First-order linear system

Equations (2.9), (2.10) and (2.11) are linear differential equations with constant coefficients. Note that the order of the differential equation is the order of the highest derivative. Systems described by such equations are called linear systems of the same order as the differential equation. For example, equation (2.9) describes a first-order linear system, equation (2.10) a second-order linear system and equation (2.11) a third-order linear system. 

Consider a first-order linear system subject to an input step change. Figure 10.4 shows the response of the system with time. In Section 10.4 we found that the response reaches 63.2% of its final value when the time elapsed is equal to one time constant xp. Also, when t = 2xp the response has reached 86.5% of the final value, at t = 3t 95%, and so on. Therefore, if the sampled representation of the response is going to be of any value, the sampling period must be smaller than one time constant. How much smaller Practical experience suggests that a sampling period between 0.1 and 0.2 of one time constant yields satisfactory results. 

The question then arises why study first order linear systems (of the form of Eq. (2)) and facilitate their incorporation into modeling software when their relevance seems at best questionable. To answer this we need to note the following. 

If the relation between one variable with time, undergoing a step input, may be described as an exponential rise to a maximum, then the kinetics of the process is typical of a first order linear system. Its corresponding equation is as follows ... 

There is a theory called the Floquet theory (Cesari, 1971) which concerns first-order linear systems with periodic coefficients. In the present context, such a system arises from the linearization of (2.1.1) about. fo(f)- By putting X(t) = X() t)- u(t), this leads to... 

The previous chapter showed how the reverse Euler method can be used to solve numerically an ordinary first-order linear differential equation. Most problems in geochemical dynamics involve systems of coupled equations describing related properties of the environment in a number of different reservoirs. In this chapter I shall show how such coupled systems may be treated. I consider first a steady-state situation that yields a system of coupled linear algebraic equations. Such a system can readily be solved by a method called Gaussian elimination and back substitution. I shall present a subroutine, GAUSS, that implements this method. 

The system of two first-order linear differential equations has the general form ... 

The concept of clearance is extremely useful in clinical pharmacokinetics because the systemic clearance of a drug is usually constant over the range of plasma concentrations of clinical interest. This is because the elimination of most drugs obeys first-order (linear) kinetics whereby a constant fraction is eliminated per unit of time. For drugs that exhibit dose-dependent elimination,... 

The advantages of using non-compartmental methods for calculating pharmacokinetic parameters, such as systemic clearance (CZg), volume of distribution (Vd(area))/ systemic availability (F) and mean residence time (MRT), are that they can be applied to any route of administration and do not entail the selection of a compartmental pharmacokinetic model. The important assumption made, however, is that the absorption and disposition processes for the drug being studied obey first-order (linear) pharmacokinetic behaviour. The first-order elimination rate constant (and half-life) of the drug can be calculated by regression analysis of the terminal four to six measured plasma... 

A system of n nonhomogeneous first order linear ODEs can be written in matrix forms as follows ... 

Convert the governing equation to finite difference form by using central difference expression accurate to the order h for the first and second derivatives in the spatial variable, x (equation (6.11)). This gives raise to N second order linear ODEs in This system of second order equations is converted to 2N first order linear ODEs in as described in equation (6.12). The variable ui(Q, i = 0..N-I-1 corresponds to the dependent variable, ui at node point i. 

With all of the above assumptions the Maxwell-Stefan relations (Eq. 2.1.16) reduce to a system of first-order linear differential equations... 

For a closed, isolated system H is time independent time dependence in the Hamiltonian enters via effect of time-dependent external forces. Here we focus on the earlier case. Equation (1) is a first-order linear differential equation that can be solved as an initial value problem. If (to) is known, a formal solution to Eq. (1) is given by... 

Equation (35) was advanced by Ginzburg and Sobyanin for superfluidity in confined finite systems [135]. This theory will be applied herein for the onset of superfluidity of He confined in a sphere of radius Rq. Adopting the step function approximation, the boundary condition for the order parameter at the free surface is taken as /(/ o) = 0. Eor low values of / the first-order linear form of Eq. (30) is... 

Fortunately, process control problems are most usually concerned with maintaining operating variables constant at particular values. Most disturbances to the process involve only small excursions of the process variables about their normal operating points with the result that the system behaves linearly regardless of how nonlinear the descriptive equations may be. Thus Eq. (1) is a nonlinear differential equation since both Cp and U are functions of 80 but for small changes in 8 average values of CP and U may be regarded as constants, and the equation becomes the simplest kind of first order linear differential equation. 

The singularity of the system (0, 0), is an unstable stationary point for e > 0. Apparently, equations (3.71), (3.72) can be immediately solved, being the system of homogeneous first-order linear equations. The solutions have... 

Only some of the important works for distributed systems control shall be reviewed here. Since Butkovskii results require the explicit solution of the system equations, this restricts the results to linear systems. This drawback was removed by Katz (1964) who formulated a general maximum principle which could be applied to first order hyperbolic systems and parabolic systems without representing the system by integral equations. Lurie (1967) obtained the necessary optimality conditions using the methods of classical calculus of variations. The optimization problem was formulated as a Mayer-Bolza problem for multiple integrals. 

These simple properties of the Laplace transform make it a very convenient tool for solving systems of first-order linear differential equations, such as the equations for growth and decay of nuclides in radioactive disintegrations and neutron inadiation. They permit these differential equations to be treated as if they were systems of simple transformed linear equations without derivatives. 

The three observations above hold not only for first-order systems but are true for any order linear system. Before we proceed with the generalization of the results above, let us make the following remarks related to the algebra of complex numbers. 

RESPONSES OF SIMPLE LINEAR SYSTEMS 2.3.1 First-Order Linear Ordinary Differential Equation... 

If N is equal to 1 the system is first order and if h(t) = 0, the system is homogeneous. This article will deal with first-order linear homogeneous systems sometimes referred to as initial value problems, viz.. 

The response of a nonlinear system at steady state (see above) to a small perturbation is linear, i.e., describable as a set of first order linear differential equations (of the form Eq. (2)). Furthermore, kinetic models represented by such a set of linear differential equations contain many helpful pieces of information concerning the host nonlinear system, viz., the number of exchanging (metabolic) pools, the rate of exchange among the (metabolic) pools, and the size of the (metabolic) pools. 

Since we are dealing with first order linear homogeneous systems the computational complexity associated with solving these systems can be considered in terms of the L matrix, though the number of solutions and the integration domain will be secondary issues as well. 

For systems where the phase ratio, O, and the properties of the peptide or protein and the sorbent surface are invariant with temperature, the values of AH, and issoc./ associated with the interaction of P, with the nonpolar ligates can be derived in the traditional manner by linear regression analysis of the log A versus 1 /T plot as the slope and intercept values, respectively. Moreover, with RPC and HIC sorbents, the extent of fit of the experimental data to the linearized form of the van t Hoff dependence can be employed to gain insight into whether first-order (linear) adsorption/desorption conditions prevail or whether additional factors are involved in the interaction of the polypeptide (or protein) with the nonpolar ligates, e.g., whether perturbation of secondary or tertiary staicture of P, occurs under the conditioiis of the interaction. 

In every case, the return to the new equilibrium value proceeds at a rate that is proportional to the distance that the system is from equilibrium. So relaxation is described by a first-order linear differential equation, the solution to which is a simple exponential function. 

Quantitative structure-activity relationships (QSAR), a concept introduced by Hansch and Fujita (1964) is a kind of formal system based on a kinetic model, which in turn is expressed in term of a first-order linear differential equation. Solution of the differential equation leads to a linear equation ( Hansch-Fujita equation ), the coefficients of which are determined by regression analysis resulting in a QSAR equation of a particular compound series. For a prediction, the dependent variable of the QSAR equation is calculated by algebraic operations. 

The present pure calculations illustrate the significance of non-Boltzmann rate coefficients for hot atom reaction systems, bxnce i (t) is a strongly varying function, Eq. 22 cannot be approximated by a first order linear differential rate expression. The mean hot atom reactive lifetime is given by Eq. 26. 

Assume that a system used in the oil and gas industry can be in any of the three states operating normally, failed safely, and failed unsafely. The following three first-order linear differential equations describe the oil and gas industry system under consideration ... 

Organic compounds with delocalized 7r-electron systems are leading candidates for nonlinear optical (NLO) materials, and interest in these materials has grown tremendously in the past decade [108-118]. Reliable structure-property relationships—where property here refers to first-order (linear) polarizability a, second-order polarizability and third-order polarizability y—are required for the rational design of optimized materials for photonic devices such as electro-optic modulators and all-optical switches. Here also, quantum-chemical calculations can contribute a great deal to the establishment of such relationships. In this section, we illustrate their usefulness in the description of the NLO response of donor-acceptor substituted polymethines, which are representative of an important class of organic NLO chromophores. We also show how much the nonlinear optical response depends on the interconnection between the geometric and electronic structures, as was the case of the properties discussed in the previous sections [ 119]. 

The PDI expression, for example, eqn [34] for the steady-state or first-order-type system applied to the B subchain, can be given by a linear function of... 

We include here a short example of linear lumping taken from Li and Rabitz (1989) in order to illustrate the approach. Consider the first-order reaction system involving reversible reactions between three species as follows ... 

The EKF has by far been the most extensively used identification algorithm, for the case of nonlinear systems, over the past 30 years, and has been applied for a number of civil engineering applications, such as structural damage identification, parameter identification of inelastic structures, and so forth. It is based on the propagation of a Gaussian random variable (GRV) through the first-order linearization of the state-space model of the system. Despite... 

A system whose dynamic behaviour is described by a first-order linear differential equation. 

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