Boundary Value Problems for Analytic Functions

The main purpose of this appendix is to summarize aspects of the theory of the linear relationship between boundary values of analytic functions on opposite sides of curves of discontinuity in the region of definition of the function, which are relevant to the solution of plane problems in Chaps. 3, 4. This topic is discussed in Sects. A2.2 und A2.3, while various introductory concepts are introduced in Sect. A2.1. 

The problem of the linear relationship may be stated as follows given the equation connecting the boundary values of an analytic function on a line of discontinuity together with some condition on the function at infinity, determine the analytic function, either uniquely or very nearly uniquely. This has been termed the Hilbert problem by Muskhelishvili (1953, 1963) and the Riemann problem by Gakhov (1966). We adopt the former terminology here. 

A closely related problem is given a linear relationship between the real and imaginary parts of the function on the boundaries of a domain in which it is analytic, find the function throughout the domain. This is termed the Riemann-Hilbert problem by Galin (1961) and also here. 

in the case of the Riemann-Hilbert problem, a function is analytic in a given domain bounded by a particular contour, and information is supplied about the limiting values of the function on this contour. In the case of the Hilbert problem, the region of analyticity contains open or closed contours on which the function is not defined. These are cuts in the region of analyticity across which the function is discontinuous. The supplied information is a relationship between the limiting values on either side of these cuts or internal boundaries. 

Plane elastic, and in a partial sense, viscoelastic boundary value problems can often be phrased in terms of either a Hilbert or a Riemann-Hilbert problem. Galin (1961) for example, bases his whole approach on the latter method. On the other hand, Muskhelishvili (1963) manages to cast various boundary value problems in the form of Hilbert problems, thereby solving them. Gakhov (1966) discusses the connection between the two problems. The Hilbert approach is adopted here, on the grounds that the theory is a little easier, though this may be a matter of taste. 

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In section 3.1.4, an analytical series solution using the matrizant was developed for the case where the coefficient matrix is a function of the independent variable. This methodology provides series solutions for Boundary value problems without resorting to any conventional series solution technique. In section 3.1.5, finite difference solutions were obtained for linear Boundary value problems as a function of parameters in the system. The solution obtained is equivalent to the analytical solution because the parameters are explicitly seen in the solution. One has to be careful when solving convective diffusion equations, since the central difference scheme for the first derivative produces numerical oscillations. 

To find the unknown functions, one writes out a system of linked boundary value problems for ordinary differential equations. The numerical solution of these problems for the first six terms of the expansion (1.8.14) is tabulated in detail in [427], The corresponding analytical expressions for the velocity components in the boundary layer can be calculated by formulas (1.1.6). 

References Courant, R., and D. Hilbert, Methods of Mathematical Physics, vol. I, Interscience, New York (1953) Linz, P., Analytical and Numerical Methods for Volterra Equations, SIAM Publications, Philadelphia (1985) Porter, D., and D. S. G. Stirling, Integral Equations A Practical Treatment from Spectral Theory to Applications, Cambridge University Press (1990) Statgold, I., Greens Functions and Boundary Value Problems, 2d ed., Interscience, New York (1997). 

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 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. 

From the standpoint of the continuum simulation of processes in the mechanics of materials, modeling ultimately boils down to the solution of boundary value problems. What this means in particular is the search for solutions of the equations of continuum dynamics in conjunction with some constitutive model and boundary conditions of relevance to the problem at hand. In this section after setting down some of the key theoretical tools used in continuum modeling, we set ourselves the task of striking a balance between the analytic and numerical tools that have been set forth for solving boundary value problems. In particular, we will examine Green function techniques in the setting of linear elasticity as well as the use of the finite element method as the basis for numerical solutions. 

The B.E.M. is thus an analytical-numerical technique for the solution of boundary value problems. The formulation of the integral equations is usually achieved using Greens Theorems and Greens Functions. For more details of the basic theory and an up-to-date account of the B.E.M., Brebbia et al (3) give an excellent review. 

IX. Causality. The requirement of Causality, namely that the current situation can be influenced only by past and contemporaneous events, may be shown to impose a constraint on the analytic structure of the complex moduli in the complex (JD plane, and also on combinations of the moduli multiplying Green s functions in the solution of non-inertial boundary value problems. These quantities can have no singularities in the lower half-plane. In a restricted sense, this can be shown directly for certain combinations of complex moduli using properties of the individual complex moduli. 

The inverse problem in this case is formulated as recovery of the unknown coefficient 7 of the elliptic operator from the known values of the field p(r, u>) in some domain or in the boundary of observations. In a number of brilliant mathematical papers the corresponding uniqueness theorems for this mathematical inverse problem have been formulated and proved The key result is that the unknown coefficient 7 (r) of an elliptic differential operator can be determined uniquely from the boundary measurements of the field, if 7 (r) is a real-analytical function, or a piecewise real-analytical function. In other words, from the physical point of view we assume that 7 (r) is a smooth function in an entire domain, or a piecewise smooth function. Note that this result corresponds well to Wcidelt s and Gusarov s uniqueness theorems for the magnetotelluric inverse problem. I would refer the readers for more details to the papers by Calderon (1980), Kohn and Vogelius (1984, 1985), Sylvester and Uhlmann, (1987), and Isakov (1993). 

As shown in Fig. 1.2, to solve this problem we need only analytical geometry. The constraints (1.29) restrict the solution to a convex polyhedron in the positive quadrant of the coordinate system. Any point of this region satisfies the inequalities (1.29), and hence corresponds to a feasible vector or feasible solution. The function (1.30) to be maximized is represented by its contour lines. For a particular value of z there exists a feasible solution if and only if the contour line intersects the region. Increasing the value of z the contour line moves upward, and the optimal solution is a vertex of the polyhedron (vertex C in this example), unless the contour line will include an entire segment of the boundary. In any case, however, the problem can be solved by evaluating and comparing the objective function at the vertices of the polyhedron. 

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