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  • 1
    Online Resource
    Online Resource
    Berlin [u.a.] : Springer
    Part of " Lecture notes in mathematics"
    Keywords: Abelsche Integralgleichung ; Abelsche Integralgleichung
    Type of Medium: Online Resource
    Pages: 1 Online-Ressource
    ISBN: 9783540536680 , 9783540469490
    Series Statement: Lecture notes in mathematics 1461
    DDC: 515
    Language: English
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 14 (1970), S. 448-467 
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Zusammenfassung Für die partielle Differentialgleichung von Fokker und Planck im ℝ n (Vorwärts-Differentialgleichung von Kolmogorov) mit orts- and zeitabhängiger Drift und Diffusionsmatrix werden, unter einigen zusätzlichen Bedingungen, explizite Differenzenschemata angegeben, die zwei wesentliche Eigenschaften des beschriebenen Diffusionsprozesses imitieren, nämlich Nichtnegativitätserhaltung und Substanzerhaltung. Diese Schemata können auch interpretiert werden als Beschreibung diskreter instationärer inhomogener Irrfahrten. Sie sind stabil in der Maximumnorm, und man kann mit ihrer Hilfe bei gegebener Anfangsbedingung die Differentialgleichung nurnerisch approximativ lösen oder aber den durch sie beschriebenen Diffusionsprozeß approximativ simulieren (Monte-Carlo) und veranschaulichen. Wesentliche Bedingungen sind die Beschränktheit der Koeffizienten der Differentialgleichung und einige ihrer Ableitungen und eine starke Diagonaldominanz der Diffusionsmatrix. Die Resultate lassen sich verallgemeinern auf allgemeine lineare parabolische Differentialgleichungen (Substanzerhaltung ist dann i. allg. nicht mehr vorhanden). Die beschriebene Methode erlaubt auch, für lineare elliptische Operatoren im ℝ n Approximationen nichtnegativen Typs anzugeben, wenn diese analogen Bedingungen genügen.
    Notes: Abstract Explicit difference schemes preserving non-negativity and mass as does the described diffusion process are given for the Fokker-Planck partial differential equation (Kolmogorov's forward differential equation) in ℝ n . These schemes can be interpreted as descriptions of discrete non-stationary inhomogeneous random walks. They are stable in the maximum norm, and if an initial condition is given, they allow approximate solution of the differential equation as well as approximate Monte-Carlo simulation of the basic diffusion process. Essential conditions are the boundedness of the coefficients of the differential equation and of some of their derivatives and a strong diagonal dominance of the diffusion matrix. The results can be generalized to general linear parabolic differential equations (conservation of mass being lost, of course). By means of the developed method it is also possible to construct difference approximations of non-negative type to linear elliptic operators in ℝ n if these operators satisfy analogous conditions.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Summary Two-dimensional analogues of the torus-like Meyer-Schmidt plasma configurations without interior field are treated by functiontheoretical methods. If the boundary of the doubly connected plasma domain consists of two closed analytic curves, the structure of the exterior magnetic field and its singularities (which determine the currents) can be found by a combination of analytic continuation and conformal mapping. An example is discussed in detail. The problem is essentially equivalent to a problem in hydrodynamics: to find the structure of an incompressible irrotational flow in two dimensions which has a given closed analytic curve as a free streamline.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 8 (1966), S. 392-406 
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The well-known integral transform $$i(r) = - \frac{1}{\pi }\int\limits_{x = r}^1 {\frac{{dI(x)}}{{\sqrt {x^2 - r^2 } }},} 0 \leqq r \leqq 1,I(1) = 0$$ arising in spectroscopy, corresponds to half-order differentiation by substitutingr 2 = 1 −s,x 2 = 1 − t. Therefore noise is amplified by transforming the measured functionI intoi. Two undesirable effects may arise: (a) lack of smoothness ini (r), (b) intervals in whichi(r) 〈 0, although for physical reasons we should havei(r) ≧ 0. After developing a heuristic theory of noise amplification we present a fitting technique for approximate computation ofi(r), using the extra informationi(r) ≧ 0 as a restriction. This leads to a quadratic programming problem.
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Archiv der Mathematik 12 (1961), S. 113-117 
    ISSN: 1420-8938
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Archiv der Mathematik 12 (1961), S. 188-192 
    ISSN: 1420-8938
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Mathematische Annalen 146 (1962), S. 226-231 
    ISSN: 1432-1807
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Acta applicandae mathematicae 24 (1991), S. 1-27 
    ISSN: 1572-9036
    Keywords: 65M30 ; 35K05 ; Inverse heat conduction ; ill-posed problems ; noncharacteristic Cauchy problem ; mollification method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The noncharacteristic Cauchy problem for the heat equation: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadwhadaWgaaWcbaGaamiEaiaadIhaaeqaaOGaaiikaiaadIha% caGGSaGaamiDaiaacMcacqGH9aqpcaWG1bWaaSbaaSqaaiaadshaae% qaaOGaaiikaiaadIhacaGGSaGaamiDaiaacMcacaGGSaqefeKCPfgB% aGqbbiaa-bcacaaIWaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDOb% cv39gaiyqacqGFKjcHcaWG4bGae4hzIqOae4ha3hJaaeymaiaabYca% caqGTaGaeuOhIuQaeuipaWJaaeiDaiabfYda8iabf6HiLkaacYcaca% WG1bGaaiikaiaaicdacaGGSaGaamiDaiaacMcacqGH9aqpcqqHvpGA% caGGOaGaamiDaiaacMcacaGGSaGaamyDamaaBaaaleaacaWG4baabe% aakiaacIcacaaIWaGaaiilaiaadshacaGGPaGaeyypa0JaaGiYdiaa% cIcacaWG0bGaaiykaiaacYcacaWFGaGaeuOhIuQaeuipaWJaamiDai% abfYda8iabf6HiLcaa!82F8!\[u_{xx} (x,t) = u_t (x,t), 0 \le x \le {\rm{1, - }}\infty 〈 {\rm{t}} 〈 \infty ,u(0,t) = \varphi (t),u_x (0,t) = \psi (t), \infty 〈 t 〈 \infty \]is considered. This problem is well-known to be ill-posed. The well-posedness class of the problem is described and some approximation schemes are proposed. For the case of inexactly given data, a mollification method is suggested.
    Type of Medium: Electronic Resource
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