Numerical solution of fractional differential equations
Date
2020-01-16
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Abstract
Murrulised tuletised (s.t. tuletised, mille järk ei ole täisarv) on pakkunud huvi juba alates ajast, millal I. Newton ja G. W. Leibniz rajasid matemaatilise analüüsi aluseks oleva diferentsiaal- ja integraalarvutuse. Kaua aega käsitleti murruliste tuletistega seotud küsimusi vaid teoreetilisest vaatepunktist, sest ei olnud näha, millised võiksid olla murruliste tuletiste rakendusvõimalused. Viimastel aastakümnetel on aga leitud, et murrulisi tuletisi sisaldavad diferentsiaalvõrrandid kirjeldavad mitmesuguste materjalide ja protsesside käitumist paremini kui täisarvulist järku tuletistega diferentsiaalvõrrandid. Kuna murruliste tuletistega diferentsiaalvõrrandite täpse lahendi leidmine ei ole enamasti võimalik, peame nende lahendeid leidma ligikaudselt. See nõuab spetsiaalsete meetodite väljatöötamist, sest murruliste tuletistega diferentsiaalvõrrandite korral ei ole reeglina rakendatavad täisarvuliste tuletistega diferentsiaalvõrrandite vallast tuntud tulemused. Käesolevas väitekirjas uuritakse murruliste tuletistega diferentsiaalvõrrandi lahendi siledust ja saadud informatsiooni alusel töötatakse välja kõrget järku täpsusega lahendusalgoritmid niisuguste võrrandite ligikaudseks lahendamiseks. Saadud teoreetilisi tulemusi kontrollitakse arvukate numbriliste eksperimentidega mitmesugustel testvõrranditel.
The concept of a fractional derivative can be traced back to the end of the seventeenth century, the time when Newton and Leibniz developed the foundations of differential and integral calculus. Despite this, for a long time, considerations regarding fractional derivatives were purely theoretical treatments for which there were no serious practical applications. It is only during the last decades that there has been a spectacular increase of studies regarding fractional derivatives and differential equations with such derivatives, mainly because of new applications of fractional derivatives in several fields of applied science. However, when working with problems stemming from real-world applications, it is only rarely possible to find the exact solution of a given fractional differential equation, and even if such an analytic solution is available, it is typically too complicated to be used in practice. Therefore numerical methods specialized for solving fractional differential equations are required. In the present thesis the regularity properties of the exact solutions of a wide class of fractional differential and integro-differential equations are investigated. Based on the obtained regularity properties, the numerical solution of the problem is discussed, the convergence of proposed algorithms is proven and their global convergence estimates are derived. The obtained theoretical results are supported by many numerical experiments with various test problems.
The concept of a fractional derivative can be traced back to the end of the seventeenth century, the time when Newton and Leibniz developed the foundations of differential and integral calculus. Despite this, for a long time, considerations regarding fractional derivatives were purely theoretical treatments for which there were no serious practical applications. It is only during the last decades that there has been a spectacular increase of studies regarding fractional derivatives and differential equations with such derivatives, mainly because of new applications of fractional derivatives in several fields of applied science. However, when working with problems stemming from real-world applications, it is only rarely possible to find the exact solution of a given fractional differential equation, and even if such an analytic solution is available, it is typically too complicated to be used in practice. Therefore numerical methods specialized for solving fractional differential equations are required. In the present thesis the regularity properties of the exact solutions of a wide class of fractional differential and integro-differential equations are investigated. Based on the obtained regularity properties, the numerical solution of the problem is discussed, the convergence of proposed algorithms is proven and their global convergence estimates are derived. The obtained theoretical results are supported by many numerical experiments with various test problems.
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splines, differential equations, integral equations