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This course introduces functional equations. First we motivate the course using the formulas for the area of a rectangle and a trapezoid, showing that they are not the most correct ones. We give the simple and compound interest formulas together with an interesting interpretation, in terms of account stability. Next, we introduce some classical functional equations, as the associative, the Cauchy's and the Pexider's equations. We continue with the sum of product equations and two examples of applications. Next, we provide a list of ten methods to solve functional equations and illustrate them with some examples. In the final part, we give interesting examples of applications to Economy, Statistics, fatigue, laws of Science and differential equations.
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In this course we introduce functional equations. Some illustrative examples of the real practice are used to present the most common functional equations and other not so common and their general solutions are given. A list of methods to solve functional equations are indicated and explained together with some illustrative examples.
Functional Equations
Enrique Castillo
Motivating functional equations
In this lesson we provide some classical functional equations. We start with an encryption equation and we continue with the bisymmetry, the associativity and the generalized associativity equations. Since we want the readers to connect these equations with functional networks, we also include some illustrative graphics and expect the reader to enrich from this use of functional equations. Finally, we end with the Cauchy's I, II and III and the Pexider's I, II and III equations. Some examples of application are given.
Some Classical functional equations
In this lesson we deal with a special functional equation that allow us to build some interesting models. The name comes from the fact that the equation is the sum of products of functions of x and functions of y. In particular, we present the problem of designing a cover of a sports building suvh that the sections parallel to the coordinate axes are second degree polynomials, and the most general bivariate distribution with normal conditionals, that is an important problem. It can be generalized to other conditional families.
The Sum of Products functional equation
In this lesson we continue with some examples of practical applications that lead to functional equations. We combine the theory and the practice to make it more enjoyable and to motivate the students."; We start with the translation equation which is motivated with damage accumulation curves that must satisfy this equation. We continue with the Euler formula for polyhedra and iterative methods. Finally, we present some functional equations that arise in expert systems and artificial intelligence and present a model for medical diagnosis.
Other functional equations examples
In this lesson we describe some methods to solve functional equations. The reader can take the opportunity to learn some techniques that are very useful to solve real life problems. The list of methods include replacing variables by given values, transforming variables, transforming functions, using more general equations, derivation, integration, inductive and iterative methods, separation of variables and mixed methods. Some illustrative examples are given to clarify how the different methods work.
Some Methods to solve functional equations
This block provides a series of economic problems that can be stated as functional equations, which reproduce important properties to be satisfied. Next, they are solved to obtain the most general models that satisfies the stated conditions. In particular we study some interest and taxation problems and some monopoly and duopoly problems.
Applications to Economy
Some probability and statistical problems are stated in terms of functional equations and solved. Some examples are the case of Bayesian conjugate distributions, the sum of a random number of random variables, the maximum stability distribution models, etc.
Applications of functional equations to Statistics
This block presents some fatigue models that are derived using functional equations. First, we give a model based on the independence assumption and later, we give three models assuming possible dependence among the strengths of the small pieces that make the whole longitudinal element. Finally, we discuss the problem of obtaining a consensus model.
Fatigue Models
In this lesson we discuss the validity of physical formulas. We discover the conditions for physical formulas to be valid, that is, invariant with respect to changes of scale and in some cases changes of origin. Functional equatio ns are the ideal tool to state ythis problem.
Laws of Science
In this lesson we show how functional equations can be an advantageous alternative to differential equations. We will use a beam example to show how the classical statement of differential equation problems, based on a balance with a differential element, can be replaced with the balance of a finite element, to get the alternative functional equation. We also will try to emphasize the similitudes of the two resulting problems
Functionals versus differential equations
Here are all the slideshows from Functional Equations
Slideshows