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The most commonly used numerical techniques in solving engineering and mathematical models are the Finite Element, Finite Difference, and Boundary Element Methods. As computer capabilities continue to impro':e in speed, memory size and access speed, and lower costs, the use of more accurate but computationally expensive numerical techniques will become attractive to the practicing engineer. This book presents an introduction to a new approximation method based on a generalized Fourier series expansion of a linear operator equation. Because many engineering problems such as the multi­ dimensional Laplace and Poisson equations, the diffusion equation, and many integral equations are linear operator equations, this new approximation technique will be of interest to practicing engineers. Because a generalized Fourier series is used to develop the approxi­ mator, a "best approximation" is achieved in the "least-squares" sense; hence the name, the Best Approximation Method. This book guides the reader through several mathematics topics which are pertinent to the development of the theory employed by the Best Approximation Method. Working spaces such as metric spaces and Banach spaces are explained in readable terms. Integration theory in the Lebesque sense is covered carefully. Because the generalized Fourier series utilizes Lebesque integration concepts, the integra­ tion theory is covered through the topic of converging sequences of functions with respect to measure, in the mean (Lp), almost uniformly IV and almost everywhere. Generalized Fourier theory and linear operator theory are treated in Chapters 3 and 4.



Chapter 1. Working Spaces

The following text introduces metric spaces, converging sequences, neighborhoods and other concepts which are necessary for the understanding of elementary approximation theory.
Theodore V. Hromadka, Chung-Cheng Yen, George F. Pinder

Chapter 2. Integration Theory

Before introducing the theory of generalized Fourier series, the key elements of integration theory need to be reviewed. This is important in order to understand the value of the approximation developed by the Best Approximation Method.
Theodore V. Hromadka, Chung-Cheng Yen, George F. Pinder

Chapter 3. Hilbert Space and Generalized Fourier Series

The subjects of inner product, Hilbert space, generalized Fourier series, and vector space representations are all used in the Best Approximation Method. To introduce these concepts let the underlying Banach space be ℝ3 where each vector (or element) ξ εℝ3 is of the form ξ = (x,y,z). Let ξ1 and ξ2 be two distinct vectors and \([\overrightarrow {0\xi }\) be the straight line through points (0,0,0) and ξ1 = (x1, y1, z1). The task is to find the point ξ* = (x*, y*, z*) in \([\overrightarrow {0\xi }\), which is closest to the point ξ2 = (x2, y2, z2). Closest is defined to mean the minimum value of the norm used in ℝ3. The norm used is the l2 norm (Euclidean norm)
$$[{\left\| \xi \right\|_2} = {\left\| {\left( {{\text{x,y,z}}} \right){\text{ }} - {\text{ }}\left( {0,0,0} \right)} \right\|_2} = {\left[ {{{\text{x}}^2} + {{\text{y}}^2} + {{\text{z}}^2}} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}$$
. Any point ξ in \([\overrightarrow {0\xi }\) has a distance from ξ2 calculated from
$$[{{\text{D}}^2}\left( {\xi ,{\text{ }}{\xi _2}} \right) = {\left\| {\xi ,{\text{ }}{\xi _2}} \right\|^2} = {\left( {{\text{x - }}{{\text{x}}_2}} \right)^2} + {\left( {{\text{y - }}{{\text{y}}_2}} \right)^2} + {\left( {{\text{z - }}{{\text{z}}_2}} \right)^2}$$
. But ξ in \([\overrightarrow {0\xi }\), can be written as ξ = λξ1 where λ εℝ. Thus
$$[{{\text{D}}^2}\left( {\xi ,{\text{ }}{\xi _2}} \right) = {\left( {\lambda {{\text{x}}_1}{\text{ - }}{{\text{x}}_2}} \right)^2} + {\left( {\lambda {{\text{y}}_1}{\text{ - }}{{\text{y}}_2}} \right)^2} + {\left( {\lambda {{\text{z}}_1}{\text{ - }}{{\text{z}}_2}} \right)^2} = {\lambda ^2}\left( {{{\text{x}}_1}^2 + {{\text{y}}_1}^2 + {{\text{z}}_1}^2} \right) - 2\lambda \left( {{{\text{x}}_1}{{\text{x}}_2} + {{\text{y}}_1}{{\text{y}}_2} + {{\text{z}}_1}{{\text{z}}_2}} \right) + \left( {{{\text{x}}_2}^2 + {{\text{y}}_2}^2 + {{\text{z}}_2}^2} \right)$$
using vector dot product notation where
$$[{\xi _1} \cdot {\xi _2} = {{\text{x}}_1}{{\text{x}}_2} + {{\text{y}}_1}{{\text{y}}_2} + {{\text{z}}_1}{{\text{z}}_2},{\text{ }}{{\text{D}}^2}\left( {\xi ,{\text{ }}{\xi _2}} \right) = {\lambda ^2}{\text{ }}{\xi _1} \cdot {\xi _1} - 2\lambda {\xi _1} \cdot {\xi _2} + {\xi _2} \cdot {\xi _2}.$$
The above result can be obtained more quickly from the definition of the norm by nothing \([{{\text{D}}^{2}}\left( \xi ,{{\xi }_{2}} \right)={{\left\| \lambda {{\xi }_{1}}-{{\xi }_{2}} \right\|}^{2}}=\left( \lambda {{\xi }_{1}}-{{\xi }_{2}} \right)\cdot \left( \lambda {{\xi }_{1}}-{{\xi }_{2}} \right)={{\lambda }^{2}}{{\xi }_{1}}\cdot {{\xi }_{1}}-2\lambda {{\xi }_{1}}\cdot {{\xi }_{2}}+{{\xi }_{2}}\cdot {{\xi }_{2}}.\)
Theodore V. Hromadka, Chung-Cheng Yen, George F. Pinder

Chapter 4. Linear Operators

Many of the mathematical relationships used in engineering analysis fall into the category of being linear operator equations. In this chapter, background theory is presented in the characteristics of linear operators. Additionally, several of the more common linear operators which are of interest in engineering studies are discussed.
Theodore V. Hromadka, Chung-Cheng Yen, George F. Pinder

Chapter 5. The Best Approximation Method

Many important engineering problems fall into the category of being linear operators, with supporting boundary conditions. In this chapter, an inner-product and norm is developed which enables the engineer to approximate such engineering problems by developing a generalized Fourier series. The resulting approximation is the “best” approximation in that a least-squares (L2) error is minimized simultaneously for fitting both the problem’s boundary conditions and satisfying the linear operator relationship (the governing equations) over the problem’s domain (both space and time). Because the numerical technique involves a well-defined inner product, error evaluation is readily available using Bessel’s inequality. Minimization of the approximation error is subsequently achieved with respect to a weighting of the inner product components, and the addition of basis functions used in the approximation.
Theodore V. Hromadka, Chung-Cheng Yen, George F. Pinder

Chapter 6. The Best Approximation Method: Applications

The theory of generalized Fourier Series as utilized in the Best Approximation Method can be applied to the approximation of linear operator relationships. To demonstrate the computational results in using this approach, several example problems where analytic solutions or quasi-analytic solutions exist are considered. Applications include two-dimensional problems involving the Laplace and Poisson equations, tests for variation in results due to inner product weighting factors, and applications to nonhomogeneous domain problems.
Theodore V. Hromadka, Chung-Cheng Yen, George F. Pinder

Chapter 7. Coupling the Best Approximation and Complex Variable Boundary Element Methods

From (5.4), a linear operation equation is solved by the Best Approximation Method by use of the inner product
$$[\left( {u,v} \right) = \int\limits_\Gamma {uvd\Gamma } + \int\limits_\Omega {Lu} Lvd\Omega$$
where the integration over Г includes both the spatial and temporal boundary conditions (i.e., initial conditions in a diffusion problem).
Theodore V. Hromadka, Chung-Cheng Yen, George F. Pinder


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