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dc.contributor.authorMirumbe, Geoffrey Ismail
dc.date.accessioned2014-08-06T11:09:52Z
dc.date.available2014-08-06T11:09:52Z
dc.date.issued2012-04
dc.identifier.citationMirumbe, G.I. (2012). Distribution solutions to ordinary differential equations with polynomial coefficients on the real line (Unpublished doctoral dissertation). Makerere University, Kampala, Ugandaen_US
dc.identifier.urihttp://hdl.handle.net/10570/3859
dc.descriptionA Dissertation submitted to the Directorate of Research and Graduate Training in fulfillment of the requirements for the award of the Degree of Doctor of Philosophy (PhD) in Mathematics of Makerere Universityen_US
dc.description.abstractIn this study we consider the Weyl algebra A1(C) that consists of differential operators P of the form P = pm(x)"m + pm−1(x)"m−1 + . . . + p1(x)" + p0(x), (1) where {p"(x)} are polynomials which in general may have complex coefficients and " is the first order derivation operator with respect to x. To the differential operator P in (1), we associate a differential equation, P(μ) = 0 where μ is either an analytic or a non-analytic function. If the leading polynomial pm(x) has no zeros on the real line (elliptic case) then Cauchy’s result gives m classical solutions. Otherwise, in the non-elliptic case, we have to take into account solutions that are generalized functions (distributions). Our main interest is to establish the dimension of the vector space of the distributional solutions. In this case, we assume that pm(x) has a finite number of real zeros with finite multiplicities and with the degree at most k i.e in the range 1 $ k $ m. With this setting, we will assume that we can re-arrange the corresponding P such that it is locally Fuchsian at each real zero a" of pm(x) withmultiplicitye" ! 1. To each such locally Fuchsian P, we demonstrate the existence of distribution solutions by use of particular cases of P. We state and prove a theorem relating the order of the operator P, the multiplicities of the real zero and the dimension of the P-kernel on the space of distributions Db. Furtherwe prove that for eachreal zero a" of pm(x) there exists a distribution μ supported by the half-line {x ! a"} such that the distribution P(μ) is the Dirac measure at a". Such a μ is called a fundamental solution, and may be used to give a general solution of the differential equation P(μ) = f where f can be either a real analytic function or a generalized function. We also use the theory of boundary values (boundary value distributions) and the Cauchy transform of distributions with compact support to prove coexistence of solutions in the form of the finite order linear combination of the Dirac delta function and its derivatives and rational function solutions to ordinary differential equations with polynomial coefficients.en_US
dc.description.sponsorshipInternational Science Program (ISP) through the East African Universities Mathematics Program (EAUMP)en_US
dc.language.isoenen_US
dc.publisherMakerere Universityen_US
dc.subjectOrdinary differential equationsen_US
dc.subjectPolynomial coefficientsen_US
dc.subjectReal lineen_US
dc.subjectDistribution solutionsen_US
dc.titleDistribution solutions to ordinary differential equations with polynomial coefficients on the real lineen_US
dc.typeThesisen_US


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