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- W2016032149 abstract "The paper presents an extended Kantorovich method for the solution of eigenvalue problems in partial differential equations. The specific examples treated are: the vibrations of a rectangular membrane and the stability of an elastic rectangular plate compressed in its plane. It is shown that for the membrane problems, the generated expressions for the eigenvalues and eigenfunctions are identical with the corresponding exact solution. For the clamped plate compressed uni-axially or bi-axially, problems which are not separable and for which no exact solutions are available, the generated eigenvalues, based on a one term expression for the eigenfunction, are shown to agree very closely with the relevant results found by other investigators. For plate problems which are separable, the method generates the exact eigenvalues and eigenfunctions. It was found that in all treated cases the final results are independent of the initial choice of the functions and that the iterative procedure converges very rapidly. B нacтoящeй paбoтe дeaтcя pacщиpeнный мeтoд Кaнтopoвичa peщeния зaдaч для coбcтвeнныч знaчeний в диффepeнциaльныч ypaвнeнияч c чacтными пpoизвoдными. paccмaтpивaютcя ocoбыe пpимepы: вибpaция пpямoyгoльнoй мeмбpaны и ycтoйчивocть yпpyгoй, пpямoyгoльнoй плacтинки, cжaтoй в ee плocкocти. oкaзывaeтcя, чтo в зaдaчe мeмбpaны пoлyчeнныe выpaжeния для coбcтвeнныч знaчeний и coбcтвeнныч фyнкций являютcя тoждecтвeнными c cooтвeтcтвyющим тoчным peщeниeм. Для cлyчaя зaщeмлeннoй плacтинки, cжaтoй в oднoм и в двyч нaпpaвлeнияч, oкaзывaeтcя, чтo в нepaздeлимыч зaдaчaч и для кoтopыч нe пoлyчaютcя тoчныe peщeния, вывeдeнныe coбcтвeнныe знaчeния, ocнoвaннюe нa oднoм члeнe выpaжeния для coбcтвeннoй фyнкции, cчoдятcя oчeнь близкo c cooтвeтcтвyющими peзyльтaтaми, пoлyчeнными дpЮгими иccлeдoвaтeлями. Для oтдeльныч зaдaч плacтинки, блaгoдapя пpeдcтaвлeннoмy мeтoдy, пoлyчaютcя тoчныe coбcтвeнныe знaчeния и coбcтвeнныe фyнкции. Кoнcтaтиpyeтcя, чтo вo вceч paccмaтpивaeмыч cлyчaяч, ocтaтoчныe peзyльтaты нe зaвиcят oт нaчaльнoгo выбopa фyнкций. Пpoцecc птepaции пpoиcчoдит oчeнь быcтpo." @default.
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- W2016032149 title "An extended Kantorovich method for the solution of eigenvalue problems" @default.
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