding and other fields.

The Bifurcations of Completely Integrable Holonomic Systems of First-Order Differential Equations

As an application of the Legendrian singularity theory, we classify the bifurcations of a holonomic first-order differential equation with a complete integral. The equations satisfy that the one-parameter integral diagrams are R+-simple and stable. Using this result, the parametric differential equation models in electrical power systems and engineering can be studied.

Jacobi Stability for T-System

In this paper will be considered a three-dimensional autonomous quadratic polynomial system of first-order differential equations with three real parameters, the so-called T-system. This system is symmetric relative to the Oz-axis and represents a special type of the generalized Lorenz system. The approach of this work will consist of the study of the nonlinear dynamics of this system through the Kosambi–Cartan–Chern (KCC) geometric theory. More exactly, we will focus on the associated system of second-order differential equations (SODE) from the point of view of Jacobi stability by determining the five invariants of the KCC theory. These invariants determine the internal geometrical characteristics of the system, and particularly, the deviation curvature tensor is decisive for Jacobi stability. Furthermore, we will look for necessary and sufficient conditions that the system parameters must satisfy in order to have Jacobi stability for every equilibrium point.

SOME QUALITATIVE PROPERTIES OF SOLUTIONS OF CERTAIN NONLINEAR THIRD-ORDER STOCHASTIC DIFFERENTIAL EQUATIONS WITH DELAY

This study considered certain nonlinear third-order stochastic differential equations with delay. The third-order equation is reduced to an equivalent system of first-order differential equations and used to construct the desired complete Lyapunov-Krasovski\v{\i} functional. Standard conditions guaranteeing stability when the forcing term is zero, boundedness of solutions when the forcing term is non-zero, and lastly the existence and uniqueness of solutions are derived. The obtained results indicated that the adopted technique is effective in studying the qualitative behaviour of solutions. The obtained results are not only new but extend the frontier of knowledge of the qualitative behaviour of solutions of nonlinear stochastic differential with delay. Finally, two special cases are given to illustrate the derived theoretical results.

The Analytical Scheme on the Inertial Drag for Buoyancy-driven Nanofluid Flow Under Convective Thermal Surface with the Soret Effect

Introduction: The two-dimensional mixed convection of nanofluid over a vertical expanding surface is analysed in the current discussion. The expanding surface is embedded in a permeable medium. Methods: In advance, Darcy Forchheimer inertial drag is considered along with the influence of Brownian and thermophoresis, which enriches the study. The novelty of the study is due to the mass concentration along with the role of volume concentration in the flow phenomena. The proposed model is designed in association with a characterizing parameter, which is attained by the use of appropriate similarity conversion. Further, the system of first-order differential equations is resolved by employing a shooting-based numerical method, in particular, the Runge-Kutta fourth-order technique. Results: The simulated results for the said parameters and their behaviour are deployed through graphs and in tabular form. Conclusion: The physical description of each parameter is deliberated briefly. Finally, the important outcomes of the proposed study reported a remarkable hike in the temperature profile that is observed for the enhanced thermophoresis and Brownian motion. Further, the shear rate also increases for the enhanced mixed convection parameter.

TRAVELING WAVES IN HEMODYNAMIC EQUATIONS WITH THE KOITER SHELL

We consider a one-dimensional model of hemodynamics for a tube defined by a Koiter shell, and investigate traveling wave solutions. The system of partial differential equations is reduced to a fourth-order ordinary differential equation for such solutions. We find a single equilibrium point and determine the condition for the changing type of the equilibria for the corresponding system of first-order differential equations. The conditions for changing the type of the equilibrium point are formulated. Then we introduce the classification of blood flow regimes relative to the type of the equilibrium point. Numerical experiments are carried out to confirm and analyze the obtained results, various regimes of blood flow are considered.

An analytical method for predicting the soil-structure interaction of axially loaded piles in sands incorporating the two-surface plasticity model

The paper proposed a novel and practical analytical method for predicting the soil-structure interaction in sands. To well capture the stress softening and soil dilatancy behaviors during loading, a developed two-surface plasticity model for interface shearing is adopted to determine the 2-D stress–strain state of the soil-structure interface and the shear band, while the 3-D elastoplastic behavior of soil around the shear band is determined by another developed two-surface plasticity model for triaxial compression. Following a standardized solving procedure, the rigorous derivation for the stress–strain relationship in plastic zone is formulated as a system of first-order differential equations, which are solved as an initial value problem. Combined with the finite difference method (FDM), the load–displacement response of the axially loaded pile and the stress–strain state around the pile are presented. To verify the proposed analytical approach, a FEM simulation is performed using a user-defined subroutine. The soil with different initial void ratios ranging from very dense to loose is also modeled to demonstrate the effect of the initial void ratio on soil-structure response. The proposed analytical approach is also validated against centrifuge tests and field tests, manifesting the validity and capability of the proposed method.

Coherent hybrid block method for approximating fourth-order ordinary differential equations

Conventionally, the most used method of solving fourth-order initial value problems of ordinary differential is to first reduce to a system of first-order differential equations. This approach affects the effectiveness and convergence of the numerical method due to the transformation. This paper comprises the derivation, analysis, and implementation of a new hybrid block method for direct solution of fourth-order equations. The method is derived by collocation and interpolation of an assumed basis function. The basic properties of the block method, including zero stability, error constants, consistency, order, and convergence, were analyzed. From the analysis, the block method derived was found to be zero-stable, consistent, and convergent. Errors were computed for the proposed method, and they were proven to produce approximations that agree with exact solutions and as such this shows improvement with those of existing works.

A Novel Inverse Simulation Method of Helicopter Maneuvering Flight

In this article, a degeneration method of inverse simulation is proposed for a helicopter system flying pull-up and slalom maneuvers. This inverse simulation method has been successfully applied, and here, the results are compared with the flight test data and the reference data, which indicate that this method has fidelity and is effective. It is different from conventional inverse simulation methods (i.e., the differentiation method and the integration method), and can improve numerical stability during inverse simulation, because the core principle is to solve the output vector in each simulation time step, by degenerating the system of first-order differential equations into nonlinear equations.

Ellipticheskie zadachi i integral'nye uravneniya v prostranstvakh razlichnoy gladkosti po peremennym

We consider a model elliptic pseudodifferential equation and the simplest boundary value problems in a quadrant in a Sobolev–Slobodetsky space of different orders of smoothness in different variables. In the case of a special representation of the symbol, we describe a general solution of the equation and consider the simplest boundary value problem with the Dirichlet and Neumann conditions on the sides of the quadrant. This boundary value problem is reduced to a system of integral equations, which, under additional assumptions about the structure of the symbol, can also be reduced to a system of first-order difference equations

Mathematical and Statistical Aspects of Estimating Small Oscillations Parameters in a Conservative Mechanical System Using Inaccurate Observations

This paper selects a set of reference points in the form of an arithmetic progression for planning an experiment to evaluate the parameters of systems of differential equations. This choice makes it possible to construct estimates of the parameters of a system of first-order differential equations based on the reversibility of the observation matrix, as well as estimates of the parameters of a system of second-order differential equations describing vibrations in a mechanical system by switching to a system of first-order differential equations. In turn, the reversibility of the observation matrix used in parameter estimation is established using the Vandermonde formula. A volumetric computational experiment has been carried out showing how, with an increase in the number of observations in the vicinity of reference points and with a decrease in the step of arithmetic progression, the accuracy of estimates of the parameters of the analyzed system increases. Among the estimated parameters, the most important are the oscillation frequencies of a conservative mechanical system, which establish its proximity to resonance, and therefore, determine the stability and reliability of the system.

Hygro-Elastic Coupling in a 3D Exact Shell Model for Bending Analysis of Layered Composite Structures

In this work, a 3D fully coupled hygro-elastic model is proposed. The moisture content profile is a primary variable of the model’s displacements. This generic fully coupled 3D exact shell model allows the investigations into the consequences arising from moisture content and elastic fields in terms of stresses and deformations on different plate and shell configurations embedded in composite and laminated layers. Cylinders, plates, cylindrical and spherical shells are analyzed in the orthogonal mixed curvilinear reference system. The 3D equilibrium equations and the 3D Fick diffusion equation for spherical shells are fully coupled in a dedicated system. The main advantage of the orthogonal mixed curvilinear coordinates is related to the degeneration of the equations for spherical shells to simpler geometries thanks to basic considerations of the radii of curvature. The exponential matrix method is used to solve this fully coupled model based on partial differential equations in the thickness direction. The closed-form solution is related to simply supported sides and harmonic forms for displacements and the moisture content. The moisture content amplitudes are directly applied at the top and bottom outer faces through steady-state hypotheses. The final system is based on a set of coupled homogeneous second-order differential equations. A first-order differential equation system is obtained by redoubling the number of variables. The moisture field implications are evaluated for the static analysis of the plates and shells in terms of displacement and stress components. After preliminary validations, new benchmarks are proposed for several thickness ratios, geometrical and material data, lamination sequences and moisture values imposed at the external surfaces. In the proposed results, there is clearly accordance between the uncoupled hygro-elastic model (where the 3D Fick diffusion law is separately solved) and this new fully coupled hygro-elastic model: the differences between the investigated variables (displacements, moisture contents, stresses and strains) are always less than 0.3%. The main advantages of the 3D coupled hygro-elastic model are a more compact mathematical formulation and lower computational costs. Both effects connected with the thickness layer and the embedded materials are included in the conducted hygro-elastic analyses.

Research on stability of deepwater drilling riser system in freestanding mode

If a floating drilling platform encounters sudden bad sea conditions during deepwater operation, it is necessary to disconnect the drilling riser and the floating platform in an emergency. A buoyancy can structure is required to be installed at the top of the riser to ensure that the disconnected drilling riser can stand alone without instability accidents. According to the main structural characteristics and environmental conditions of the freestanding riser and the buoyancy can at the top of the riser, the compressive buckling stability of deepwater drilling riser in freestanding mode under the condition of insufficient top tension is studied in this paper, and the main problem is simplified to the buckling problem of an equal section beam under uniformly distributed axial load and top concentrated force. Firstly, the elastic stability control equation of the riser is established according to the mechanical balance, geometry and constitutive relationship of the riser's micro-element, and the equation as well as the corresponding top and bottom boundary conditions are made dimensionless. Secondly, the elastic stability equation of the micro-bending riser is linearized, the power series solution is obtained according to the perturbation theory, and the influence of different parameters on linear buckling is analyzed. Finally, we convert the elastic stability equation with large deflection into a two-point boundary value problem of the first-order differential equation system under the boundary conditions and two other given conditions. The shooting method program of the two-point boundary value problem including the changes of the top concentrated load and the distributed axial load parameters is developed by using MATLAB, and the calculation analysis is carried out, which reveals the rules of the buckling load and buckling shape after the buckling. The research results of this paper can provide scientific basis and technical guidance for the structure and size design of drilling riser and buoyancy can.

A Charge Conserving Exponential Predictor Corrector FEMPIC Formulation for Relativistic Particle Simulations

The state of art of charge conserving electromagnetic finite element particle-in-cell (EM-FEMPIC) has grown by leaps and bounds in the past few years. These advances have primarily been achieved for leap-frog time stepping schemes for Maxwell solvers, in large part, due to the method strictly following the proper space for representing fields, charges, and measuring currents. Unfortunately, leap-frog-based solvers (and their other incarnations) are only conditionally stable. Recent advances have made it possible to construct EM-FEMPIC methods built around unconditionally stable time-stepping methods while conserving charge. Together with the use of a quasi-Helmholtz decomposition, these methods were both unconditionally stable and satisfied Gauss’ laws to machine precision. However, this architecture was developed for systems with explicit particle integrators where fields and velocities were off by a time step. While completely self-consistent methods exist in the literature, they follow the classic rubric: collect a system of first-order differential equations (Maxwell and Newton equations) and use an integrator to solve the combined system. These methods suffer from the same side effect as earlier—they are conditionally stable. Here, we propose a different approach; we pair an unconditionally stable Maxwell solver to an exponential predictor corrector (PC) method for Newton’s equations. As we will show via numerical experiments, the proposed method conserves energy within a particle-in-cell (PIC) scheme, has an unconditionally stable electromagnetic (EM) solve, solves Newton’s equations to much higher accuracy than a traditional Boris solver, and conserves charge to machine precision. We further demonstrate benefits compared with other polynomial methods to solve Newton’s equations, like the well-known Boris push.

НОВОЕ ВЕРОЯТНОСТНОЕ РАСПРЕДЕЛЕНИЕ С НЕЧЁТКИМИ ПАРАМЕТРАМИ

The aim of the study is to obtain a new probabilistic distribution of the random moment T of complete transition of patients from cohort S (susceptible to infection) to cohort R (recovered and cured). Methodology. The distribution is constructed under the assumption that in the SIR Kermack – McKendrick model, the nonlinear system of firstorder differential equations has an approximate solution in the form of logistic functions based on the profile of the normal probability distribution F. The obtained distribution AN(t) has an analytical form and is threeparameter, and all these parameters of the distribution can be represented in a fuzzy form (fuzzified). Thus, T is a fuzzy random variable (FRV) and AN(t) takes the form of a fuzzy function of general form. Results. The form of the distribution function AN(t) and the density distribution an(t) with fuzzy parameters of transition frequencies β and γ from cohort S to cohort I (infected) and from cohort I to cohort R, respectively, are obtained. Conclusion. Similar distributions, given on the time axis t, can be obtained for all other cohorts.

Fixed point theorems in generalized locally convex spaces and applications

In this paper, we present the concept of generalized locally convex spaces, by introducing the socalled family of vector-valued seminorms. Some extensions of classical fixed point theorems in Hausdorff complete generalized locally convex spaces are given. These results are formulated in terms of continuity and L-contractions. As applications, we study the existence and uniqueness of solutions for systems of first order differential equations with impulsive.

The phase-space view of non-local gravity cosmology

We consider non-local Integral Kernel Theories of Gravity in a homogeneous and isotropic universe background as a possible scenario to drive the cosmic history. In particular, we investigate the cosmological properties of a gravitational action containing the inverse d'Alembert operator of the Ricci scalar proposed to improve Einstein's gravity at both high and low-energy regimes. In particular, the dynamics of a physically motivated non-local exponential coupling is analyzed in detail by recasting the cosmological equations as an autonomous system of first-order differential equations with dimensionless variables. Consequently, we study the phase-space domain and its critical points, investigating their stability and main properties. In particular, saddle points and late-time cosmological attractors are discussed in terms of the free parameters of the model. Finally, we discuss the main physical consequences of our approach in view of dark energy behavior and the ΛCDM model.

Stability, cost-effectiveness, and global sensitivity analysis of COVID-19 model incorporating non-pharmaceutical interventions and indirect transmission

Covid-19 is an ongoing pandemic caused by SARS-CoV-2. Some interventions are implemented to control the spread of the disease. In Indonesia, there is a campaign related to non-pharmaceutical approach called 3M. This campaign is carried out so that people use masks, wash their hands, and keep their distance. In this paper, we propose a mathematical model considering non-pharmaceutical interventions and indirect transmission. The non-pharmaceutical interventions studied are the implementation of mask-wearing, handwashing, and social distancing. The model is presented as a system of first-order differential equations. The basic reproduction number is determined. The system has two equilibrium points, namely the disease-free equilibrium point and the endemic equilibrium point. The local stability condition of the disease-free equilibrium point is proved using the Lienard-Chipart criterion. Center manifold theory is used to prove the local stability condition of the endemic equilibrium point. We also study the optimal control strategy related to mask-wearing, handwashing, and social distancing. Furthermore, cost-effectiveness analysis of intervention strategies is also conducted by studying the average cost-effectiveness ratio of each intervention strategy. Our results show that the most effective strategy to control covid-19 spread is the combination of mask-wearing, handwashing, and social distancing. Moreover, the most cost-effective strategy is mask-wearing intervention. Global sensitivity analysis is performed by studying the partial rank correlation coefficient. The results show that mask-wearing intervention is the most influential intervention on basic reproduction number compared to social distancing and handwashing

A solution for the quasi-one-dimensional linearised Euler equations with heat transfer

The unsteady response of nozzles with steady heat transfer forced by acoustic and/or entropy waves is modelled. The approach is based on the quasi-one-dimensional linearised Euler equations. The equations are cast in terms of three variables, namely the dimensionless mass, stagnation temperature and entropy fluctuations, which are invariants of the system at zero frequency and with no heat transfer. The resulting first-order system of differential equations is then solved using the Magnus expansion method, where the perturbation parameters are the normalised frequency and the volumetric heat transfer. In this work, a measure of the flow non-isentropicity (in this case the steady heat transfer) is used for the first time as an expansion parameter. The solution method was applied to a converging–diverging nozzle with constant heat transfer for both subcritical and supercritical flow cases, showing good agreement with numerical predictions. It was observed that the acoustic and entropy transfer functions of the nozzle strongly depend on the frequency and heat transfer.