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Scope and Basic Concepts
Published in William G. Gray, Anton Leijnse, Randall L. Kolar, Cheryl A. Blain, of Physical Systems, 2020
William G. Gray, Anton Leijnse, Randall L. Kolar, Cheryl A. Blain
This is also a book on the use of generalized functions in the context of theorem development. In short, a systematic framework may be defined in which generalized functions are used to derive integral theorems. In a very real sense, generalized functions are mathematical catalysts - they facilitate the derivations but they do not appear in the end product. As in the context of chemistry wherein a reaction would not take place without the presence of a catalyst, the derivations here rely on the catalytic activity of the generalized functions. The advantage of this approach is that theorems for complicated geometries, such as contact lines in a multiphase fluid, become tractable. In fact, the mechanics of the proofs of each theorem are the same regardless of the complexity of the geometry or the dimensionality of the derivative operator, although the derivations become lengthier with system complexity. The exposition on generalized functions extends the familiar one-dimensional Dirac delta function and Heaviside step function, neither of which is a “function” in the strict sense of the word, to their multidimensional analogs. Additionally, the derivative and integral properties of these quantities are provided with emphasis on their power to interconvert integrals over curves, surfaces, and volumes.
Transforms: Laplace, Fourier, Z, and Hilbert
Published in A. David Wunsch, ® Companion to Complex Variables, 2018
The Dirac function has, if we permit ourselves the use of generalized functions, a first derivative. Generalized functions include the Dirac function and/or any of its derivatives. In the language of MATLAB, we have that ddtdirac(t) = dirac(t,1), where the notation dirac(t,1) means simply the first derivative of the Dirac function. MATLAB can handle higher-order derivatives of the Dirac function and uses dirac(t, n) to mean the nth derivative of dirac(t). If the number n is left unspecified, it is taken as zero—no derivative is taken. Just as the function dirac(t − τ), where τ is real, samples the value of a function that is continuous at τ when we do an integration (see Equation 6.5), the function dirac(t−τ, 1) will sample the negative of the first derivative of a function with a continuous first derivative. Thus, () ∫−∞∞dirac(t−τ,1)f(t)dt=−f′(τ)
Basic equations for electromagnetic fields
Published in G. Someda Carlo, Electromagnetic Waves, 2017
The theory to be presented here deals only with phenomena on a macroscopic scale, where consequences of the discrete nature of the electric charge are irrelevant. Therefore, we shall model charge in terms of a function of spatial coordinates P and time t, which we denote as ρ(P, t) and refer to as electric charge density. Throughout this book, the phrase “charge density” will mean the so-called free charge density, i.e., the local imbalance between densities of positive and negative charges, even though, on a microscopic scale, charges of opposite signs cannot be located exactly at the same points. In order not to make the text unnecessarily cumbersome, the term “function” (of spatial coordinates and/or time) will also implicitly include the case of a generalized function. Those cases where this is not acceptable will be outlined explicitly. A typical example of generalized function, that we will use soon, is a point charge q(P0) = qδ(P − P0), where δ(r) stands for a three-dimensional Dirac delta function.
Approximate solutions of time and time-space fractional wave equations with variable coefficients
Published in Applicable Analysis, 2018
Miloš Japundžić, Danijela Rajter-Ćirić
Also, we solve the problem in the framework of the Colombeau theory of generalized functions since our intention is to treat these equations by using an operator approach in contrast to commonly used numerical methods, that is, applying the solution operators as a generalization of semigroup of operators and cosine operators. Primarily, the theory of Colombeau generalized functions is developed in order to make possible studying some nonlinear differential equations that cannot be treated neither classically (there is no classical solution) nor in distributional sense (nonlinear problems include the multiplication and the multiplication of distribution is not well defined). For the Colombeau theory in general we refer, for example, to [2–5].
Universal approximation with neural networks on function spaces
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2022
Wataru Kumagai, Akiyoshi Sannai, Makoto Kawano
Thus, is exactly represented by a bounded affine map, if the Dirac delta function is allowed. However, the Dirac delta function is not a function but a generalised function. Here, the Dirac delta can be approximated by a smooth function called a mollifier, with any precision. Thus, instead of , we take as . We can verify that is approximated by a bounded affine map with precision.