Kiired ja kvaasikiired lahendusmeetodid nõrgalt singulaarsete Fredholmi teist liiki integraalvõrrandite jaoks

Doktoritöös käsitletakse lineaarsete Fredholmi teist liiki integraalvõrrandite ligikaudse lahendamisega seotud probleeme situatsioonis, kus võrrandi tuum võib argumentide kokkulangemise korral olla iseärane (nõrgalt singulaarne). Tuuma iseärasus toob reeglina kaasa integraalvõrrandi lahendi iseärase käitumise integreerimispiirkonna raja lähedal ning raskused kiirete lahendusmeetodite konstrueerimisel niisuguste võrrandite jaoks. Töö põhitulemuseks on kiirete ja kvaasikiirete meetodite väljatöötamine nimetatud võrrandite korral. Kiire meetod tähendab siin meetodit võrrandi lähislahendite leidmiseks, mis antud ülesannete klassi korral annab lähislahenditele optimaalset järku täpsuse võimalikult väikese aritmeetiliste tehete arvu korral. Vajalikud veahinnangud on saadud lähteülesande periodiseerimise kaudu, mille puhul integraalvõrrandi lähislahendite leidmine taandub perioodiliste funktsioonide aproksimeerimisele trigonomeetriliste polünoomide abil.In the present thesis the bounds of fast solving Fredholm integral equations of the second kind with a possible weak diagonal singularity of the kernel and certain boundary singularities of the derivatives of the free term has been discussed in a situation when the information about the smooth coefficient functions in the kernel and about the free term is restricted to a given number of their sample values. In a fast solver, the conditions of optimal accuracy and minimal arithmetical operations (complexity of the solver) are met. We mean the order optimality and order minimal work on a class of problems; the class of problems is defined by the smoothness conditions which have been set on the kernel and free term of the underlying problem.https://www.ester.ee/record=b536058

Fractional Calculus and Special Functions with Applications

The study of fractional integrals and fractional derivatives has a long history, and they have many real-world applications because of their properties of interpolation between integer-order operators. This field includes classical fractional operators such as Riemann–Liouville, Weyl, Caputo, and Grunwald–Letnikov; nevertheless, especially in the last two decades, many new operators have also appeared that often define using integrals with special functions in the kernel, such as Atangana–Baleanu, Prabhakar, Marichev–Saigo–Maeda, and the tempered fractional equation, as well as their extended or multivariable forms. These have been intensively studied because they can also be useful in modelling and analysing real-world processes, due to their different properties and behaviours from those of the classical cases.Special functions, such as Mittag–Leffler functions, hypergeometric functions, Fox's H-functions, Wright functions, and Bessel and hyper-Bessel functions, also have important connections with fractional calculus. Some of them, such as the Mittag–Leffler function and its generalisations, appear naturally as solutions of fractional differential equations. Furthermore, many interesting relationships between different special functions are found by using the operators of fractional calculus. Certain special functions have also been applied to analyse the qualitative properties of fractional differential equations, e.g., the concept of Mittag–Leffler stability.The aim of this reprint is to explore and highlight the diverse connections between fractional calculus and special functions, and their associated applications

Singularly perturbed volterra integral equations

This thesis studies singularly perturbed Volterra integral equations of the form eu(t)=/(t,e)+fg(t,s,11(5)) ds, 00 is a small parameter The function f(t,e) is defined for 00. The aim is to find asymptotic approximations l to these solutions. This work is restricted to problems where there is an imtial-layer, various hypotheses are placed on g(tts,u) to exclude other behaviour. A major part of this work is that formal solutions of the nonlinear problem are determined and rigorously proved to be asymptotic approximations to the exact solutions Formal approximate solutions: £Mi,e) NEn-0 Enu(t,E,),=un(t,e)=0(1) as e->0, are obtained using the additive decomposition method Algorithms which improve the method used m Angell and Olmstead (1987), are presented for obtaining these solutions Assuming a stability condition in the boundary layer, it shown that there is a constant Cjv such ,that \u(t,e)-Ujsf{t,e)\ <cN eN+1 as e 0, uniformly for t € [0,T], thus establishing that £/jv(i,£) is an asymptotic solution SIkinner (1995) has proved similar results, but almost all the theorems here were discovered before Skinner’s work was found and are largely independent of it Lange and Smith (1988) prove results for the case g(t, syu) = k(tys)u, where k(t, s) is continuous and satisfies a stability condition in the boundary layer These results are carefully developed here and similar results for linear integrodifferential equations The problem of extending these to the class of weakly singular equations with g(t,s,u)= k(t,s)/(t-s)BU-0</3<1 is discussed An interesting aspect of this problem and others for which the boundary 1 layer stability condition fails, is that the solutions decay algebraically rather exponentially within ¡the boundary layer

On the Influence of Multiplication Operators on the Ill-posedness of Inverse Problems: Zum Einfluss von Multiplikationsoperatoren auf die Inkorrektheit Inverser Probleme

In this thesis we deal with the degree of ill-posedness of linear operator equations in Hilbert spaces, where the operator may be decomposed into a compact linear integral operator with a well-known decay rate of singular values and a multiplication operator. This case occurs for example for nonlinear operator equations, where the local degree of ill-posedness is investigated via the Frechet derivative. If the multiplier function has got zeroes, the determination of the local degree of ill-posedness is not trivial. We are going to investigate this situation, provide analytical tools as well as their limitations. By using several numerical approaches for computing the singular values of the operator we find that the degree of ill-posedness does not change through those multiplication operators. We even provide a conjecture, verified by several numerical studies, how these multiplication operators influence the singular values of the operator equation. Finally we analyze the influence of those multiplication operators on the opportunities of Tikhonov regularization and corresponding convergence rates. In this context we also provide a short summary on the relationship between nonlinear problems and their linearizations.Diese Arbeit beschaeftigt sich mit dem Grad der Inkorrektheit linearer Operatorgleichungen in Hilbertraeumen, die sich als Komposition eines vollstetigen linearen Integraloperators mit bekannter Abklingrate der Singulaerwerte und eines Multiplikationsoperators darstellen lassen. Dieser Fall tritt beispielsweise bei nichtlinearen Operatorgleichungen auf, wobei der lokale Inkorrektheitsgrad ueber die Frechetableitung bestimmt wird. Falls die Multiplikatorfunktion Nullstellen hat, so ist die Bestimmung des lokalen Grades der Inkorrektheit nicht einfach. Moeglichkeiten und Grenzen der Analysis fuer diese Situation werden betrachtet. Unterschiedliche numerische Ansaetze fuer die Bestimmung der Singulaerwerte liefern, dass der Grad der Inkorrektheit durch die Multiplikationsoperatoren nicht veraendert wird. Es wird sogar ein Zusammenhang angegeben, wie Multiplikationsoperatoren die Singulaerwerte beeinflussen. Schliesslich werden Moeglichkeiten der Tikhonov-Regularisierung unter Einfluss der Multiplikationsoperatoren untersucht. In diesem Zusammenhang wird auch eine kurze Zusammenfassung zur Beziehung von nichtlinearen Problemen und ihren Linearisierungen gegeben

Spectral geometry of partial differential operators

Access; Differential; Durvudkhan; Geometry; Makhmud; Michael; OA; Open; Operators; Partial; Ruzhansky; Sadybekov; Spectral; Suraga