Single First-order Differential Equation Research Articles

The Schwarzian Approach in Sturm–Liouville Problems

A novel method for finding the eigenvalues of a Sturm–Liouville problem is developed. Following the minimalist approach, the problem is transformed to a single first-order differential equation with appropriate boundary conditions. Although the resulting equation is nonlinear, its form allows us to find the general solution by adding a second part to a particular solution. This splitting of the general solution into two parts involves the Schwarzian derivative: hence, the name of the approach. The eigenvalues that correspond to acceptable solutions can be found by requiring the second part to correct the asymptotically diverging behavior of the particular solution. The method can be applied to many different areas of physics, such as the Schrödinger equation in quantum mechanics and stability problems in fluid dynamics. Examples are presented.

Mathematical model of insulin kinetics accounting for the amino acids effect during a mixed meal tolerance test.

Amino acids (AAs) are well known to be involved in the regulation of glucose metabolism and, in particular, of insulin secretion. However, the effects of different AAs on insulin release and kinetics have not been completely elucidated. The aim of this study was to propose a mathematical model that includes the effect of AAs on insulin kinetics during a mixed meal tolerance test. To this aim, five different models were proposed and compared. Validation was performed using average data, derived from the scientific literature, regarding subjects with normal glucose tolerance (CNT) and with type 2 diabetes (T2D). From the average data of the CNT and T2D people, data for two virtual populations (100 for each group) were generated for further model validation. Among the five proposed models, a simple model including one first-order differential equation showed the best results in terms of model performance (best compromise between model structure parsimony, estimated parameters plausibility, and data fit accuracy). With regard to the contribution of AAs to insulin appearance/disappearance (kAA model parameter), model analysis of the average data from the literature yielded 0.0247 (confidence interval, CI: 0.0168 – 0.0325) and -0.0048 (CI: -0.0281 – 0.0185) μU·ml-1/(μmol·l-1·min), for CNT and T2D, respectively. This suggests a positive effect of AAs on insulin secretion in CNT, and negligible effect in T2D. In conclusion, a simple model, including single first-order differential equation, may help to describe the possible AAs effects on insulin kinetics during a physiological metabolic test, and provide parameters that can be assessed in the single individuals.

A General Theory of Injection Locking and Pulling in Electrical Oscillators—Part I: Time-Synchronous Modeling and Injection Waveform Design

A general model of electrical oscillators under the influence of a periodic injection is presented. Stemming solely from the autonomy and periodic time variance inherent in all oscillators, the model’s underlying approach makes no assumptions about the topology of the oscillator or the shape of the injection waveform. A single first-order differential equation is shown to be capable of predicting a number of important properties, including the lock range, the relative phase of an injection-locked oscillator, and mode stability. The framework also reveals how the injection waveform can be designed to optimize the lock range. A diverse collection of simulations and measurements, performed on various types of oscillators, serve to verify the proposed theory.

On the Averaging in Strongly Damped Systems: The General Approach and its Application to Asymptotic Analysis of the Sommerfeld Effect

Passage through resonance of an unbalanced rotor mounted on a strongly damped oscillating system that is excited by an asynchronous motor of limited power is investigated using an averaging procedure for a partially strongly damped system. The approach is closely related to a singular perturbation technique. It is demonstrated that the system's dynamics can be reduced on the slow manifold to a single first-order differential equation predicting both the stationary and transient solutions of the original system. Furthermore, the technique provides insight into the system's dynamics, enabling an outline of two simple methods to avoid capture into resonance. The first method is based on an active compensation of the small averaged vibrational torque, and the second method is a passive mechanical device reducing vibration amplitudes. The approximate results obtained were compared with the direct numerical simulations of the full system and demonstrate very good accuracy everywhere except in the vicinity of the bifurcation point. In this vicinity, the asymptotically long time interval is not sufficient for predicting capture into resonance.

Core-envelope and regular models in Einstein-Maxwell fields

New classes of exact solutions which could serve as sources for the Reissner-Nordstrom metric representing the exterior gravitational field of an isolated charged sphere are derived. Firstly, we sacrifice the requirement that the stellar centre is free of a singularity and then obtain a core-envelope model. The charged fluid envelope is matched suitably to the neutral core and the vacuum exterior solution. Next we investigate models of charged stars that are regular at the stellar centre. The Einstein-Maxwell system of partial differential equations is reduced to the study of a single first-order differential equation in which two of the matter or geometrical variables must be specified at the outset. In each case mentioned above, new exact models are found by choosing functional forms for the electric field intensity and one of the gravitational potentials. A Riccati equation is then solved to obtain the remaining potential. The charged spherical shell model as well as the non-singular models are shown to display necessary qualitative features that are demanded for physical acceptability. It is shown that the regular model has a vanishing pressure-free hypersurface. The density and pressure profiles are positive and monotonically decreasing outwards from the centre of the sphere for a chosen set of parameters. The weak strong and dominant energy conditions are also satisfied. A drawback of the model is that the causality criterion is not satisfied within the fluid boundary.

Ion acoustic traveling waves

AbstractModels for traveling waves in multi-fluid plasmas give essential insight into fully nonlinear wave structures in plasmas, not readily available from either numerical simulations or from weakly nonlinear wave theories. We illustrate these ideas using one of the simplest models of an electron–proton multi-fluid plasma for the case where there is no magnetic field or a constant normal magnetic field present. We show that the traveling waves can be reduced to a single first-order differential equation governing the dynamics. We also show that the equations admit a multi-symplectic Hamiltonian formulation in which both the space and time variables can act as the evolution variable. An integral equation useful for calculating adiabatic, electrostatic solitary wave signatures for multi-fluid plasmas with arbitrary mass ratios is presented. The integral equation arises naturally from a fluid dynamics approach for a two fluid plasma, with a given mass ratio of the two species (e.g. the plasma could be an electron–proton or an electron–positron plasma). Besides its intrinsic interest, the integral equation solution provides a useful analytical test for numerical codes that include a proton–electron mass ratio as a fundamental constant, such as for particle in cell (PIC) codes. The integral equation is used to delineate the physical characteristics of ion acoustic traveling waves consisting of hot electron and cold proton fluids.

First-order partial differential equations in classical dynamics

Carathèodory’s classic work on the calculus of variations explores in depth the connection between ordinary differential equations and first-order partial differential equations. The n second-order ordinary differential equations of a classical dynamical system reduce to a single first-order differential equation in 2n independent variables. The general solution of first-order partial differential equations touches on many concepts central to graduate-level courses in analytical dynamics including the Hamiltonian, Lagrange and Poisson brackets, and the Hamilton–Jacobi equation. For all but the simplest dynamical systems the solution requires one or more of these techniques. Three elementary dynamical problems (uniform acceleration, harmonic motion, and cyclotron motion) can be solved directly from the appropriate first-order partial differential equation without the use of advanced methods. The process offers an unusual perspective on classical dynamics, which is readily accessible to intermediate students who are not yet fully conversant with advanced approaches.

ELASTOPLASTIC ANALYSIS OF CYLINDRICAL CAVITY PROBLEMS IN GEOMATERIALS

A large-strain elastoplastic analysis is presented for a cylindrical cavity embedded in an infinite medium under uniform radial pressure. The investigation employs invariant, non-associated deformation-type theories for Mohr-Coulomb (M-C) and Drucker-Prager (D-P) solids, accounting for arbitrary hardening, with the equivalent stress as the independent variable. The M-C model results in a single first-order differential equation, whereas for the D-P solid an algebraic constraint supplements the governing differential equation. Material parameters and response characteristics were determined by calibrating the models with data from triaxial compression tests on Castlegate sandstone and on Jurassic shale. A comparison is presented between predictions obtained from the two models and experimental data from hollow cylinder tests under external loading. A sensitivity of the results to material parameters, like friction and dilation angles, is provided for the case of a cavity subjected to internal pressure in terms of limit pressure predictions. In all cases it has been found that the results of the D-P inner cone model are in close agreement with those obtained from the M-C model.

Spinup dynamics of axial dual-spin spacecraft

We consider spinup dynamics of axial dual-spin spacecraft composed of two rigid bodies: an asymmetric platform and an axisymmetric rotor parallel to a principal axis of the platform. The system is free of external torques, and spinup of the rotor is effected by a small constant internal axial torque. The dynamics are described by four first-order differential equations. Conservation of angular momentum and the method of averaging are used to reduce the problem to a single first-order differential equation which is studied numerically. This reduction has a geometric counterpart that we use to simplify the investigation of spinup dynamics. In particular, a resonance condition due to platform asymmetry and associated with an instantaneous separatrix crossing is clearly identified using our approach.

Perturbation Techniques for Models of Bursting Electrical Activity in Pancreatic $\beta $-Cells

Pancreatic $\beta $-cells exhibit periodic bursting electrical activity consisting of active and silent phases. Experimentally, the ratio, $\rho _f $, of the active phase duration to the overall period is correlated to the insulin response of these cells to glucose concentration. Several different mathematical models of the $\beta $-cell have been developed to describe changes in the intracellular ionic concentrations and the ionic flow through the cellular membrane. The membrane potential in each of these models exhibits bursting patterns similar to those observed experimentally. Values of the plateau fraction, $\rho _f $, can be computed from these models. The Sherman–Rinzel–Keizer (SRK) model of this phenomenon consists of three, coupled, first-order nonlinear differential equations that describe the dynamics of the membrane potential, the activation parameter for the voltage-gated potassium channel, and the intracellular calcium concentration. These equations are transformed into a Liénard differential equation coupled to a single first-order differential equation for the slowly changing nondimensional calcium concentration. Leading-order perturbation problems for the silent phase and the transition regions are reduced to quadrature. The solution of the leading-order active phase problem is a limit cycle which depends on the value of the intracellular calcium concentration. Since the active phase equations exhibit weak damping, Melnikov’s method can be applied to determine the bifurcation point of these equations. Thus, an explicit expression for the active phase duration is obtained. Together with the silent phase analysis, an approximation of the plateau fraction $\rho _f $ is derived and its value compared to the plateau fraction numerically obtained from the SRK model.

Chain-length-dependent termination rate processes in free-radical polymerizations. 1. Theory

A complete set of rate equations describing the kinetics of free-radical polymerization is deduced, in which termination rate coefficients are allowed to depend on chain length. The rate equations are applicable to bulk, solution, and heterogeneous (e.g., emulsion) polymerization systems. They incorporate the following kinetic processes: initiation, propagation, transfer, chain-length-dependent termination, and, for emulsion systems, exit and reentry (i.e., aqueous-phase kinetic processes are taken into account). It is shown that, despite their apparent complexity, the full set of population balance equations can, to a good approximation, be reduced to a single first-order differential equation. This equation and that for the overall rate of polymerization form a pair of coupled differential equations that can be solved easily and are thus suitable for routine modeling of experiment. The key parameters required in the model are the rate coefficients for transfer and propagation and the diffusion coefficient of the monomer as a function of the polymer weight fraction (conversion), this variation being used in expressions for the dependence of the termination rate coefficient on the length of a growing chain. In accord with previous experiment and theory (Adams, M. E.; Russell, G. T.; Casey, B. S.; Napper, D. H.; Gilbert, R. G.; Sangster, D. F. Macromolecules 1990, 23, 4624), the present theory indicates that, at intermediate conversions, termination is dominated by interactions between short chains formed by transfer and entangled long chains. In the regime in which propagation is diffusion-controlled, most termination events involve two highly entangled chains, whose ends move by the reaction-diffusion process (Russell, G. T.; Napper, D. H.; Gilbert, R. G. Macromolecules 1988, 21, 2133). The mathematical treatments of this paper are of very general applicability and should inter alia be useful in addressing practical problems such as minimizing the residual monomer content of polymer products.

A mathematical model of the rat proximal tubule.

The equations of mass conservation and electroneutrality are used to extend a nonequilibrium thermodynamic model of the rat proximal tubule epithelium to a representation of a 0.5-cm segment of tubule. The output of the tubule model includes the luminal profiles and absolute proximal reabsorption of Na, K, Cl, HCO3, HPO4, H2PO4, glucose, and urea, generated by the epithelial model. Transport rates and permeabilities, chosen in agreement with those of the rat, result in luminal glucose and bicarbonate depletion and a transition from an electronegative to positive lumen. Despite the development of significant transepithelial osmotic driving forces (a transepithelial glucose gradient and Cl-HCO3 asymmetry), intraepithelial solute-solvent coupling remains an important force for water reabsorption along the proximal tubule length. In particular, this means that when osmotic gradients that appear under free-flow conditions are used in the calculation of the epithelial water permeability, a substantial overestimate of this permeability will be obtained. A single first-order differential equation has been derived in conjunction with an approximate nonelectrolyte model of the proximal tubule that represents both coupled and gradient-driven water reabsorption. In the present work, this equation is shown to yield an accurate description of water transport by the comprehensive tubule model.

An approximate calculation of the oxygen activity profile in solid-solution scales formed during the oxidation of binary alloys

A simplified method to calculate the oxygen activity profiles in solid-solution scales formed on binary alloys is proposed. This is based on Wagner's theoretical analysis but involves the solution of a single first-order differential equation by a process of successive approximations using the experimental data on the profiles of the scale composition. The agreement between the oxygen activity profiles calculated by this procedure and those obtained by the more general treatment is satisfactory for the systems examined. The method can therefore be used to obtain rapidly an approximate oxygen activity profile when the conditions for the application of the theory are satisfied.

Toroidal free oscillations by the method of factorization

The eigenvalue problem for the toroidal free oscillations of the earth may be reduced to a single first-order differential equation, which is readily integrated numerically even at large wave numbers, with arbitrary radial variation of the model parameters. The technique seems particularly suited to inclusive programs for earth model inversion which incorporate data from body waves, surface waves, and free oscillations.

Runge-Kutta methods with minimum error bounds

1. Introduction. Numerical methods for the solution of ordinary differential equations may be put in two categories-numerical integration (e.g., predictorcorrector) methods and Runge-Kutta methods. The advantages of the latter are that they are self-starting and easy to program for digital computers but neither of these reasons is very compelling when library subroutines can be written to handle systems of ordinary differential equations. Thus, the greater accuracy and the error-estimating ability of predictor-corrector methods make them desirable for systems of any complexity. However, when predictor-corrector methods are used, Runge-Kutta methods still find application in starting the computation and in changing the interval of integration. If, then, Runge-Kutta methods are considered in the context of using them for starting and for changing the interval, matters such as stability [2], [3] and minimization of roundoff errors [4] are not significant. Also, simplifying the coefficients so that the computation will be speeded up is not important and, on modern computers, minimization of storage [4] is seldom important. In fact, the only criterion of significance in judging Runge-Kutta methods in this context is minimization of truncation error. It is the purpose of this paper to derive Runge-Kutta methods of second, third and fourth order which have minimum truncation error bounds of a specified type. We will consider only the case of integrating a single first-order differential equation because this is the only tractable case analytically. But it seems reasonable to assume that methods which are best in a truncation error sense for one equation will be at least nearly best for systems of equations. 2. The General Equations. For the solution of the equation

On gas flow in one dimension following a normal shock of variable strength

Unsteady non-homentropic flow of a gas in one dimension is studied by taking a form of the Lagrangian equations of motion. An ‘exact’ solution representing progressive waves is found, and this is applied to the problem of a shock advancing into a region in which the pressure is constant, but the density (and temperature) varies according to a simple power law. The problem is shown to depend upon a single first-order differential equation of standard type, and it is indicated how numerical solutions could be constructed if desired. For convenience in presentation, however, the discussion is limited to the case of a very strong shock, and only qualitative conclusions are offered at this stage.