Finite difference approximation of the boundary-value problem

2 Finite difference approximation of the boundary-value problem 

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Finite difference approximation of the boundary-value problem In this case, matrix D has a septa-block-diagonal structure ... 

Eq. (5.248) and its modification for a deformed surface [154], together with the corresponding equations for AT [152] and Eq. (5.243) are the only analytical results obtained as solution of the boundary value problem for the diffusion equations of micelles and monomers. An approximate relation for Ay can be also obtained without integration of the diffusion equations with the help of the penetration theory [155], In this case the derivative on the right hand side of Eq. (5.237) is replaced by the ratio of finite differences... 

Recall that we are interested in the behavior of the error in the approximate solution for various values of the parameter e. To compute the solution of the boundary value problem (2.16), (2.18), we use a classical finite difference scheme. We now describe this scheme. On the set G the uniform rectangular grid... 

Thus, by numerical experiments we verify that the approximate solution of the Dirichlet problem (2.16), found by the classical finite difference scheme (2.28), (2.27), and the computed normalized diffusion fiux converge for N, Nq respectively, to the solution of the boundary value problem and the real normalized diffusion flux for fixed e. However, we can also see that they do not converge e-uniformly. The solution of the grid problem approaches the solution of the boundary value problem uniformly in e qualitatively well. The normalized flux computed according to the solution of the difference problem does not approach e-uniformly the real normalized flux (i.e., the flux related to the solution of the boundary value problem) even qualitatively. Nevertheless, if the solution of the singularly perturbed boundary value problem is smooth and e-uniformly bounded, the approximate solution and the computed normalized flux converge e-uniformly (when N, Nq oo) to the exact solution and flux. 

We describe a finite difference scheme, the solution of which approximates the solution of the boundary value problem -uniformly on the whole grid set in G. 

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. 

The theory of finite difference solution for Boundary value problems was developed in section 3.1.5. When finite difference approximations are used, the given nonlinear boundary value problem is converted to a system of nonlinear algebraic equations. This resulting system is solved in this section using Maple s fsolve command. 

In section 3.2.3, finite difference solutions were obtained for nonlinear boundary value problems. This is a straightforward and easy technique and can be used to obtain an initial guess for other sophisticated techniques. This technique is important because it forms the basis for the method of lines technique for solving linear and nonlinear partial differential equations (chapter 5 and 6). However, for stiff boundary value problems, this technique may not work and might demand prohibitively large number of node points. In addition, approximate initial guess should be provided for all the node points for stiff boundary value problems. 

Once we have stated or derived the mathematical equations which define the physics of the system, we must figure out how to solve these equations for the particular domain we are interested in. Most numerical methods for solving boundary value problems require that the continuous domain be broken up into discrete elements, the so-called mesh or grid, which one can use to approximate the governing equation (s) using the particular numerical technique (finite element, boundary element, finite difference, or multigrid) best suited to the problem. 

There are many numerical approaches one can use to approximate the solution to the initial and boundary value problem presented by a parabolic partial differential equation. However, our discussion will focus on three approaches an explicit finite difference method, an implicit finite difference method, and the so-called numerical method of lines. These approaches, as well as other numerical methods for aU types of partial differential equations, can be found in the literature [5,9,18,22,25,28-33]. 

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. 

However, on the parts with sharp variation of the solution, that is, in the neighborhood of the boundary layer, the following situation is realized. The parameter e can take arbitrarily small values. Therefore, for the chosen grid with very condensed nodes (or for the chosen step size in the case of uniform grids) a value of the parameter e can be found that is comparable to the distance between neighboring nodes in the boundary layer. In this case, the exact solution of the problem varies considerably between the nodes mentioned. With this argument we cannot expect that the finite difference equation and the grid solution approximate well the differential equation and the exact solution, respectively, at these nodes. 

In the case of Dirichlet problem (2.37) considered in Section 2.3, the unknown solution and the solution of the finite difference scheme take the same given values on the boundary. Thus, in the Dirichlet problem no error appears on the boundary. The error is generated only by an error in the approximation of the differential equation by the difference equation. 

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 return to the study of the normalized diffusion fluxes for boundary value problems with Dirichlet boundary condition. In Section II.D the e-uniformly convergent finite difference schemes (2.74), (2.76) and (2.67), (2.72) were constructed for the Dirichlet problems (2.12), (2.13) and (2.14), (2.15), respectively. For these problems, we now construct and analyze the approximations of the normalized diffusion fluxes. We consider the normalized diffusion fluxes for problem (2.14), (2.15) in the form... 

Note, in particular, one feature in the behavior of the approximate solutions of boundary value problems with a concentrated source. It follows from the results of Section II that, in the case of the Dirichlet problem, the solution of the classical finite difference scheme is bounded 6-uniformly, and even though the grid solution does not converge s-uniformly, it approximates qualitatively the exact solution e-uniformly. But now, in the case of a Dirichlet boundary value problem with a concentrated source, the behavior of the approximate solution differs sharply from what was said above. For example, in the case of a Dirichlet boundary value problem with a concentrated source acting in the middle of the segment D = [-1,1], when the equation coefficients are constant, the right-hand side and the boundary function are equal to zero, the solution is equivalent to the solution of the problem on [0,1] with a Neumann condition at x = 0. It follows that the solution of the classical finite difference scheme for the Dirichlet problem with a concentrated source is not bounded e-uniformly, and that it does not approximate the exact solution uniformly in e, even qualitatively. 

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 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]). 

A differential equation that has data given at more than one value of the independent variable is a boundary-value problem (BVP). Consequently, the differential equation must be of at least second order. The solution methods for BVPs are different compared to the methods used for initial-value problems (IVPs). An overview of a few of these methods will be presented in Sections 6.2.1. 2.3. The shooting method is the first method presented. It actually allows initial-value methods to be used, in that it transforms a BVP to an IVP, and finds the solution for the IVP. The lack of boundary conditions at the beginning of the interval requires several IVPs to be solved before the solution converges with the BVP solution. Another method presented later on is the finite difference method, which solves the BVP by converting the differential equation and the boundary conditions to a system of linear or non-hnear equations. Finally, the collocation and finite element methods, which solve the BVP by approximating the solution in terms of basis functions, are presented. 

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