Ordinary differential equations, initial value numerical solution

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 ). 

Figure E.l represents a highly simplified view of an ideal structure for an application program. The boxes with the rounded borders represent those functions that are problem specific, while the square-comer boxes represent those functions that can be relegated to problem-independent software. This structure is well-suited to problems that are mathematically systems of nonlinear algebraic equations, ordinary differential equation initiator boundary-value problems, or parabolic partial differential equations. In these cases the problem-independent mathematical software is usually written in the form of a subroutine that in turn calls a user-supplied subroutine to define the system of equations. Of course, the analyst must write the subroutine that describes the particular system of equations. Moreover, for most numerical-solution algorithms, the system of equations must be written in a discrete form (e.g., a finite-volume representation). However, the equation-defining sub-...

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

This equation must be solved for yn +l. The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL [Ascher, U. M., and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia (1998) and Brenan, K. E., S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North Holland Elsevier (1989)]. 

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. 

The reason for constructing this rather complex model was that even though the mathematical equations may be easily set up using the dispersion model, the numerical solutions are quite involved and time consuming. Deans and Lapidus were actually concerned with the more complicated case of mass and heat dispersion with chemical reactions. For this case, the dispersion model yields a set of coupled nonlinear partial differential equations whose solution is quite formidable. The finite-stage model yields a set of differential-double-difference equations. These are ordinary differential equations, which are easier to solve than the partial differential equations of the dispersion model. The stirred-tank equations are of an initial-value type rather than the boundary-value type given by the dispersion model, and this fact also simplifies the numerical work. 

Engineers develop mathematical models to describe processes of interest to them. For example, the process of converting a reactant A to a product B in a batch chemical reactor can be described by a first order, ordinary differential equation with a known initial condition. This type of model is often referred to as an initial value problem (IVP), because the initial conditions of the dependent variables must be known to determine how the dependent variables change with time. In this chapter, we will describe how one can obtain analytical and numerical solutions for linear IVPs and numerical solutions for nonlinear IVPs. 

Steady state mass or heat transfer in solids and current distribution in electrochemical systems involve solving elliptic partial differential equations. The method of lines has not been used for elliptic partial differential equations to our knowledge. Schiesser and Silebi (1997)[1] added a time derivative to the steady state elliptic partial differential equation and applied finite differences in both x and y directions and then arrived at the steady state solution by waiting for the process to reach steady state. [2] When finite differences are applied only in the x direction, we arrive at a system of second order ordinary differential equations in y. Unfortunately, this is a coupled system of boundary value problems in y (boundary conditions defined at y = 0 and y = 1) and, hence, initial value problem solvers cannot be used to solve these boundary value problems directly. In this chapter, we introduce two methods to solve this system of boundary value problems. Both linear and nonlinear elliptic partial differential equations will be discussed in this chapter. We will present semianalytical solutions for linear elliptic partial differential equations and numerical solutions for nonlinear elliptic partial differential equations based on method of lines. 

With the exception of a few cases (simple reaction orders, isothermal condition), the solution of the above system of ordinary differential equations can be obtained only numerically. For this, there are several numerical software tools available, (e.g., Matlab, Mathematica, GNU Octave, etc.) that provide advanced solvers for such initial value problems. A good overview can be found, e.g., in Press et al., 2002. 

In the previous chapter, we discussed analytical and approximate methods to obtain solutions to ordinary differential equations. When these approaches fail, the only remaining course of action is a numerical solution. This chapter and the next consider numerical methods for solving ODEs, which may not be obtainable by techniques presented in Chapters 2, 3, and 6. ODEs of initial value type (i.e., conditions are specified at one boundary in time or space) will be considered in this chapter, whereas ODEs of boundary value type (conditions are specified at two boundary points) will be considered in Chapter 8. 

In [165] a study of a new methodology for development of efficient methods for the numerical solution of second-order periodic initial value problems (IVPs) of ordinary differential equations is presented. The methodology is based on the development of nonlinear numeircal methods. In this paper the authors study the following nonlinear scheme ... 

Abstract. A class (m,k)-methods is discussed for the numerical solution of the initial value problems for impHcit systems of ordinary differential equations. The order conditions and convergence of the numerical solution in the case of implementation of the scheme with the time-lagging of matrices derivatives for systems of index 1 are obtained. At A < 4 the order conditions are studied and schemes optimal computing costs are obtained. 

The modern methods for numerical solution of the initial-value problem for systems of ordinary differential equations (ODE) suppose usually the explicit dependence of the derivative of the solution [7]... 

The solution to Eq. 10.79 must be obtained numerically as it is a nonlinear ordinary differential equation. (See Problem 10C.6 for a solution.) The solution is obtained for arbitrary values of L. The thickness profile can be obtained by means of Eq. 10.75 using the initial thickness estimated from die swell data. One must assume that die swell is unaffected by the weight of the hanging parison. The complete solution of the problem can only be obtained using numerical techniques. ... 

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