Polynomial solutions of certain differential equations

Sezer, Mehmet; Keşan, Cenk

doi:10.1080/00207160008805011

Cited by 6 publications

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“…Soluções polinomiais para os PVIs já foram propostas e implementadas [Sezer & Kesan, 2000], [Kesan, 2003], [Babolian & Fattahzadeh, 2007] x ⋅ + + = ∈ , tal que (0) 1 e '(0) 0 y y = = .…”

Section: Motivaçãounclassified

“…A aproximação de funções por séries de polinômios de Legendre, ou outras bases ortogonais, demonstrou ser uma ferramenta efetiva na resolução de PVIs, como denotam várias publicações, por exemplo, as referências [Babolian & Fattahzadeh, 2007], [Elbarbary, 2002], [Sezer & Kesan, 2000], [Psihoyios & Simos, 2003].…”

Section: Integral De Uma Função Expressa Na Base De Legendreunclassified

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Computação evolutiva na resolução de equações diferenciais ordinárias não lineares no espaço de Hilbert.

Guimarães¹

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This thesis shows a new method to get polynomial solutions to the initial value problems (IVP), with an error margin comparable to the consecrate numerical methods (NM), for both the function and its derivatives. The method works with differential equations (DEs) linear or not, beeing the developed tolls available until 4 th order, whose can be expanded to higher orders. The solution is a polynomial high degree expression with coefficients expressed by the ratio between two integers. The method behaves efficiently even in some cases that NM cannot get started. The resolutions are gotten considering that, the solution space is a Hilbert space, equipped with a complete set basis of Legendre Polynomials. Due the method here developed, the error's majoratives for the function and its derivatives are found analytically by a matrix calculus, also derived in this thesis. Beside all analytical foundation, a software (SAM) was developed to automate the whole process, joining all the tasks involved in the search for solutions to the IVP. This thesis proposes, verifies and validates a new error criterion, which takes in account simultaneously the local and global errors. As sub-products of the results described before, also integrated to the SAM, the following achievements should be highlighted: (1) An objective criterion to analyze the quality of any NM, despite of the knowledge of its algorithm; (2) A tool for a polynomial approximation, of high precision, for functions whose square is integrable in a given limited domain, with an error's majorative; (3) A tool-kit for a generically transpose (linear or not) of the IVPs' domain and form, taking into account its derivatives, until the 4th order; (4) The generic matrices for integration and differentiation for all the polynomial basis of the Hilbert space.

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Section: Motivaçãounclassified