### thinkBS 305 Innovative Mathematical Modeling Techniques: Fractional Calculus, Wavelet Analysis, and Estimating of Nonlinearities

• Offered by: Universitatea Politehnica Din Bucurresti

This course is composed of three theoretical parts with applications that present: two sections which contain the notions of fractional calculus and wavelet analysis, as well as a section of methods for estimating nonlinearities.

Fractional Calculus

The last decades proved that derivatives and integrals of arbitrary order are very convenient for describing properties of real materials (for example, polymers). The new fractional-order models are more satisfying than former integer-order ones. Fractional derivatives are a remarkable tool for describing the memory and hereditary properties of various materials and processes while in integer-order models such effects are neglected. The fractional calculus has significant applications in different fields of science, including the theory of fractals, numerical analysis, physics, engineering, biology, economics, and finance. Also, fractional derivatives and integrals, respectively fractional differential equations are used in the theory of control of dynamical systems to describe the controlled system and the controller.

Wavelet Analysis

As many notions - dynamic system, Fourier analysis, processing algorithms, Shannon information, etc., the notion of signal is claimed not only by communications engineering, but also by mathematics, in communion with computer techniques. The theory of waves ("wavelets") became the subject of scientific research and also of the disciplines to be learned, after 1980; this theory does not replace Fourier analysis, as an extension of it, with special virtues, we mention in this sense the good localization of signals (in time, frequency and scale); decomposition of signals and 2D images into "rocks", with the creation of zooms; digital analysis (A/D), directly related to the digital age in which we have irreversibly entered. Wavelet transform is a mathematical approach widely used for signal processing applications. It can decompose special patterns hidden in mass of data. Regarding the prediction issue through time series and neural networks, we need modeling task. Wavelet transform has the ability to simultaneously display functions and manifest their local characteristics in time-frequency domain.

Estimating of nonlinearities

Some topics in this section that will be discussed: functions approximations by polynomial interpolation, iterative methods for calculating the eigenvalues and eigenvectors of a matrix, estimation methods of probability densities functions, estimation methods for solutions of nonlinear ODEs.

###### References:

1. 1.Podlubny, I. (1999). Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, 198, 1999.
2. 2. Puscasu, S. V., Bibic, S. M., Rebenciuc, M., Toma, A. and Nicolescu, D. St. (2018). Aspects of fractional calculus in RLC circuits, Proc. Of the 3rd International Symposium on Fundamentals of Electrical Engineering (ISFEE), 1-3 November 2018, Bucharest, Romania, DOI: 10.1109/ISFEE.2018.8742421 .
3. 3. Newland, D. E. (1993). Harmonic wavelet analysis, Proc. Royal Society of London, Series A, 443, 203-225.
4. 4. Cattani, C. (2008). Shannon wavelet theory, Mathematical Problems in Engineering, 1-24.
5. 5. Rebenciuc, M., Bibic, S. M. and Toma, A. (2020). Assessment of structural monitoring by analyzing some modal parameters: an extended inventory of methods and developments, Archives of Computational Methods in Engineering, Springer Verlag, 1-16, DOI: 10.1007/s11831-020-09433-1.
6. 6. Bibic, S. M. and Malureanu, E. S. (2013). Wavelet solution of the time independent Scrodinger equation for a rectangular potential barrier, Proc. Of the 8th International Symposium on Advanced Topics in Electrical Engineering (ATEE), 2-25 May2013, Bucharest, Romania, DOI: 10.1109/ATEE.2013.6563472.
7. 7. Bibic, S. M. (2011). Harmonic wavelet analysis – connection coefficients for nonlinear PDE, UPB, Sci. Bull,Series A - Applied Mathematics and Physics, 73(1), 12-35.
8. 8. Cipu, E. C., Bibic, S. M. and Toma, A. (2020). Some applications of mathematical modeling of risk assessment, University Politehnica of Bucharest Scientific Bulletin-Series A - Applied Mathematics and Physics, 82(4), 69-82.
9. 9. Cipu, E. C. (2006). Economics and finance for engineers. Mathematical models. University Publishing House, Bucharest, ISBN 973-749-052-5, ISBN 978-973-052-0.
10. 10. Dautray, R. and Lions, J. L. (2000). Mathematical Analysis and Numerical Methods for Science and Technology, Springer Verlag, 2000.

###### Teachers:

The lecturers of this course are as follows: Simona Mihaela BIBIC, Elena Corina CIPU, Mihai Rebenciuc, Carmina GEORGESCU, Emil SIMION and Antonela TOMA from the Center for Research and Training in Innovative Techniques of Applied Mathematics in Engineering “Traian Lalescu” (CiTi), University Politehnica of Bucharest.