Special Boundary Value Problems

We have shown that multiple travelling front waves can occur in a reaction-diffusion-convection system. These waves can be studied in an unbounded system by using a wave transformation and solving a special boundary value problem with the use of continuation methods. These results provide various parameter dependences of the velocity of the wave. Moreover, in a bounded system the waves move back and forth through the system and form remarkable zig-zag patterns. 

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. 

Example 4. The three-layer difference scheme for the heat conduction equation. A special attention is being paid to the first boundary-value problem... 

We call the nodes, at which equation (1) is valid under conditions (2), inner nodes of the grid uj is the set of all inner nodes and ui = ui + y is the set of all grid nodes. The first boundary-value problem completely posed by conditions (l)-(3) plays a special role in the theory of equations (1). For instance, in the case of boundary conditions of the second or third kinds there are no boundary nodes for elliptic equations, that is, w = w. 

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 boundary value problem posed by the differential equation (2.166) and the two boundary conditions (2.168) and (2.169) leads to the class of Sturm-Liouville eigenvalue problems for which a series of general theorems are valid. As we will soon show the solution function F only satisfies the boundary conditions with certain discrete values /q of the separation parameter. These special values /q are called eigenvalues of the boundary value problem, and the accompanying solution functions Fi are known as eigenfunctions. The most important rules from the theory of Sturm-Liouville eigenvalue problems are, cf. e.g. K. Janich [2.33] ... 

Maple s dsolve command can be used to solve linear boundary value problems. One of the advantages of using Maple s dsolve command is Maple can give Bessel and other special function solutions to linear boundary value problems. However, the analytical solution obtained from the dsolve command may not be in simplified or elegant form. The syntax for using the dsolve command is follows. 

In section 3.1.6, linear Boundary value problems were solved using Maple s dsolve command. The solution obtained may not be in the simplified form. Maple gives the Bessel function and other special function solutions for linear... 

It is usually considered that boundary value problems do not differ essentially from free systems. To some extent this is true, and the discussion above confirms this statement. However, even a quick consideration demonstrates that existence of nontrivial boundary conditions requires special attention to some details that may be ignored, or supposed to be almost trivial, for the case when the region 2 is the whole space. In this section we describe some situations, where the details mentioned essentially change the physical picture and physical intuition. 

The Dirichlet boundary value problem for a large scale region 2 (A.) has special interest for our discussions, as it seems to be a reasonable approximation to the free problem. To some extent, this is a simple problem, when some properties of 2 (A.) are supposed to be satisfied. We suppose that enlargement of A means extension of 2 (A) and any point of R3 belongs to some 2 (A) for a large enough A values. One may also suppose that for any A the distance between boundaries of 2 (A) and 2 (A + 8) is not less then KS for any 8 > 0 and some constant K. 

In the models described in Sections 7.2. through 7.5., equilibrium between the adsorption layer and the adjacent subsurface is assumed. A generalisation, taking into account a Henry transfer mechanism as the relation between surface and subsurface concentration (cf Section 4.4), is given in Section 7.6. The special problems connected with the adsorption model of ionic surfactants as well as macro-ions is discussed in Sections 7.7. and 7.8. and an attempt to solve the boundary value problem numerically is demonstrated in Section 7.9. The few experimental results on ionic adsorption kinetics are reported in Section 7.10. 

As outlined above, the FD, FE, and BE methods can aU be used to approximate the boundary value problems which arise in biomedical research problems. The choice depends on the nature of the problem. The FE and FD methods are similar in that the entire solution domain must be discretized, while with the BE method only the bounding surfaces must be discretized. For regular domains, the FD method is generally the easiest method to code and implement, but the FD method usually requires special modifications to define irregular boundaries, abrupt changes in material properties, and complex boundary conditions. While typically more difficult to implement, the BE and FE methods are preferred for problems with irregular, inhomogeneous domains, and mixed boundary conditions. The FE method is superior to the... 

The Galerkin method is particularly efficient for some special structured boundary value problems (Burnett, 1987). 

Boundary value problems are encountered so frequently in modelling of engineering problems that they deserve special treatment because of their importance. To handle such problems, we have devoted this chapter exclusively to the methods of weighted residual, with special emphasis on orthogonal collocation. The one-point collocation method is often used as the first step to quickly assess the behavior of the system. Other methods can also be used to treat boundary value problems, such as the finite difference method. This technique is considered in Chapter 12, where we use this method to solve boundary value problems and partial differential equations. 

Considering chemical application problems, a large number of them yields mathematical models that consist of initial-value problems (IVPs) for ordinary differential equations (ODEs) or of initial-boundary-value problems (IBVPs) for partial differential equations (PDEs). Special problems of this kind, which we have treated, are diffusion-reaction processes in chemical kinetics (various polymerizations), polyreactions in microgravity environment (photoinitiated polymerization with laser beams) and drying procedures of hygroscopic porous media. 

In this section, using an example of a boundary value problem for a singularly perturbed ordinary differential equation, we discuss some principles for constructing special finite difference schemes. In Section II.D, these principles will be applied to the construction of special schemes for singularly perturbed equations of the parabolic type. 

We examine the efficiency of special finite difference schemes for the model boundary value problem (2.16), (2.18). On the set G j.i ) we construct the special grid... 

From the results given in Tables XIV-XVI it follows that the error in the solution of the finite difference scheme (2.78), (2.77) does not exceed the sum of the errors in the solutions of the finite difference schemes (2.79), (2.77) (2.80), (2.77), and (2.81), (2.77). Therefore, the solution of the special scheme (2.78), (2.77) converges e-uniformly to the solution of the boundary value problem (2.16). Thus, numerical experiments illustrate the efficiency of the constructed finite difference scheme. 

In Section II (see Sections II.B, II.D) we considered finite difference schemes in the case when the unknown function takes given values on the boundary. The boundary value problem for the singularly perturbed parabolic equation on a rectangle, that is, a two-dimensional problem, is described by Eqs. (2.12), while the boundary value problem on a segment, that is, a one-dimensional problem, is described by equations (2.14). In Section II.B classical finite difference schemes were analyzed. It was shown that the error in the approximate solution, as a function of the perturbation parameter, is comparable to the required solution for any fine grid. For the above mentioned problems special finite difference schemes were constructed. The error in the approximate solution obtained by the new scheme does not depend on the parameter value and tends to zero as the number of grid nodes increases. 

In this section, we consider singularly perturbed diffusion equations when the diffusion flux is given on the domain boundary. We show (see Section III.B) that the error in the approximate solution obtained by a classical finite difference scheme, depending on the parameter value, can be many times greater than the magnitude of the exact solution. For the boundary value problems under study we construct special finite difference schemes (see Sections III.C and III.D), which allow us to find the solution and diffusion flux. The errors in the approximate solution for these schemes and the computed diffusion flux are independent of the parameter value and depend only on the number of nodes in the grid. 

Thus, for the Neumann boundary value problem (3.23), a special finite difference scheme has been constructed. Its solution z(x),xE.D,, and the function xED allow us to approximate the solution of the... 

Now we study the efficiency of the special finite difference scheme in the case of the Neumann boundary value problem (3.7), (3.9). On the set G(3 g) we introduce the special grid... 

Finally, we have also analyzed the convergence of the discrete fluxes for the Dirichlet boundary value problem (2.16). Here we used the special finite difference scheme. The grid fluxes were considered separately for each component in the representation (2.24). In Tables XXX and XXXI, we give the errors Q(e,N),Q(N) computed from the solutions of the difference schemes (2.79), (2.77) and (2.80), (2.77) corresponding to problems (2.20) and (2.25). Table XXXII shows the... 

The construction and investigation of special difference schemes for a particular singularly perturbed boundary value problem with a discontinuous initial condition were examined in [16-19]. 

The analysis of heat exchange processes, in the case of the plastic shear of a material, leads us to singularly perturbed boundary value problems with a concentrated source. Problems such as these were considered in Section IV, where it was shown that classical difference schemes give rise to errors, which exceed the exact solution by many orders of magnitude if the perturbation parameter is sufficiently small. Besides, a special finite difference scheme, which allows us to approximate both the solution and... 

The above examples permit us to reach the following conclusion. For the singularly perturbed boundary value problems arising in the numerical analysis of heat transfer for various technologies, we have constructed special e-uniformly convergent finite difference schemes. These schemes allow us to compute heat fluxes and the quantity of heat transferred across the interfaces of bodies in contact during the processes. Numerical experiments show the efficiency of the new schemes in comparison with classical schemes. 

In the case of singularly perturbed boundary value problems, for which it is required to find diffusion fluxes, computational difficulties arise. These lead to theoretical and applied problems that require special numerical methods allowing us to approximate both the problem solution and the... 

The special finite difference schemes constructed here allow one to approximate solutions of boundary value problems and also normalized di sion fluxes. They can be used to solve effectively applied problems with boundary and interior layers, in particular, equations with discontinuous coefficients and concentrated factors (heat capacity, sources, and so on). Methods for the construction of the special schemes developed here can be used to construct and investigate special schemes for more general singularly perturbed boundary value problems (see, e.g., [4, 17, 18, 24, 35-39]). 

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