Solution of Ordinary Differential Equations

Following the traditional approach, this section will be discussed under two subheadings (1) initial value problems and (2) boundary value problems. Supporting information, such as the algebra or calculus of finite differences, can be found in the literature [5,9]. 

The discussion that follows assumes that the problem under consideration for a numerical approach will be well posed [5,21]. 

The issue now is, how does Equation 9.44 describe the solution given by Equation 9.45, when all we know are Equation 9.44 and the initial value  

Generally, the procedures will rely on the fact that given any point (t, y) on the solution curve, we can obtain the direction of the curve through that point. In principle, we have a starting point and a set of directions given by Equation 9.44 and without any other information, we must follow the directions to the desired final destination, the approximate solution. 

Formally, the procedure is as follows we wish to generate an approximation y corresponding to the point t such that 

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Shampine S 1994 Numerical Solutions of Ordinary Differential Equations (New York Chapman and Hall)... 

Numerical Solution of Ordinary Differential Equations as Initial... 

Hindmarsh, A. C. GEARB Solution of Ordinary Differential Equations Having Banded Jacobian, UCID-.30059, Rev. 1 Computer Documentation, Lawrence Livermore Laboratory, University of California (1975). 

Lapidus, L., and J. Seinfeld. Numerical Solution of Ordinary Differential Equations, Academic, New York (1971). 

NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS AS INITIAL VALUE PROBLEMS... 

Once the initial state x(f = 0) of the system is specified, future states, x(t), are uniquely defined for all times t . Moreover, the uniqueness theorem of the solutions of ordinary differential equations guarantees that trajectories originating from different initial points never intersect. 

Section 5.3. There, the operator T> s d/dx was used in the solution of ordinary differential equations. In Chapter 5 the vector operator del , represented by the symbol V, was introduced. It was shown that its algebraic form is dependent on the choice of curvilinear coordinates. 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements. One simply combines the methods for ordinary differential equations (see Ordinary Differential Equations—Boundary Value Problems ) with the methods for initial-value problems (see Numerical Solution of Ordinary Differential Equations as Initial Value Problems ). Fast Fourier transforms can also be used on regular grids (see Fast Fourier Transform ). 

Absorption columns can be modeled in a plate-to-plate fashion (even if it is a packed bed) or as a packed bed. The former model is a set of nonlinear algebraic equations, and the latter model is an ordinary differential equation. Since streams enter at both ends, the differential equation is a two-point boundary value problem, and numerical methods are used (see Numerical Solution of Ordinary Differential Equations as Initial-Value Problems ). 

Dynamic simulations are also possible, and these require solving differential equations, sometimes with algebraic constraints. If some parts of the process change extremely quickly when there is a disturbance, that part of the process may be modeled in the steady state for the disturbance at any instant. Such situations are called stiff, and the methods for them are discussed in Numerical Solution of Ordinary Differential Equations as Initial-Value Problems. It must be realized, though, that a dynamic calculation can also be time-consuming and sometimes the allowable units are lumped-parameter models that are simplifications of the equations used for the steady-state analysis. Thus, as always, the assumptions need to be examined critically before accepting the computer results. 

Table 1.2. Solutions of Ordinary Differential Equations of the First Order... 

An implementation of this algorithm, using the sequential procedure within the MATLAB environment, was proposed by Figueroa and Romagnoli (1994). To solve step 2, the constr function from the MATLAB Optimization Toolbox has been used. The numerical integration necessary in this step has been performed via the function ode45 for the solution of ordinary differential equations. 

Figure 3.6 Numerical solution of ordinary differential equations sketch of the four steps of the Runge-Kutta method to the order four giving the n+1 th estimate y(n+1) from the nth estimate y ". ...

For a discussion of the algorithm, see Shampine, L. F. Gordon, M. K. "Computer Solution of Ordinary Differential Equations" Freeman San Francisco, 1975. 

As discussed in the introduction to this chapter, the solution of ordinary differential equations (ODEs) on a digital computer involves numerical integration. We will present several of the simplest and most popular numerical-integration algorithms. In Sec, 4.4.1 we will discuss explicit methods and in Sec. 4.4.2 we will briefly describe implicit algorithms. The differences between the two types and their advantages and disadvantages will be discussed. 

Solution of ordinary differential equations fourth order Runga-Kutta method 7000 7058... 

M71 Solution of ordinary differential equations predictor-corrector method of Milne 7100 7188... 

REM t SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS I REN t FOURTH ORDER RUN6A-KUTTA NETHOD I... 

SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS t 4 PREDICTOR-CORRECTOR HETHOD OF HILNE 4 444444444444444444444444444444444444444444444444444... 

Seydel, R., 1981, Numerical computation of periodic orbits that bifurcate from stationary solutions of ordinary differential equations. Appl. Math. Comput. 9, 257-271. 

Hoppensteadt, F. (1971). Properties of solutions of ordinary differential equations with small parameters. Commun. Pure Appl. Math., XXIV, 807-840. 

Hindmarsh, A.C., nGEARB Solution of Ordinary Differential Equations Having Banded Jacobians," Lawrence Livermore Laboratory Report UCID-30059, June 1977, Rev. 2. 

Solutions for the integration of ODEs such as those given in Equation 7.5 are not always readily available. For nonspecialists, it is difficult to determine whether there is an explicit solution at all. MATLAB s symbolic toolbox provides a very convenient means of producing the results and also of testing for explicit solutions of ordinary differential equations, e.g., for the reaction 2A — B, as seen in MATLAB Example 7.2. (Note that MATLAB s symbolic toolbox demands lowercase characters for species names.)... 

In this chapter, the numerical solution of ordinary differential equations (odes) will be described. There is a direct connection between this area and that of partial differential equations (pdes), as noted in, for example [558]. The ode field is large but here we restrict ourselves to those techniques that appear again in the pde field. Readers wishing greater depth than is presented here can find it in the great number of texts on the subject, such as the classics by Lapidus k Seinfeld [351], Gear [264] or Jain [314] there is a very clear chapter in Gerald [266]. 

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