Functional Equations in Applied Sciences
The book provides the reader with the different types of functional equations that s/he can find in practice, showing, step by step, how they can be solved.
A general methodology for solving functional equations is provided in Chapter 2. The different types of functional equations are described and solved in Chapters 3 to 8. Many examples, coming from different fields, as geometry, science, engineering, economics, probability, statistics, etc, help the reader to change his/her mind in order to state problems as functional equations as an alternative to differential equations, and to state new problems in terms of functional equations or systems.
An interesting feature of the book is that it deals with functional networks, a powerful generalization of neural networks that allows solving many practical problems. The second part of the book, Chapters 9 to 13, is devoted to the applications of this important paradigm.
The book contains many examples and end of chapter exercises, that facilitates the understanding of the concepts and applications.
ˇ A general methodology for solving functional equations is provided in Chapter 2.
ˇ It deals with functional networks, a powerful generalization of neural networks.
ˇ Many examples, coming from different fields, as geometry, science, engineering, economics, probability, statistics, etc, illustrate the concept of functional equation.
ˇ Functional equations are presented as a powerful alternative to differential equations.
ˇ The book contains end of chapter exercises.
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Equations with Several Functions in One Variable
Equation for One Function of Several Variables
Equations with Functions of Several Variables
Functional Equations and Differential Equations
Vector and Matrix Equations
Applications of Functional Equations
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Aczél applied approximate arbitrary constants arbitrary continuous arbitrary function B-spline Bézier Bézier surface bivariate Castillo Cauchy’s equation Chapter characterize consider constant coefficients continuous and strictly continuous solution continuous-at-a-point Corollary cumulative distribution function curves depends difference equation differential equation domain equa errors Example Finally following theorem formula func function f functional equation functional equation f(x functional network given homogeneous I(po implies income Initial topology inputs knot vectors leads Lemma linearly independent matrix methods monotonic function network in Figure neural functions neural networks neuron Note obtain output polynomial probability density function problem Proof random variables real numbers rectangle respectively RMSE satisfies Section sets of functions shows solution of Equation Solve the functional Step strictly monotonic function substitution surfaces survivor function system of functional Table tax function tion transformation uniqueness of representation units unknown functions
Page 10 - R,R+ and R++ are the sets of real numbers, the set of non-negative real numbers and the set of positive real numbers, respectively. • Definition 1.2 (System of functional equations). A system of functional equations is a set of n > 2 functional equations. • Example 1.3 (Systems of functional equations). Examples of systems of functional equations are...