Second order boundary value problems

One approach to second-order boundary value problems is a matrix formulation. Given... 

Numerical Solution Equations 6.40 and 6.41 represent a nonlinear, coupled, boundary-value system. The system is coupled since u and V appear in both equations. The system is nonlinear since there are products of u and V. Numerical solutions can be accomplished with a straightforward finite-difference procedure. Note that Eq. 6.41 is a second-order boundary-value problem with values of V known at each boundary. Equation 6.40 is a first-order initial-value problem, with the initial value u known at z = 0. 

The orthogonal collocation technique is a simple numerical method which is easy to program for a computer and which converges rapidly. Therefore it is useful for the solution of many types of second order boundary value problems. This method in its simplest form as presented in this section was developed by Villadsen and Stewart (1967) as a modification of the collocation methods. In collocation methods, trial function expansion coefficients are typically determined by variational principles or by weighted residual methods (Finlayson, 1972). The orthogonal collocation method has the advantage of ease of computation. This method is based on the choice of suitable trial series to represent the solution. The coefficients of trial series are determined by making the residual equation vanish at a set of points called collocation points , in the solution domain. 

Listing 11.14. Code for solving second order boundary value problem using finite... 

Listing 11.22 Code for coupled second order boundary value problems by the finite difference algorithm. 

Although the specific boundary conditions depend on the details of the flow situation and the domain, there are common elements in the boundary conditions that are considered here. Equations 6.159 through 6.163 represent a ninth-order boundary-value problem, requiring nine boundary conditions. The continuity equation is a first-order equation that requires only a boundary condition on u at the stagnation surface. The second-order transport equations demand boundary conditions at each end of the domain, 0 < z < zend-... 

Equation (1), with the associated boundary conditions, is a nonlinear second-order boundary-value ODE. This was solved by the method of collocation with piecewise cubic Hermite polynomial basis functions for spatial discretization, while simple successive substitution was adequate for the solution of the resulting nonlinear algebraic equations. The method has been extensively described before [9], and no problems were found in this application. 

Equations (11.111) - (11.113) define a boundary value problem for a pair of simultaneous second order differential equations in and x, subject... 

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. 

The second-order difference equations. The Cauchy problem. Boundary-value problems. The second-order difference equation transforms into a more transparent form... 

For the second-order difference equations capable of describing the basic mathematical-physics problems, boundary-value problems with additional conditions given at different points are more typical. For example, if we know the value for z = 0 and the value for i = N, the corresponding boundary-value problem can be formulated as follows it is necessary to find the solution yi, 0 < i < N, of problem (6) satisfying the boundary conditions... 

Difference equations with a symmetric matrix are typical in numerical solution of boundary-value problems associated with self-adjoint differential equations of second order. In what follows we will show that the condition Bi = is necessary and sufficient for the operator [yj] be self-adjoint. As can readily be observed, any difference equation of the form... 

Example 1. The third boundary-value problem for an ordinary second-order differential equation ... 

In this way, the third kind difference boundary-value problem (2)-(4) of second-order approximation on the solution of the original problem is put in correspondence with the original problem (1). 

Difference Green s function. Further estimation of a solution of the boundary-value problem for a second-order difference equation will involve its representation in terms of Green s function. The boundary-value problem for the differential equation... 

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... 

The difference boundary-value problem associated with the difference equation (7) of second order can be solved by the standard elimination method, whose computational algorithm is stable, since the conditions Ai 0, Ci > Ai -f Tj+i are certainly true for cr > 0. 

Thus, in order to determine the potential we have to know at the surface S both the potential and its normal derivative. At the same time, as follows from the first and second boundary value problems, in order to find the potential it is sufficient to know only one of these quantities on S. This apparent contradiction can be easily resolved by the appropriate choice of Green s function and we will illustrate this fact in the two following sections. 

Theoretical chemists learn about a number of special functions, the Hermite functions in connection with the quantisation of the harmonic oscillator, Legendre and associated Legendre functions in connection with multipole expansions, Bessel functions in connection with Coulomb Greens functions, the Coulomb wave functions and a few others. All these have in common that they are the solutions of second order linear equations with a parameter. It is usually the case that solutions of boundary value problems for these equations only exist for countable sets of values of the parameter. This is how quantisation crops up in the Schrddinger picture. Quantum chemists are very comfortable with this state of affairs, but rarely venture outside the linear world where everything seems to be ordered. 

The first situation involves two algebraic equations, the second involves an algebraic equation (the mixed phase) and a first-order ordinary differential equation (the unmixed phase), and the third situation involves two coupled differential equations. Countercurrent flow is in fact more compHcated than cocurrent flow because it involves a two-point boundary-value problem, which we will not consider here. 

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,... 

The steady-state stagnation-flow equations represent a boundary-value problem. The momentum, energy, and species equations are second order while the continuity equation is first order. Although the details of boundary-condition specification depend in the particular problem, there are some common characteristics. The second-order equations demand some independent information about V,W,T and Yk at both ends of the z domain. The first-order continuity equation requires information about u on one boundary. As developed in the following sections, we consider both finite and semi-infinite domains. In the case of a semi-infinite domain, the pressure term kr can be determined from an outer potential flow. In the case of a finite domain where u is known on both boundaries, Ar is determined as an eigenvalue of the problem. 

It is instructive to study a much simpler mathematical equation that exhibits the essential features of boundary-layer behavior. There is a certain analogy between stiffness in initial-value problems and boundary-layer behavior in steady boundary-value problems. Stiffness occurs when a system of differential equations represents coupled phenomena with vastly different characteristic time scales. In the case of boundary layers, the governing equations involve multiple physical phenomena that occur on vastly different length scales. Consider, for example, the following contrived second-order, linear, boundary-value problem ... 

Solution of the second-order two-point boundary value problem (5.24) for a second-order reaction... 

Let us investigate how MATLAB handles boundary value problems such as (5.24) with the boundary conditions (5.25) and (5.26) for first- and second-order reactions. The MATLAB program linquadbvp.m has been designed for this purpose. 

In terms of the concentration vector, Eq. (A10) is a nonlinear differential equation of the second order. The boundary-value problem [Eqs. (A10) and (All)] is usually solved numerically. However, it is also possible to linearize the reaction term using the method suggested in Ref. 181 ... 

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