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Approximative-Iterative Method for Solving Non-Linear Differential and Integral Equations

  • Chapter
Algebraic and Geometric Methods in Mathematical Physics

Part of the book series: Mathematical Physics Studies ((MPST,volume 19))

Abstract

In 1890 Ch. Picard developed a method to obtain existence and uniqueness theorems for a wide class of differential and integral equations of the type

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaafa % Gaeyypa0JaamOzamaabmaabaGaamiEaiaacYcacaWG5baacaGLOaGa % ayzkaaGaaiilaiaadMhadaqadaqaaiaadIhadaWgaaWcbaGaaGimaa % qabaaakiaawIcacaGLPaaacqGH9aqpcaWG5bWaaSbaaSqaaiaaicda % aeqaaOGaeyi1HSTaamyEamaabmaabaGaamiEaaGaayjkaiaawMcaai % abg2da9iaadMhadaWgaaWcbaGaaGimaaqabaGccqGHRaWkdaWdXbqa % aiaadAgadaWadaqaaiaadshacaGGSaGaamyEamaabmaabaGaamiDaa % GaayjkaiaawMcaaaGaay5waiaaw2faaaWcbaGaamiEamaaBaaameaa % aeqaaaWcbaGaamiEaaqdcqGHRiI8aOGaamizaiaadshacaGGSaaaaa!5E2E!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$y' = f\left( {x,y} \right),y\left( {{x_0}} \right) = {y_0} \Leftrightarrow y\left( x \right) = {y_0} + \int\limits_{{x_{}}}^x {f\left[ {t,y\left( t \right)} \right]} dt,$$
(1)

where the function f is sufficiently smooth.

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References

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Authors and Affiliations

  1. Institute of Mathematics, Tereschenkivska str. 3, 252601, Kiev, Ukraine

    V. K. Dzyadyk

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  1. V. K. Dzyadyk
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Editors and Affiliations

  1. Université Paris 7, Paris, France

    Anne Boutet de Monvel

  2. Institute for Low Temperature Physics, Kharkov, Ukraine

    Vladimir Marchenko

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Dzyadyk, V.K. (1996). Approximative-Iterative Method for Solving Non-Linear Differential and Integral Equations. In: de Monvel, A.B., Marchenko, V. (eds) Algebraic and Geometric Methods in Mathematical Physics. Mathematical Physics Studies, vol 19. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0693-3_23

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  • DOI: https://doi.org/10.1007/978-94-017-0693-3_23

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4663-5

  • Online ISBN: 978-94-017-0693-3

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