System Of First-order Differential Equations Research Articles

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.

Undrained Elastoplastic Solution for Cylindrical Cavity Expansion in Structured Cam Clay Soil Considering the Destructuration Effects

Soil structure has significant influences on the mechanical behaviors of natural soils, although it is rarely considered in previous cavity expansion analyses. This paper presents an undrained elastoplastic solution for cylindrical cavity expansion in structured soils, considering the destructuration effects. Firstly, a structural ratio was defined to denote the degree of the initial structure, and the Structured Cam Clay (SCC) model was employed to describe the subsequent stress-induced destructuration, including the structure degradation and crushing. Secondly, combined with the large strain theory, the considered problem was formulated as a system of first-order differential equations, which can be solved in a simplified procedure with the introduced auxiliary variable. Finally, the significance and efficiency of the present solution was demonstrated by comparing with the previous solutions, and parametric studies were also conducted to investigate the effects of soil structure and destructuration on the cavity expansion process. The results show that the soil structure has pronounced effects on the mechanical behavior of structured soils around the cavity. For structured soils, a cavity pressure that is larger than the corresponding reconstituted soils when the cavity expands to the same radius is required, and the effective stresses first increase to a peak value before decreasing rapidly with soil structure degradation and crushing. The same final critical state is reached for soils with different degrees of the initial structure, which indicates that the soil structure is completely destroyed during the cavity expansion. With the increase of the destructuring index, the soil structure was destroyed more rapidly, and the stress release during the plastic deformation became more significant. Moreover, the present solution was applied in the jacking of a casing during the sand compact pile installation and in situ self-boring pressuremeter (SBPM) tests, which indicates that the present solution provides an effective theoretical tool for predicting the behavior of natural structured soils around the cavity.

Development of an analytical model for copper heap leaching from secondary sulfides in chloride media in an industrial environment

In multivariate analysis, a predictive model is a mathematical/statistical model that relates a set of independent variables to dependent or response variable(s). This work presents a descriptive model that explains copper recovery from secondary sulfide minerals (chalcocite) taking into account the effects of time, heap height, superficial velocity of leaching flow, chloride concentration, particle size, porosity, and effective diffusivity of the solute within particle pores. Copper recovery is then modelled by a system of first-order differential equations. The results indicated that the heap height and superficial velocity of leaching flow are the most critical independent variables while the others are less influential under operational conditions applied. In the present study representative adjustment parameters are obtained, so that the model could be used to explore copper recovery in chloride media as a part of the extended value chain of the copper sulfides processing.

On thermoelastic deformation of a transversally isotropic medium in 3D construction printing

A three-dimensional dynamic model of a thermoelastic transversely isotropic medium is used to describe the thermoelastic deformation of materials with anisotropy of elastic properties with a selected direction of anisotropy. Such materials are layered and composite materials used in construction, mechanical engineering, aircraft and shipbuilding, soils in permafrost conditions, glaciers, as well as rocks (basalt, sandstone, marble, limestone, shale, and others). The study of this model, in particular, is relevant in connection with the use of 3D printers in construction. This is due to the fact that it is necessary to select the heating mode of the 3D printer head, in which cracks will not form during the cooling of the polystyrene concrete layers.We study this model using the group analysis methods, which is one of the most powerful and effective tools for obtaining exact solutions. The group stratification of the system of second-order differential equations defining this model is carried out. A system of first-order differential equations is obtained, which is equivalent to the equations of the original model. The solution describing a traveling wave for this system is obtained, that depends on arbitrary elements: parameters and function. For the specific sets of these elements, we study a deformation of a sphere and cube located inside a thermoelastic transversely isotropic medium with increasing time is found. The corresponding graphs are given.

Reformulation of elliptic equations for heat transfer and diffusion in solids with space-time algebra

In this paper, we demonstrate the application of non-commutative space-time algebra of sedeons to generalize the system of equations describing heat transfer and impurity diffusion in solids at finite velocity. It is shown that by analogy with electrodynamics, these transfer processes can be described using a compact second-order sedeonic equation for generalized scalar and vector potentials. On the one hand, this equation is reduced to the system of first-order differential equations for vortex-less mass and heat flows, and on the other hand, it can be transformed to the second-order elliptical equations for the profiles of temperature and impurity concentration. The comparison of peculiarities in transfer within the frames of parabolic and elliptic equations is discussed.

A system of first-order differential equations to calculate the Jost functions

Theoretical or experimental studies of the interactions between quantum particles are very important in physical sciences because from them it is possible to know the properties of the systems, therefore, the Jost functions are very important in Quantum Scattering Theory because they allow to carry out unified studies on bound states, scattered states, virtual states and resonance states. In this work a new method to calculate the Jost functions is presented, this method is based on a system of first order differential equations and has the advantage that the solutions of the problem can be calculated by solving the system of differential equations from the origin or from infinity, therefore, it facilitates calculations when systems are modeled with singular potentials at the origin. We tested this method by modeling the collision of an electron with a hydrogen atom in its ground state at low energies (10 eV ) and found very good results.

Dynamic modeling of bone remodeling, osteolytic metastasis and PK/PD therapy: introducing variable order derivatives as a simplification technique.

Bone is constantly being renewed: in the adult skeleton, bone resorption and formation are in a tightly coupled balance, allowing for a constant bone density to be maintained. Yet this micro-environment provides the necessary conditions for the growth and proliferation of tumor cells, and thus bone is a common site for the development of metastases, mainly from primary breast and prostate cancer. Mathematical and computational models with differential equations can replicate this bone remodeling process. These models have been extended to include the effects of disruptive tumor pathologies in the bone dynamics, as metastases contribute to the decoupling between bone resorption and formation and to the self-perpetuating tumor growth cycle. Such models may also contemplate the counteraction effects of currently used therapies, and, in the case of treatments with drugs, their pharmocokinetics and pharmacodynamics. We present a thorough overview of biochemical models for bone remodeling, in the presence of a tumour together with anti-cancer and anti-resorptive therapy, formulated as systems of first-order differential equations, or simplified using variable order derivatives. The latter models, of which some are new to this paper, result in equations with fewer parameters, and allow accounting for anomalous diffusion processes. In this way, more compact and parsimonious models, that promptly highlight tumorous bone interactions, are achieved, providing an effective framework to counteract the loss of bone integrity on the affected areas.

Numerical solution of the system of Marchenko integral equations

In this paper, inverse scattering problems for a system of differential equations of the first order are considered. The Marchenko approach is used to solve the inverse scattering problem. The system of Marchenko integral equations is reduced to a linear system of algebraic equations such that the solution of the resulting system yields to the unknown coefficients of the system of first-order differential equations. Illustrative examples are provided to demonstrate the preciseness and effectiveness of the proposed technique. The results are compared with the exact solution by using computer simulations.

Stability of sign patterns from a system of second order ODEs

The stability and inertia of sign pattern matrices with entries in {+,−,0} associated with dynamical systems of second-order ordinary differential equations x¨=Ax˙+Bx are studied, where A and B are real matrices of order n. An equivalent system of first-order differential equations has coefficient matrix C=[ABIO] of order 2n, and eigenvalue properties are considered for sign patterns C=[ABDO] of order 2n, where A,B are the sign patterns of A,B respectively, and D is a positive diagonal sign pattern. For given sign patterns A and B where one of them is a negative diagonal sign pattern, results are determined concerning the potential stability and sign stability of C, as well as the refined inertia of C. Applications include the stability of such dynamical systems in which only the signs rather than the magnitudes of entries of the matrices A and B are known.

A General Scheme for Solving Systems of Linear First-Order Differential Equations Based on the Differential Transform Method

In this study, we develop the differential transform method in a new scheme to solve systems of first-order differential equations. The differential transform method is a procedure to obtain the coefficients of the Taylor series of the solution of differential and integral equations. So, one can obtain the Taylor series of the solution of an arbitrary order, and hence, the solution of the given equation can be obtained with required accuracy. Here, we first give some basic definitions and properties of the differential transform method, and then, we prove some theorems for solving the linear systems of first order. Then, these theorems of our system are converted to a system of linear algebraic equations whose unknowns are the coefficients of the Taylor series of the solution. Finally, we give some examples to show the accuracy and efficiency of the presented method.

Reliability modeling and performance evaluation of solar photovoltaic system using Gumbel–Hougaard family copula

PurposeSolar photovoltaic (PV) is commonly used as a renewable energy source to provide electrical power to customers. This research establishes a method for testing the performance reliability of large grid-connected PV power systems. Solar PV can turn unrestricted amounts of sunlight into energy without releasing carbon dioxide or other contaminants into the atmosphere. Because of these advantages, large-scale solar PV generation has been increasingly incorporated into power grids to meet energy demand. The capability of the installation and the position of the PV are the most important considerations for a utility company when installing solar PV generation in their system. Because of the unpredictability of sunlight, the amount of solar penetration in a device is generally restricted by reliability constraints. PV power systems are made up of five PV modules, with three of them needing to be operational at the same time. In other words, three out of five. Then there is a charge controller and a battery bank with three batteries, two of which must be consecutively be in operation. i.e. two out of three. Inverter and two distributors, all of which were involved at the same time. i.e. two out of two. In order to evaluate real-world grid-connected PV networks, state enumeration is used. To measure the reliability of PV systems, a collection of reliability indices has been created. Furthermore, detailed sensitivity tests are carried out to examine the effect of various factors on the efficiency of PV power systems. Every module's test results on a realistic 10-kW PV system. To see how the model works in practice, many scenarios are considered. Tables and graphs are used to show the findings.Design/methodology/approachThe system of first-order differential equations is formulated and solved using Laplace transforms using regenerative point techniques. Several scenarios were examined to determine the impact of the model under consideration. The calculations were done with Maple 13 software.FindingsThe authors get availability, reliability, mean time to failure (MTTF), MTTF sensitivity and gain feature in this research. To measure the reliability of PV systems, a collection of reliability indices has been created. Furthermore, detailed sensitivity tests are carried out to examine the effect of various factors on the efficiency of PV power systems.Originality/valueThis is the authors' original copy of the paper. Because of the importance of the study, the references are well-cited. Nothing from any previously published articles or textbooks has been withdrawn.

An elastoplastic solution for cylindrical cavity expansion under constant water content conditions in anisotropic unsaturated soils

In this paper, an elastoplastic solution is developed for cylindrical cavity expansion in anisotropic unsaturated soils under constant water content conditions. The most prominent features of the solution can be attributed to 1) the stress–strain behaviour of unsaturated soil is subtly coupled with the saturation-suction behaviour; 2) both the mechanical and hydraulic behaviours are modelled as elastoplastic processes; and 3) an anisotropic critical state soil model for saturated soil is extended to an unsaturated situation to consider the effects of the initial stress anisotropy and stress-induced anisotropy of the natural unsaturated soils. The problem is formulated as a system of first-order differential equations and solved as an initial value problem. A numerical code that can judge the hydraulic yield condition of soil particles during the cavity expansion process is developed based on the Runge-Kutta algorithm to solve the governing equations. Parametric analyses, which cover a wide range of suctions, overconsolidation ratios, degrees of initial stress anisotropy, are conducted to investigate the effects of these important factors on the expansion responses in unsaturated soils. The proposed solution could potentially be applied to interpret pressuremeter tests and model pile installation effects in anisotropic unsaturated soils.

Regularized classical optimality conditions in iterative form for convex optimization problems for distributed Volterra-type systems

We consider the regularization of the classical optimality conditions (COCs) — the Lagrange principle and the Pontryagin maximum principle — in a convex optimal control problem with functional constraints of equality and inequality type. The system to be controlled is given by a general linear functional-operator equation of the second kind in the space $L^m_2$, the main operator of the right-hand side of the equation is assumed to be quasinilpotent. The objective functional of the problem is strongly convex. Obtaining regularized COCs in iterative form is based on the use of the iterative dual regularization method. The main purpose of the regularized Lagrange principle and the Pontryagin maximum principle obtained in the work in iterative form is stable generation of minimizing approximate solutions in the sense of J. Warga. Regularized COCs in iterative form are formulated as existence theorems in the original problem of minimizing approximate solutions. They “overcome” the ill-posedness properties of the COCs and are regularizing algorithms for solving optimization problems. As an illustrative example, we consider an optimal control problem associated with a hyperbolic system of first-order differential equations.

Mean Square Geodesic Deviation in the Zeldovich Problem on Light Propagation in a Universe with Inhomogeneities

In 1964, Ya.B. Zeldovich formulated the problem of light propagation in the Universe under the influence of inhomogeneities. It is reduced to describing the divergence of two close geodesics in a Riemannian space and is described by the geodesic deviation equation (Jacobi equation) with the curvature along the geodesic line varying randomly. Assuming the curvature to be constant on segments of small but finite length, the problem is reduced to studying the product of random matrices and makes it possible to apply the appropriate well-developed mathematical theory, which, however, did not allow calculating the mean-square growth rate of the geodesic deviation. In our paper, we propose a way to solve this problem by introducing a bilinear quantity, one component of which coincides with the square of the Jacobi field. The system of first-order differential equations for the bilinear quantity is explicitly written out, and the solution, same as the growth rate, is again expressed through the product of matrices. Such a technique can be used in the study of a wide range of problems and is naturally generalized to higher-order moments.