Rational spline collocation for boundary value problems
Date
2013-05-24
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Abstract
Paljud matemaatika, füüsika ja teiste teadusalade probleemid on formuleeritavad rajaülesannete kujul. Traditsioonilised meetodid rajaülesannete lahendamiseks on võrgumeetod, mis annab ainult diskreetse lahendi, ja polünomiaalsete splainidega kollokatsioonimeetod. Viimast on teist järku harilike diferentsiaalvõrrandite rajaülesannete lahendamisel suhteliselt hästi uuritud. Interpoleerimise korral on teada, et ratsionaalsplainid lähendavad mõningaid funktsioone paremini kui polünomiaalsed splainid. Seetõttu võivad ratsionaalsplainid anda mõnedes rajaülesannetes paremaid tulemusi. Erinevalt ruut- ja kuupsplainidest säilitavad ratsionaalsplainid ka geomeetrilisi omadusi, täpsemalt lineaar/lineaar ratsionaalsplainid monotoonsust ja ruut/lineaar ratsionaalsplainid kumerust. Viimasel paaril aastakümnel on rakendusmatemaatikute suuremat tähelepanu pälvinud just ülesanded, kus täpsel lahendil on mingid geomeetrilised omadused nagu positiivsus, monotoonsus, kumerus ja samu omadusi nõutakse ka lähislahendilt. Käesoleva töö põhiprobleemiks ongi uurida hariliku teist järku diferentsiaalvõrrandi rajaülesande lahendamist lineaar/lineaar ja ruut/lineaar ratsionaalsplainidega kollokatsioonimeetodil, mis on oma olemuselt mittelineaarsed meetodid. Tulemusi võrreldakse vastavalt ruut- ja kuupsplain-kollokatsioonimeetodite tulemustega. Eelnevalt on töös uuritud ratsionaalsplainidega interpoleerimist, sest ratsionaalsplainidega kollokatsioonimeetodi tulemused põhinevad spetsiaalsete ratsionaalsplaininterpolantide teatud omadustel. Toodud on ka illustreerivad arvutuseksperimendid.
Boundary value problems arise in several branches of scientific and engineering problems. Traditional methods for approximate solution of boundary value problems are finite difference method which only gives a discrete solution, and collocation method with polynomial splines. The latter one has been quite well studied. It is known, that in interpolation in some cases the rational splines may have better results compared to polynomial ones. In such circumstances, it is natural to pose the question about rational spline collocation method for boundary value problems. In recent years the shape preserving problems have been considered by several authors. Polynomial splines do not preserve the monotonicity or convexity. On the other hand, the linear/linear rational spline is constant or strictly monotone and the quadratic/linear rational spline is always convex (or concave). Therefore, it is a reasonable approximate solution only if the exact solution of the problem has same properties. The main purpose of the thesis is to study the linear/linear and quadratic/linear rational spline collocation method for boundary value problems and also compare them to quadratic and cubic spline cases, respectively. The collocation method with rational splines is, in nature, a nonlinear method because it leads to a nonlinear system with respect to the spline parameters. Also the rational spline interpolation has been studied because the results of rational spline collocation method are based on certain convergence properties of interpolating rational splines. Numerical examples support the obtained theoretical results.
Boundary value problems arise in several branches of scientific and engineering problems. Traditional methods for approximate solution of boundary value problems are finite difference method which only gives a discrete solution, and collocation method with polynomial splines. The latter one has been quite well studied. It is known, that in interpolation in some cases the rational splines may have better results compared to polynomial ones. In such circumstances, it is natural to pose the question about rational spline collocation method for boundary value problems. In recent years the shape preserving problems have been considered by several authors. Polynomial splines do not preserve the monotonicity or convexity. On the other hand, the linear/linear rational spline is constant or strictly monotone and the quadratic/linear rational spline is always convex (or concave). Therefore, it is a reasonable approximate solution only if the exact solution of the problem has same properties. The main purpose of the thesis is to study the linear/linear and quadratic/linear rational spline collocation method for boundary value problems and also compare them to quadratic and cubic spline cases, respectively. The collocation method with rational splines is, in nature, a nonlinear method because it leads to a nonlinear system with respect to the spline parameters. Also the rational spline interpolation has been studied because the results of rational spline collocation method are based on certain convergence properties of interpolating rational splines. Numerical examples support the obtained theoretical results.
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Väitekirja elektrooniline versioon ei sisalda publikatsioone.
Keywords
harilikud diferentsiaalvõrrandid, rajaülesanded, arvutusmeetodid, splainid, ordinary differential equations, boundary value problems, numerical methods, splines