Systems of First-Order Ordinary Differential Equations

Solve the following system of first-order ordinary differential equations and prepare a state-space plot where x is plotted against y using the solution. 

This appendix presents two methods of obtaining an analytical solution to a system of first order ordinary differential equations. Both methods (power series and the Laplace transform) yield a solution in terms of the matrix exponential. That is, we seek a solution to... 

Linear Second-Order and Systems of First-Order Ordinary Differential Equations... 

Systems of first-order ordinary differential equations... 

At pesudo-steady state (i.e., dNJdt = 0), three limiting cases can be obtained from this system of first-order ordinary differential equations (Figure 4.2). 

Equation 3.23 together with the kinetic model expressions gives a system of first-order ordinary differential equations (ODEs), which can be solved numerically with respect to reactor length using a fourth-order Runge-Kutta method. For the case of isothermal operation, this solution can also be done in analytic manner. For instance, for the 5-lump kinetic model reported by Singh et al. (2005) at the following initial conditions ... 

Equation (31) is the system of second order ordinary differential equations which will be solved. But, in order to solve these equations, they must first be rewritten as a set of... 

State-space models provide a convenient representation of dynamic models that can be expressed as a set of first-order, ordinary differential equations. State-space models can be derived from first principles models (for example, material and energy balances) and used to describe both linear and nonlinear dynamic systems. 

In Chapter 3, the analytieal method of solving kinetie sehemes in a bateh system was eonsidered. Generally, industrial realistie sehemes are eomplex and obtaining analytieal solutions ean be very diffieult. Beeause this is often the ease for sueh systems as isothermal, eonstant volume bateh reaetors and semibateh systems, the designer must review an alternative to the analytieal teehnique, namely a numerieal method, to obtain a solution. Eor systems sueh as the bateh, semibateh, and plug flow reaetors, sets of simultaneous, first order ordinary differential equations are often neeessary to obtain die required solutions. Transient situations often arise in die ease of eontinuous flow stirred tank reaetors, and die use of numerieal teehniques is die most eonvenient and appropriate mediod. 

The solution of boundary value problems depends to a great degree on the ability to solve initial value problems.) Any n -order initial value problem can be represented as a system of n coupled first-order ordinary differential equations, each with an initial condition. In general... 

Kunii and Levenspiel(1991, pp. 294-298) extend the bubbling-bed model to networks of first-order reactions and generate rather complex algebraic relations for the net reaction rates along various pathways. As an alternative, we focus on the development of the basic design equations, which can also be adapted for nonlinear kinetics, and numerical solution of the resulting system of algebraic and ordinary differential equations (with the E-Z Solve software). This is illustrated in Example 23-4 below. 

These relationships define the equations of motion of the atoms, which can be written as a system of 6N first-order ordinary differential equations ... 

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 ... 

We now have to solve the following system of two nonlinear coupled first-order ordinary differential equations for the given initial conditions ... 

Using the boundary conditions (equations (5.7) and (5.8)) the boundary values uo and Un+1 can be eliminated. Hence, the method of lines technique reduces the linear parabolic ODE partial differential equation (equation (5.1)) to a linear system of N coupled first order ordinary differential equations (equation (5.5)). Traditionally this linear system of ordinary differential equations is integrated numerically in time.[l] [2] [3] [4] However, since the governing equation (equation (5.5)) is linear, it can be written as a matrix differential equation (see section 2.1.2) ... 

Steady state heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a linear elliptic partial differential equation. For linear parabolic partial differential equations, finite differences can be used to convert to any given partial differential equation to system of linear first order ordinary differential equations in time. In chapter 5.1, we showed how an exponential matrix method [3] [4] [5] could be used to integrate these simultaneous equations... 

A secondary benefit from the manipulations that are performed to yield solvable equations is that the resulting equations may allow a more efficient calculation by the SOLVER module. For example, in the reaction system given in Section 2.4., a system of four differential equations was transformed into a system of two first order ordinary differential equations and two linear algebraic equations. 

Ordinary differential equations govern systems that vary either with time or space, but not both. Examples are equations that govern the dynamics of a CSTR or the steady state of mbular reactors. Both the dynamics of a CSTR and the steady state of a plug-flow reactor are governed by first-order ordinary differential equations with prescribed initial conditions. The steady-state tubular reactors with axial dispersion are governed by a second-order differential equation with the boundary conditions spec-... 

AUT097 (http //indy.cs.concordia.ca/auto) A code for tracking by continuation the solution of systems of nonlinear algebraic and/or first-order ordinary differential equations as a function of a bifurcation parameter (available only for UNIX-based computers)... 

Although there are many definitions of chaos (Gleick, 1987), for our purposes a chaotic system may be defined as one having three properties deterministic dynamics, aperiodicity, and sensitivity to initial conditions. Our first requirement implies that there exists a set of laws, in the case of homogeneous chemical reactions, rate laws, that is, first-order ordinary differential equations, that govern the time evolution of the system. It is not necessary that we be able to write down these laws, but they must be specifiable, at least in principle, and they must be complete, that is, the system cannot be subject to hidden and/or random influences. The requirement of aperiodicity means that the behavior of a chaotic system in time never repeats. A truly chaotic system neither reaches a stationary state nor behaves periodically in its phase space, it traverses an infinite path, never passing more than once through the same point. 

This is especially true in our case, as q> is the solution of a first-order ordinary differential equations system [Lorton et al. 2013]. 

In all the above examples, the systems were chosen so that the models resulted in sets of simultaneous first-order ordinary differential equations. These are the most commonly encountered types of problems in the analysis of multicomponent and/or multistage operations. Closed-form solutions for such sets of equations are not usually obtainable. However, numerical methods have been thoroughly developed for the solution of sets of simultaneous differential equations. In this chapter, we discuss the most useful techniques for the solution of such problems. We first show that higher-order differential equations can be reduced to first order by a series of substitutions. 

Numerical integration of ordinary differential equations is most conveniently performed when the system consists of a set of n simultaneous first-order ordinary differential equations of the form ... 

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. 

There are several ways to solve a third-order ordinary-differential-equation boundary-value problem. One is shooting, which is discussed in Section 6.3.4.1. Here, we choose to separate the equation into a system of two equations—one second-oider and one first-order equation. The two-equation system is formed in the usual way by defining a new variable g = /, which itself serves as one of the equations,... 

If the system is isothermal with Tout=Tim Equation (14.3) is unnecessary. Unsteady behavior in an isothermal perfect mixer is governed by a maximum of N + 1 ordinary differential equations. Except for highly complicated reactions such as polymerizations (where N is theoretically infinite), solutions are usually straightforward. Numerical methods for unsteady CSTRs are similar to those used for steady-state PFRs, and analytical solutions are usually possible when the reaction is first order. 

This is a system of four coupled, second-order ordinary differential equations. The solution of each equation must satisfy two boundary conditions. The first is dictated by the symmetry of the crystallite (sphere) which requires the concentration gradients of all species to disappear at the center of the crystallite ... 

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