Ordinary Differential Equations Boundary Value Problems

So-called boundary value problems (BVPs) occur most often when the system model is a second-order ODE and the known information is available for two different values of the independent variable. For example, consider the following ODE problem  

This problem differs from ODE-IVP since two initial conditions are not given. The two values of the independent variable where information is available are usually at a physical boundary of the system, and the problem is referred to as a boundary value problem. Here is a more specific example  

A copper rod of length 1 m is placed between two tanks, one containing boiling water and the other containing ice. The rod is exposed to the air. The mathematical model for this system can be expressed as follows  

The methods covered in Chapter 5 cannot be used directly for this problem. The standard procedure would be to convert Equation 6.2 into two first-order ODEs, which would require two initial conditions. In the present example, 0 jc 1, and only one condition is available at x=0. Therefore, some strategy must be used so that available methods can be applied. 

ORDINARY DIFFERENTIAL EQUATIONS-BOUNDARY VALUE PROBLEMS 

Diffusion problems in one dimension lead to boundary value problems. The boundary conditions are applied at two different spatial 

Boundary value methods provide a description of the solution either by providing values at specific locations or by an expansion in a series of functions. Thus, the key issues are the method of representing the solution, the number of points or terms in the series, and how the approximation converges to the exact answer, i.e., how the error changes with the number of points or number of terms in the series. These issues are discussed for each of the methods finite difference, orthogonal collocation, and Galerkin finite element methods. 

The truncation error in the first two expressions is proportional to Ax, and the methods are said to be first-order. The truncation error in the third expression is proportional to Ax2, and the method is said to be second-order. Usually the last equation is used to insure the best accuracy. The finite difference representation of the second derivative is  

The truncation error is proportional to Ax2. To solve a differential equation, it is evaluated at a point i and then these expressions are inserted for the derivatives. 

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Ordinary Differential Equations-Boundary Value Problems. 3-57... 

ORDINARY DIFFERENTIAL EQUATIONS-BOUNDARY VALUE PROBLEMS... 

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 problem is described mathematically as an ordinary-differential-equation boundary-value problem. After discretization (Eq. 4.27) a system of algebraic equations must be solved with the unknowns being the velocities at each of the nodes. Boundary conditions are also needed to complete the system of equations. The most straightforward boundary-condition imposition is to simply specify the values of velocity at both walls. However, other conditions may be appropriate, depending on the particular problem at hand. In some cases a balance equation may be required to describe the behavior at the boundary. 

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. 

There is a class of flow situations, first identified by Jeffery [201] and Hamel [163], for which the flow has self-similar behavior. To realize the similar behavior leading to ordinary-differential-equation boundary-value problems, the analysis is restricted to steady-state, incompressible, constant property flows. After first discussing the classic analysis,... 

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,... 

Overall the system of equations (continuity and momentum) is third order, nonlinear, ordinary-differential equation, boundary-value problem. The boundary conditions require no-slip at the plates and specified wall-injection velocities,... 

Stagnation flows represent a very important class of flow configurations wherein the steady-state Navier-Stokes equations, together with thermal-energy and species-continuity equations, reduce to systems of ordinary-differential-equation boundary-value problems. Some of these flows have great practical value in applications, such as chemical-vapor-deposition reactors for electronic thin-film growth. They are also widely used in combustion research to study the effects of fluid-mechanical strain on flame behavior. 

This appendix provides a detailed description of how to build Excel spreadsheet solutions for several of the problems that were presented and solved in Chapter 4. Generically, these include an ordinary-differential-equation boundary-value problem, a one-dimensional parabolic partial differential equation, and a two-dimensional elliptic partial differential equation. 

In this chapter, analytical solutions were obtained for parabolic and elliptic partial differential equations in semi-infinite domains. In section 4.2, the given linear parabolic partial differential equations were converted to an ordinary differential equation boundary value problem in the Laplace domain. The dependent variable was then solved in the Laplace domain using Maple s dsolve command. The solution obtained in the Laplace domain was then converted to the time domain using Maple s inverse Laplace transform technique. Maple is not capable of inverting complicated functions. Two such examples were illustrated in section 4.3. As shown in section 4.3, even when Maple fails, one can arrive at the transient solution by simplifying the integrals using standard Laplace transform formulae. 

Ordinary Differential Equations (Boundary Value Problems)... 

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