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# Mathematical modeling with generalized function

## Citation

Kilicman, Adem (2011) Mathematical modeling with generalized function. [Inaugural Lecture]

## Abstract / Synopsis

In recent years there has been a growing interest in setting up the modeling and solving mathematical problems in order to explain numerous experimental findings which are relevant to industrial applications. Distributions also known as generalized functions which generalize classical functions and allow us to extend the concept of derivative to all continuous functions. The theory of distributions have applications in various fields especially in science and engineering where many non-continuous phenomena that naturally lead to differential equations whose solutions are distributions, such as the delta distribution therefore distributions can help us to develop an operational calculus in order to investigate linear ordinary differential equations as well as partial differential equations with constant coefficients through their fundamental solutions. Further, some regular operations which are valid for ordinary functions such as addition, multiplication by scalars are extended into distributions. Other operations can be defined only for certain restricted subclasses; these are called irregular operations. They allow us to extend the concept of derivative to all continuous functions and beyond and are used to formulate generalized solutions of partial differential equations.They are important in physics and engineering where many non-continuous problems naturally lead to differential equations whose solutions are distri- butions, such as the Dirac delta distribution. In this work we aim to show how differential equations arise in the mathematical modeling of certain problems in industry. The focus of the presentations will be on the use of mathematics to advance the understanding of specific problems that arise in industry. For this purpose we let D be the space of infinitely differentiable functions with compact support and let D′ be the space of distributions defined on D then we provide some particular examples how to use the generalized functions in Statistics and Economics. At the end of the study we relate the Tomography and The Radon Transform on using the generalized functions. In mathematical analysis, distributions also known as generalized functions are objects which generalize functions and probability distributions. We apply the distributions to the some mathematical problems. For this purpose we let ρ be a fixed infinitely differentiable function having the following properties: (i) ρ(x) = 0 for|x|≥1, (ii) ρ(x)≥0, (iii) ρ(x) =ρ(−x), (iv) ∫1−1ρ(x)dx= 1 . Define, the function δn by putting δn(x) =nρ(nx) forn= 1,2,...,it follows that{δn(x)} is a regular sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x). Now let D be the space of infinitely differentiable functions with compact support and let D′ be the space of distributions defined on D. Then if f is an arbitrary distribution in D′, we define fn(x) = (f∗δn)(x) =⟨f(t),δn(x−t)⟩for n= 1,2,.... It follows that{fn(x)} is a regular sequence of infinitely differentiable functions converging to the distribution f(x) PDF (Cover)
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