Ordinary differential equations, boundary value numerical solutions

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 ). 

This is a linear ordinary-differential-equation boundary-value problem that can be solved analytically (see Bird, Stewart, and Lightfoot, Transport Phenomena, Wiley, 1960). Here, however, proceed directly to numerical finite-difference solution, which can be implemented easily in a spreadsheet. Assuming a cone angle of a = 2° and a rotation rate of 2 = 30 rpm, determine f(0) — v /r. 

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 ). 

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. 

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-...

Mathematical modeling of mass or heat transfer in solids involves Pick s law of mass transfer or Fourier s law of heat conduction. Engineers are interested in the steady state distribution of heat or concentration across the slab or the material in which the experiment is performed. This steady state process involves solving second order ordinary differential equations subject to boundary conditions at two ends. Whenever the problem requires the specification of boundary conditions at two points, it is often called a two point boundary value problem. Both linear and nonlinear boundary value problems will be discussed in this chapter. We will present analytical solutions for linear boundary value problems and numerical solutions for nonlinear boundary value problems. 

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. 

To find the unknown functions, one writes out a system of linked boundary value problems for ordinary differential equations. The numerical solution of these problems for the first six terms of the expansion (1.8.14) is tabulated in detail in [427], The corresponding analytical expressions for the velocity components in the boundary layer can be calculated by formulas (1.1.6). 

In the case of the other boundary-value problem mentioned, we can demonstrate mathematically that such a substitution is correct here we cannot make such a demonstration, so the resulting solution rests on this additional assumption. This assumption converts the set of two partial differential equations to a single, ordinary differential equation, which Blasius was able to solve in numerical form. The details of this calculation are shown by Schlichting [1, p. 135] see Prob. 11.4. The result is in the form of a curve of VJV versus r, shown in Fig. 11.3. 

Application of numerical methods have been rather seldom in studies of adsorption kinetics from micellar solutions. The main difficulties are probably connected with the large number of independent parameters. The first work belongs to Miller [146]. Fainerman and Rakita also published numerical results of the solution of the boundary value problem (5.236), (5.237), (5.245) [85]. Recently Danov et al. proposed an original method for solving the boundary value problem for the diffusion of micelles and monomers [92]. The system of equations was reduced to a system of ordinary differential equations by using a model concentration profile in the bulk phase. The obtained results agree better with dynamic surface tensions of micellar solutions than equation (5.248). 

Ascher, U.R., Mattheij, R.M.M., and Russell, R.D. (1987) Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Society for Industrial and Applied Mathematics. 

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. 

Now we analyze the quality of the approximation of the boundary value problem solution by the grid solution. By using a simple example of an ordinary differential equation, we demonstrate some problems arising from its numerical solution. 

In Section I we obtained an intuitive impression of the numerical problems appearing when one uses classical finite difference schemes to solve singularly perturbed boundary value problems for ordinary differential equations. In this section, for a parabolic equation, we study the nature of the errors in the approximate solution and the normalized diffusion flux for a classical finite difference scheme on a uniform grid and also on a grid with an arbitrary distribution of nodes in space. We find distributions of the grid nodes for which the solution of the finite difference scheme approximates the exact one uniformly with respect to the parameter. The efficiency of the new scheme for finding the approximate solution will be demonstrated with numerical examples. 

In [169] the authors study the numerical solution of two point boundary value problems (BVPs) for second-order Ordinary Differential Equations (ODEs). They solve the above problems using a direct backward difference and the shooting technique. 

Liu SL (1967) Numerical solution of two-point boundary value problems in simultaneous second-order nonlinear ordinary differential equations. Chem Eng Sci 22 871-881... 

Since the first derivative of X(t) on the interval [T, k N exists, X(t) is continuous on such interval. If X(t) is monotoimus in the interval [7., 7(. j], the extreme values of X(t) are reached at the boundary points of the interval. Otherwise, the extreme values of X(t) can be obtained by comparing all the local extreme values found by Fermat theorem (Bronshtein et al. 1985). To calculate the value of X(t), Runge-Kutta methods can be applied for the numerical solution of the ordinary differential equations (Hairer et al. 1993, Hairer et al. 1996). 

For a CSTR the stationary-state relationship is given by the solution of an algebraic equation for the reaction-diffusion system we still have a (non-linear) differential equation, albeit ordinary rather than partial as in eqn (9.14). The stationary-state profile can be determined by standard numerical methods once the two parameters D and / have been specified. Figure 9.3 shows two typical profiles for two different values of )(0.1157 and 0.0633) with / = 0.04. In the upper profile, the stationary-state reactant concentration is close to unity across the whole reaction zone, reflecting only low extents of reaction. The profile has a minimum exactly at the centre of the reaction zone p = 0 and is symmetric about this central line. This symmetry with the central minimum is a feature of all the profiles computed for the class A geometries with these symmetric boundary conditions. With the lower diffusion coefficient, D = 0.0633, much greater extents of conversion—in excess of 50 per cent—are possible in the stationary state. 

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