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Über dieses Buch

This book presents numerical and other approximation techniques for solving various types of mathematical problems that cannot be solved analytically. In addition to well known methods, it contains some non-standard approximation techniques that are now formally collected as well as original methods developed by the author that do not appear in the literature.

This book contains an extensive treatment of approximate solutions to various types of integral equations, a topic that is not often discussed in detail. There are detailed analyses of ordinary and partial differential equations and descriptions of methods for estimating the values of integrals that are presented in a level of detail that will suggest techniques that will be useful for developing methods for approximating solutions to problems outside of this text.

The book is intended for researchers who must approximate solutions to problems that cannot be solved analytically. It is also appropriate for students taking courses in numerical approximation techniques.



Chapter 1. Interpolation and Curve fitting

Often one is presented with numerical values of a function f(x) at specified values of x. Experimental results are often presented in a table as a set of discrete data points. When data is presented in this way, the values of the function at points not given in the table must be found by some numerical technique.
Harold Cohen

Chapter 2. Zeros of a Function

In this chapter, we present methods for finding the zeros of f(x) when f(x) is a polynomial. By the time one has finished high school, the methods for finding the roots of first- and second order polynomials have been learned. It is well known that it is not possible to solve for the roots of a polynomial in f(x) in terms of the coefficients of x for a polynomial of order N ≥ 5.
Harold Cohen

Chapter 3. Series

We consider a sum of terms written as
$$ S(z) \equiv \sum\limits_{{n = {n_0}}}^N {{\sigma_n}(z)} = {\sigma_{{{n_0}}}}(z) + {\sigma_{{{n_0} + 1}}}(z) +... + {\sigma_N}(z) $$
When n 0 and N are both finite integers, S(z) is a finite sum or simply a sum. If n 0 and N are not both finite integers (that is n 0=–∞ and/or N=∞), then S(z) is called an infinite series or simply a series.
Harold Cohen

Chapter 4. Integration

In this chapter, we present methods for approximating an integral that cannot be evaluated exactly.
Harold Cohen

Chapter 5. Determinants and Matrices

It is assumed that the reader has been introduced to the fundamental properties of determinants. Appendix 4 presents many of these properties that will be used to develop methods of evaluating determinants.
Harold Cohen

Chapter 6. Ordinary First Order Differential Equations

The general form of an ordinary first order differential equation is
$$ F\left( {x,y,\frac{{dy}}{{dx}}} \right) = F\left( {x,y,y^{\prime}} \right) = 0 $$
Harold Cohen

Chapter 7. Ordinary Second Order Differential Equations

The general form of an ordinary second order differential equation is
$$ F\left( {x,y,y^{{\prime}},y^{{\prime}{\prime}}} \right) = 0 $$
Harold Cohen

Chapter 8. Partial Differential Equations

Many of the partial differential equations that describe physical systems involve derivatives with respect to space variables (x,y,z) or with respect to space and time variables (x,y,z,t). Such equations include the diffusion equation which describes the spread (diffusion) of energy throughout a material medium with diffusion factor K
$$ \frac{{\partial \psi (x,y,z,t)}}{{\partial t}} - {\rm K}{\nabla^2}\psi (x,y,z,t) = f(x,y,z,t) $$
the wave equation which describes the propagation of a wave traveling at speed c
$$ \frac{{{\partial^{ 2}}\psi (x,y,z,t)}}{{\partial {t^2}}} - {c^2}{\nabla^2}\psi (x,y,z,t) = f(x,y,z,t) $$
and Poisson’s equation which describes the electrostatic potential at any point in space due to a distribution of charge, the properties of which are embodied in ρ(x,y,z), the charge density (charge per unit volume).
$$ {\nabla^2}\psi (x,y,z) = \frac{{\rho (x,y,z)}}{{{\varepsilon_0}}} $$
Harold Cohen

Chapter 9. Linear Integral Equations in One Variable

The general form of an integral equation for the unknown function of one variable ψ(x) is
$$ A(x)\psi (x) = {\psi_0}(x) + \lambda \int_{a(x)}^{b(x)} {J\left[ {x,y,\psi (y)} \right]dy} $$
Harold Cohen


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