First-order ordinary differential linear

Equation (A4) is a first order, linear, ordinary differential equation which can be solved analytically for [PJ assuming X, and X, are constant over a small increment in time. Solving for [PJ from some time ti to tj gives Equation (1) (1). When X, is considered a function of time (i.e., initiator concentration is allowed to vary through the small time increment) while maintaining X, constant over the increment. Equation (A4) can again be solved analytically to give Equation (3). 

J.1 First-Order Linear Ordinary Differential Equation... 

First-Order Linear Ordinary Differential Equation / 2.3.2 Second-Order Linear ODEs with Constant Coefficients / 2.3.3 Nth-Order Linear ODEs with Constant Coefficients... 

This is now a first-order linear ordinary differential equation with... 

We assume a constant stretch rate, in which case these are simply uncoupled first-order linear ordinary differential equations, which are easily integrated. When the modes are summed to give the total stress, we then obtain the equation... 

Equation (25) is Bernoulli s equation that can be transformed into a first-order linear ordinary differential equation via the following transformations ... 

Since the orthogonal collocation or OCFE procedure reduces the original model to a first-order nonlinear ordinary differential equation system, linearization techniques can then be applied to obtain the linear form (72). Once the dynamic equations have been transformed to the standard state-space form and the model parameters estimated, various procedures can be used to design one or more multivariable control schemes. 

The mass balance with diffusion and first-order chemical reaction, given by (24-12), is classified as a frequently occurring second-order linear ordinary differential equation (i.e., ODE) with constant coefficients. It is a second-order equation because diffusion is an important mass transfer rate process that is included in the mass balance. It is linear because the kinetic rate law is first-order or pseudo-first-order, and it is ordinary because diffusion is considered only in one coordinate direction—normal to the interface. The coefficients are constant under isothermal conditions because the physicochemical properties of the fluid don t change... 

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. 

Linear first order hyperbolic partial differential equations are solved using Laplace transform techniques in this section. Hyperbolic partial differential equations are first order in the time variable and first order in the spatial variable. The method involves applying Laplace transform in the time variable to convert the partial differential equation to an ordinary differential equation in the Laplace domain. This becomes an initial value problem (IVP) in the spatial direction with s, the Laplace variable, as a parameter. The boundary conditions in x are converted to the Laplace domain and the differential equation in the Laplace domain is solved by using the techniques illustrated in chapter 2.1 for solving linear initial value problems. Once an analytical solution is obtained in the Laplace domain, the solution is inverted to the time domain to obtain the final analytical solution (in time and spatial coordinates). This is best illustrated with the following example. 

In this subsection, some commonly used numerical schemes that involve difference equations to solve ordinary differential equations are presented along with their stability characteristics. Simple examples to illustrate the effects of step size on the convergence of numerical methods are shown. A simple discretization of the first-order linear differential equation... 

PSEUDO-FIRST-ORDER KINETIC RATE EXPRESSIONS THAT CAN REPLACE HOUGEN-WATSON MODELS AND GENERATE LINEARIZED ORDINARY DIFFERENTIAL EQUATIONS FOR THE MASS BALANCE... 

Differential equations may be either first order, or second order, (third order and others are also possible, but less likely), depending on the derivative levels that appear in the equation. For instance, a differential eqnation commonly appearing in biological systems is the first order constant coefficient linear ordinary differential equation... 

This system of equations can be reduced to four first-order linked linear ordinary differential equations with split boundary conditions. The transition point where the two-phase fluid is completely vaporized is unknown. Furthermore, the wall thickness is discontinuous, which means that is also discontinuous. 

Assume that all steps are first-order reactions and write the set of linear ordinary differential equations that describe the kinetics of these reactions. Solve the problem numerically for the following values of the kinetic rate constants ... 

Calculates the concentration profile of a system of first-order chemical reactions by solving the set of linear ordinary differential equations (LinearODE.m). 

In the following section, we only consider the integration of the equation of linear motion Eq. (20) the procedure for the equation of rotational motion, Eq. (21), will be completely analogous. Mathematically, Eq. (20) represents an initial-value ordinary differential equation. The evolution of particle positions and velocities can be traced by using any kind of method for ordinary differential equations. The simplest method is the first-order integrating scheme, which calculates the values at a time t + 5t from the initial values at time t (which are indicated by the superscript 0 ) via ... 

The mathematical problem posed is the solution of the simultaneous differential equations which arise from the mass-action treatment of the chemistry. For the homogeneous, well-mixed reactor, this becomes a set of ordinary, non-linear, first-order differential equations. For systems that are not... 

The problem in obtaining a state space model for the dynamics of the CSD from this physical model is that the population balance is a (nonlinear) first-order partial differential equation. Consequently, to obtain a state space model the population balance must be transformed into a set of ordinary differential equations. After this transformation, the state space model is easily obtained by substitution of the algebraic relations and linearization of the ordinary differential equations. 

EXAMPLE 4.3 Plagan Poiseuille flow - laminar, steady incompressible flow in a long pipe with a linear pressure gradient (first-order, nonlinear solution to an ordinary differential equation)... 

In simplifying the packed bed reactor model, it is advantageous for control system design if the equations can be reduced to lit into the framework of modern multivariable control theory, which usually requires a model expressed as a set of linear first-order ordinary differential equations in the so-called state-space form ... 

After the kinetic model for the network is defined, a simulation method needs to be chosen, given the systemic phenomenon of interest. The phenomenon might be spatial. Then it has to be decided whether in addition stochasticity plays a role or not. In the former case the kinetic model should be described with a reaction-diffusion master equation [81], whereas in the latter case partial differential equations should suffice. If the phenomenon does not involve a spatial organization, the dynamics can be simulated either using ordinary differential equations [47] or master equations [82-84]. In the latter case but not in the former, stochasticity is considered of importance. A first-order estimate of the magnitude of stochastic fluctuations can be obtained using the linear noise approximation, given only the ordinary differential equation description of the kinetic model [83-85, 87]. 

The above ordinary differential equations (ODEs), Eqs. (19-11) and (19-12), can be solved with an initial condition. For an isothermal first-order reaction and an initial condition, C(0) = 0, the linear ODE may be solved analytically. At steady state, the accumulation term is zero, and the solution for the effluent concentration becomes... 

Process Transfer Function Models In continuous time, the dynamic behaviour of an ideal continuous flow stirred-tank reactor can be modelled (after linearization of any nonlinear kinetic expressions about a steady-state) by a first order ordinary differential equation of the form... 

Many drugs undergo complex in vitro drug degradations and biotransformations in the body (i.e., pharmacokinetics). The approaches to solve the rate equations described so far (i.e., analytical method) cannot handle complex rate processes without some difficulty. The Laplace transform method is a simple method for solving ordinary linear differential equations. Although the Laplace transform method has been used for more complex applications in physics, engineering, and other research areas, here it will be applied to ordinary differential equations of first-order rate processes. 

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