Arpad Takači

A note on the distributional Stieltjes transformation

In this note we use the notion of the quasiasymptotic behaviour of distributions (introduced in [ 2 ]) in order to obtain a final value Abelian theorem for the distributional Stieltjes transformation. At the end of the note we give a few examples in which two different concepts of the asymptotic behaviour of distributions are compared.

On the operational solutions of fuzzy fractional differential equations

Fuzzy fractional differential equations with fuzzy coefficients are analyzed in the frame of Mikusiński operators. Systems of fuzzy operational algebraic equations are obtained, in view of the definition of fuzzy derivatives. Their exact and approximate solutions are constructed and their characters are analyzed, considering them as the corresponding solutions of the given problem. The described procedure of the construction of solutions is illustrated on an example and the obtained approximate solutions of the considered problems are visualized by using the GeoGebra software package.

Partial Differential Equations through Examples and Exercises

Preface. List of Symbols. 1. Introduction. 2. First Order PDEs. 3. Classification of the Second Order PDEs. 4. Hyperbolic Equations. 5. Elliptic Equations. 6. Parabolic Equations. 7. Numerical Methods. 8. Lebesgues Integral, Fourier Transform. 9. Generalized Derivative and Sobolev Spaces. 10. Some Elements from Functional Analysis. 11. Functional Analysis Methods in PDEs. 12. Distributions in the Theory of PDEs. Bibliography. Index.

The g-operational calculus

Using the g-calculus, introduced earlier by E. Pap, the g-operator field Fg is introduced analogously to the classical field of Mikusinski operators F. The obtained results enable the solving of a certain nonlinear PDE of the Burgers type in a wider framework of the generalized functions, which is a generalization of the earlier obtained result.

ON THE DISTRIBUTIONAL STIELTJES TRANSFORMATION

This paper is concerned with some general theorems on the distributional Stieltjes transformation. Some Abelian theorems are proved.

On convolutions and neutrix convolutions involving the incomplete gamma function

The incomplete Gamma function γ(α, x) and its associated functions γ(α, x +) and γ(α, x −) are defined as locally summable functions on the real line for α > 0 and as distributions for α < 0, α ≠ −1, −2, …. Some convolutions and neutrix convolutions of γ(α, λ x −) and the functions eμ x xr and are then found. E-mail: takaci@im.ns.ac.yu E-mail: biljana@etf.ukim.edu.mk

CHAPTER 30 – Generalized Derivatives*

This chapter elaborates the different types of generalized derivatives. The types of generalized derivatives include Sobolev type, distributional one, and the one appearing in the Mikusinski operational calculus. Some elements from the theory of Sobolev and distribution spaces, as well as the construction of the Mikusinski operator field are discusses in the chapter. It is found that if all first-order generalized derivatives of a function exist and are zero, then this function is equal to a constant almost everywhere. It is observed that things are different with the generalized derivative giving an essential difference between the classical and generalized derivatives. A necessary and sufficient condition for a function to have a generalized derivative is that it is absolutely continuous. The last supposition of the existence of the derivative does not imply the absolute continuity of the function. In the upper analysis, one can take the space whose elements are square integrable on every compact set. The statements giving relations between Sobolev spaces and spaces of continuously differentiable functions are usually called “imbedding theorems.” The Mikusinski operational calculus is also elaborated in the chapter.

Elementary analysis through examples and exercises

Preface. 1. Real numbers. 2. Functions. 3. Sequences. 4. Limits of functions. 5. Continuity. 6. Derivatives. 7. Graphs of functions. Bibliography. Index.

Wavelet transforms in generalized Fock spaces

A generalized Fock space is introduced as it was developed by Schmeelk [1-5], also Schmeelk and Takaci [6-8]. The wavelet transform is then extended to this generalized Fock space. Since each component of a generalized Fock functional is a generalized function, the wavelet transform acts upon the individual entry much the same as was developed by Mikusinski and Mort [9] based upon earlier work of Mikusinski and Taylor [10]. It is then shown that the generalized wavelet transform applied to a member of our generalized Fock space produces a more appropriate functional for certain appfications.

Operational and approximate solutions of a fractional integro-differential equation

We dedicate this paper to the 100th anniversary of the birth of Professor Jan Mikusinski. A time-fractional integro-differential equation is analyzed in the frames of the Mikusinski calculus. We present a method for obtaining the exact and the approximate operational solution. A numerical example and some plots are provided as illustration.