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Konstrukcija spektralnih metod z neperiodičnimi trigonometričnimi vrstami : doktorska disertacija
Perne, Andrej (Author), Orel, Bojan (Mentor) More about this mentor... This link opens in a new window

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Abstract
Ortogonalni polinomi so poleg Fourierove vrste eno izmed orodij, ki se v teoriji aproksimacije najpogosteje uporablja. Posebne lastnosti trigonometričnih funkcij in polinomov zagotavljajo učinkovito računanje ter stabilne in konvergentne numerične rešitve. Spektralne metode so poleg metod končnih razlik in končnih elementov pomembno orodje za reševanje robnih problemov tako pri navadnih kot pri parcialnih diferencialnih enačbah. V prvem delu doktorske disertacije so opisane osnovne lastnosti Fourierove vrste in ortogonalnih polinomov ter nekateri osnovni pristopi za konstrukcijo spektralnih metod z osnovnimi orodji za analizo konvergence in napake. V nadaljevanju sta predstavljeni dve neklasični družini ortogonalnih polinomov, tj. poldomenski polinomi Čebiševa prve in druge vrste ter pripadajoča poldomenska Čebišev-Fourierova vrsta. Obe družini sta konstruuirani z uporabo modificiranega algoritma Čebiševa za izračun rekurzivnih koeficientov v tričlenski rekurzivni formuli. Aproksimacija s kvadratom integrabilnih funkcijs poldomensko Čebišev-Fourierovo vrsto vrne primerljive rezultate kot aproksimacija s Fourierovo vrsto ali z vrsto Čebiševa. V osrednjem delu doktorske disertacije je konstruiran nov razred Čebišev-Fourierovih kolokacijskih spektralnih metod za reševanje linearnih dvo- točkovnih robnih problemov z Dirichletovimi robnimi pogoji, kjer numerično rešitev problema iščemo v obliki odrezane poldomenske Čebišev-Fourierove vrste, spektralne koeficiente pa izračunamo z metodo kolokacije. Analiza konvergence in napake pokaže, da so te metode primerljive s standardnimi, kjer iščemo rešitev v obliki Fourierove vrste za periodične ali v obliki vrste Čebiševa za neperiodične probleme. Nov razred metod konstruiramo tudi za nekatere evolucijske robne probleme, tj. za posplošene toplotne in valovne enačbe. Numerični zgledi potrjujejo teoretične rezultate in prikazujejo primerljivost napake numerične rešitve dobljene z novimi ali s standardnimi metodami. Kljub temu pa je računska zahtevnost neprimerljiva, saj v primeru poldomenske Čebišev-Fourierove vrste ni na voljo orodja za izračun koeficientov, ki bi bilo primerljivo s hitro Fourierovo transformacijo.

Language:Slovenian
Keywords:numerična analiza, spektralne metode, ortogonalni polinomi, dvotočkovni robni problemi, kolokacija, aproksimacija, poldomenski polinomi Čebiševa prve vrste, poldomenski polinomi Čebiševa druge vrste, poldomenska Čebišev-Fourierova vrsta, posplošena toplotna enačba, posplošena valovna enačba
Work type:Doctoral dissertation (mb31)
Tipology:2.08 - Doctoral Dissertation
Organization:FMF - Faculty of Mathematics and Physics
Year:2012
Publisher:[A. Perne]
Number of pages:139 str.
UDC:519.62(043.3)
COBISS.SI-ID:16539993 This link opens in a new window
Views:1529
Downloads:386
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Secondary language

Language:English
Abstract:
Orthogonal polynomials are, along with Fourier series, one of the most widely used tools in the theory of approximation. Specific properties of trigonometric functions and polynomials assure effcient computation as well as stable and convergent numerical solutions. Spectral methods are, besides finite difference and finite element methods, an important tool for solving boundary value problems for ordinary as well as partial differential equaations. In the first part of the doctoral thesis, some basic properties of the Fourier series and orthogonal polynomials as well as some basic approaches for the construction of spectral methods with fundamental tools for convergence and error analysis are described. In the sequel, two non-classical families of orthogonal polynomials are presented, i.e., the half-range Chebyshev polynomials of the first and second kind as well as the corresponding half-range Chebyshev-Fourier series. Both families are constructed via the modified Chebyshev algorithm used for the computation of the recursive coefficients for the three-term recurrence relation. The approximation of square integrable functions with half-range Chebyshev-Fourierseries yields comparable results to the approximation with Fourier or Chebyshev series. In the central part of the doctoral thesis, a new class of Chebyshev-Fourier collocation spectral methods for solving linear two-point boundary value problems with Dirichlet boundary conditions is constructed. We seek for the numerical solution in the form of the truncated half-range Chebyshev-Fourier series, where spectral coefficients are computed using the collocation method. Convergence and error analysis shows that these methods are comparable with standard ones, where the solution is approximated with the Fourier series for periodic or with the Chebyshev series for non-periodic problems. We construct a new class of methods also for some evolutive boundary value problems, i.e., for generalized heat and wave equations. Numerical examples confirm theoretical results and show the comparability of the error of the numerical solution obtained with the new or the standard methods. Yet, computational costs are not comparable, because in the case of half-range Chebyshev-Fourier series there does not exist a tool for the computation of coefficients being comparable with fast Fourier transform.

Keywords:numerical analysis, spectral methods, orthogonal polynomials, two-point boundary value problems, collocation, approximation, half-range Chebyshev polynomials of the first kind, half-range Chebyshev polynomials of the second kind, half-range Chebyshev-Fourier series, generalized heat equation, generalized wave equation

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