Boundary value problem direction

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. 

The axial stress is the only stress component which can be determined directly from measurement data. Hence, we have the boundary-value problem with equations (27), (29)-(31) and the boundary conditions (34)-(36). 

In integrated photoelasticity it is impossible to achieve a complete reconstruction of stresses in samples by only illuminating a system of parallel planes and using equilibrium equations of the elasticity theory. Theory of the fictitious temperature field allows one to formulate a boundary-value problem which permits to determine all components of the stress tensor field in some cases. If the stress gradient in the axial direction is smooth enough, then perturbation method can be used for the solution of the inverse problem. As an example, distribution of stresses in a bow tie type fiber preforms is shown in Fig. 2 [2]. 

Diffusion problems in one dimension lead to boundaiy value problems. The boundaiy conditions are applied at two different spatial locations at one side the concentration may be fixed and at the other side the flux may be fixed. Because the conditions are specified at two different locations, the problems are not initial value in character. It is not possible to begin at one position and integrate directly because at least one of the conditions is specified somewhere else and there are not enough conditions to begin the calculation. Thus, methods have been developed especially for boundary value problems. 

Elliptic Equations Elhptic equations can be solved with both finite difference and finite element methods. One-dimensional elhptic problems are two-point boundary value problems. Two- and three-dimensional elliptic problems are often solved with iterative methods when the finite difference method is used and direct methods when the finite element method is used. So there are two aspects to consider howthe equations are discretized to form sets of algebraic equations and howthe algebraic equations are then solved. 

They convert the initial value problem into a two-point boundary value problem in the axial direction. Applying the method of lines gives a set of ODEs that can be solved using the reverse shooting method developed in Section 9.5. See also Appendix 8.3. However, axial dispersion is usually negligible compared with radial dispersion in packed-bed reactors. Perhaps more to the point, uncertainties in the value for will usually overwhelm any possible contribution of D. ... 

In the present section a direct method for solving the boundary-value problems associated with second-order difference equations will be the subject of special investigations. 

In the case of the second boundary-value problem with dv/dn = 0, the boundary condition of second-order approximation is imposed on 7, as a first preliminary step. It is not difficult to verify directly that the difference eigenvalue problem of second-order approximation with the second kind boundary conditions is completely posed by... 

In Section 1 we confine ourselves to direct economical methods available for solving boundary-value problems associated with Poisson s equation in a rectangle such as the decomposition method and the mathod of separation of variables. 

Direct and iterative methods. Recall that the final results of the difference approximation of boundary-value problems associated with elliptic equations from Chapter 4 were various systems of linear algebraic equations (difference or grid equations). The sizes of the appropriate matrices are extra large and equal the total number N of the grid nodes. For... 

When solving difference boundary-value problems for Poisson s equation in rectangular, angular, cylindrical and spherical systems of coordinates direct economical methods are widely used that are known to us as the decomposition method and the method of separation of variables. The calculations in both methods for two-dimensional problems require Q arithmetic operations, Q — 0 N og. N), where N is the number of the grid nodes along one of the directions. 

Equation (1.84) form Dirichlet s boundary value problem, which can be either exterior or internal one. Fig. 1.8a, and it has several important applications in the theory of the gravitational field of the earth. It is worth to notice that in accordance with Equation (1.83) we can say that along any direction tangential to the boundary surface, the component of the field is also known, since = dU/dt. Consequently, the boundary value problem can be written in terms of the field as... 

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. 

The diffusion equation is the partial-differential equation that governs the evolution of the concentration field produced by a given flux. With appropriate boundary and initial conditions, the solution to this equation gives the time- and spatial-dependence of the concentration. In this chapter we examine various forms assumed by the diffusion equation when Fick s law is obeyed for the flux. Cases where the diffusivity is constant, a function of concentration, a function of time, or a function of direction are included. In Chapter 5 we discuss mathematical methods of obtaining solutions to the diffusion equation for various boundary-value problems. 

For initial value problems such as the ones we have encountered in Chapter 4, there is only one possible direction of integration, namely from the initial value ujstart onwards. But for two-point BVPs we have function information at both ends ujstart < atend and we could integrate forwards from uj start to ujend, or backwards from uiend to uj start-Regardless of the direction of integration, each boundary value problem on [uJstart, uJend]... 

The first one is based on a classical variation method. This approach is also known as an indirect method as it focuses on obtaining the solution of the necessary conditions rather than solving the optimization directly. Solution of these conditions often results in a two-point boundary value problem (TPBVP), which is accepted that it is difficult to solve [15], Although several numerical techniques have been developed to address the solution of TPBVP, e.g. control vector iteration (CVI) and single/multiple shooting method, these methods are generally based on an iterative integration of the state and adjoint equations and are usually inefficient [16], Another difficulty relies on the fact that it requires an analytical differentiation to derive the necessary conditions. 

The approximating scheme converts the system of partial differential equations to a set of ordinary differential equations in the axial spatial coordinate. The detailed equations are contained in Yu et al. (2). The advantage of reduction in this manner is that the transient location of the combustion zone does not have to be known a priori, but can be found in the course of the integrations. Gear integration, which is designed for stiff systems, is used to solve the two point boundary value problem in the axial direction. 

The disadvantage of including axial dispersion is that an exit boundary condition must be specified, and in cases where an analytical solution is not available, a numerical boundary-value problem must be solved in the axial direction, rather than an initial-value problem. 

This set of hyperbolic partial differential equations for the gasifier dynamic model represents an open or split boundary-value problem. Starting with the initial conditions within the reactor, we can use some type of marching procedure to solve the equations directly and to move the solution forward in time based on the specified boundary conditions for the inlet gas and inlet solids streams. 

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