Krajova zadacha zadacha teoriyi diferencialnih rivnyan v yakij mezhovi umovi zadayutsya v riznih tochkah Napriklad pri kolivannyah struni iz zakriplenemi kincyami zmishennya na kozhnomu z kinciv dorivnyuye nulyu Krajovi zadachi skladnishe rozv yazuvati nizh zadachi Koshi osoblivo chiselno Krajovi zadachi vinikayut yak v teoriyi zvichajnih diferencijnih rivnyan tak i v teoriyi diferencijnih rivnyan iz chastkovimi pohidnimi osoblivo rivnyan eliptichnogo tipu Osoblivij vid krayevoyi zadachi vimoga pevnoyi povedinki fuknciyi skinchennosti pri pryamuvanni argumentu do neskinchennosti abo v okoli osoblivih tochok Nehaj W displaystyle Omega oblast na ploshini x y displaystyle x y iz mezheyu G displaystyle Gamma Vazhlivimi zadachami ye u x y x y G f x y displaystyle u x y x y in Gamma varphi x y persha krajova zadacha zadacha Dirihle u x y n x y G f x y ps x y displaystyle frac partial u x y partial n vert x y in Gamma varphi x y psi x y druga krajova zadacha zadacha Nejmanaa u x y b u x y n x x y displaystyle au x y b frac partial u x y partial n chi x y dlya x y displaystyle x y na G displaystyle Gamma tretya krajova zadacha zadacha Robina Zmist 1 Metodi rozv yazannya krajovih zadach 1 1 Metod sitok 1 2 Chiselni 2 Div takozh 3 DzherelaMetodi rozv yazannya krajovih zadach RedaguvatiMetod sitok Redaguvati Rozglyadayetsya ne kontinuum tochok ploshini x y displaystyle x y nbsp a zlichenna mnozhina diskretnih tochok P i j x i y i displaystyle Pi ij x i y i nbsp nbsp Yaksho oblast W displaystyle Omega nbsp rozmistiti na sitci to odni tochki sitki popadut vseredinu a inshi viyavlyatsya nazovni oblasti Diskretna W displaystyle Omega nbsp oblast skladayetsya z tochok sitki yaki lezhat vseredini oblasti W displaystyle Omega nbsp tochki sitki najblizhchi do mezhi j yaki lezhat abo vseredini abo zzovni ce zalezhit vid postanovki zadachi rozrahovuyut yak tochki diskrednoyi mezhi G displaystyle Gamma nbsp U comu vipadku diskretna oblast W W G displaystyle tilde Omega Omega bigcup Gamma nbsp skladayetsya lishe z tochok sitki nbsp Druga mozhlivist polyagaye u tomu sho dodayut tochki peretinu G displaystyle Gamma nbsp iz pryamimi sitki yak neregulyarni granichni tochki nbsp Pohidni yaki zustrichayutsya u rozglyaduvanomu diferencialnomu rivnyanni zaminyuyutsya u kozhnij tochci sitki P i j x i y i displaystyle Pi ij x i y i nbsp na vidpovidni riznisni vidnoshennya Napriklad u n i j 1 2 h u i 1 j u i 1 j O h 2 displaystyle frac partial u partial n vert ij frac 1 2h u i 1 j u i 1 j O h 2 nbsp Taki virazi nazivayutsya takozh molekulami j pishuyutsya u viglyadi naochnih strukturnih formul P yatitochkovi molekuli dlya operatora Laplasa kvadratna sitka nbsp Yaksho oblast W displaystyle Omega nbsp taka sho dlya dostatno prostoyi sitki za vidpovidno obranogo roztashuvannya mezha G displaystyle Gamma nbsp skladayetsya lishe z sitkovih pryamih to krajovi znachennya zadayutsya u granichnih sitkovih tochkah j uvodyatsya vidpovidni molekuli yaksho voni vklyuchayut taki tochki Napriklad rivnyannya Puasona u pryamokutniku 1 W x y 0 x 4 k 0 y 3 h displaystyle Omega x y 0 leq x leq 4k 0 leq y leq 3h nbsp Sitka x i y i i k j h displaystyle x i y i ik jh nbsp W i k j h 0 i 4 4 j 3 displaystyle Omega ik jh 0 leq i leq 4 4 leq j leq 3 nbsp regulyarna mezha Nehaj W displaystyle Omega nbsp ye oblastyu na ploshini x y displaystyle x y nbsp iz mezheyu G displaystyle Gamma nbsp Potribno vidnajti funkciyu u x y displaystyle u x y nbsp yaka zadovilnyaye W displaystyle Omega nbsp rivnyannyu PuasonaD u u x x u y y g x y displaystyle Delta u u xx prime prime u yy prime prime g x y nbsp Pri zastosuvanni molekuli livoruchU i 1 j U i 1 j U i j 1 U i j 1 4 U i j h 2 f i j displaystyle U i 1 j U i 1 j U i j 1 U i j 1 4U ij h 2 f ij nbsp yak diskretnij analog rivnyannya Puasona cherez U i j displaystyle U ij nbsp poznachene nablizhennya dlya u x i y j u i j displaystyle u x i y j u ij nbsp Yaksho zapisati usi rivnyannya dlya yakih centralnij element u i j displaystyle u ij nbsp ye vnutrishnoyu tochkoyu tobto 1 i 3 1 j 2 displaystyle 1 leq i leq 3 1 leq j leq 2 nbsp toU 01 U 21 U 10 U 12 4 U 11 h 2 f 11 displaystyle underline U 01 U 21 underline U 10 U 12 4U 11 h 2 f 11 nbsp U 11 U 31 U 20 U 22 4 U 21 h 2 f 21 displaystyle U 11 U 31 underline U 20 U 22 4U 21 h 2 f 21 nbsp U 21 U 41 U 30 U 32 4 U 31 h 2 f 31 displaystyle U 21 underline U 41 underline U 30 U 32 4U 31 h 2 f 31 nbsp U 02 U 22 U 10 U 13 4 U 12 h 2 f 12 displaystyle underline U 02 U 22 U 10 underline U 13 4U 12 h 2 f 12 nbsp U 12 U 32 U 20 U 23 4 U 22 h 2 f 22 displaystyle U 12 U 32 U 20 underline U 23 4U 22 h 2 f 22 nbsp U 22 U 42 U 30 U 33 4 U 32 h 2 f 32 displaystyle U 22 underline U 42 U 30 underline U 33 4U 32 h 2 f 32 nbsp Pidkresleni znachennya mozhut buti pereneseni pravoruch Todi v yakosti diskretnogo analogu zadachi ye sistema linijnih rivnyan 4 1 0 1 0 0 1 4 1 0 1 0 0 1 4 0 0 1 1 0 0 4 1 0 0 1 0 1 4 1 0 0 1 0 1 4 U 11 U 21 U 31 U 12 U 22 U 32 h 11 2 U 01 U 10 h 2 f 21 U 20 h 2 f 31 U 41 U 30 h 2 f 12 U 02 U 13 h 2 f 22 U 23 h 2 f 32 U 42 U 33 displaystyle begin pmatrix 4 amp 1 amp 0 amp 1 amp 0 amp 0 1 amp 4 amp 1 amp 0 amp 1 amp 0 0 amp 1 amp 4 amp 0 amp 0 amp 1 1 amp 0 amp 0 amp 4 amp 1 amp 0 0 amp 1 amp 0 amp 1 amp 4 amp 1 0 amp 0 amp 1 amp 0 amp 1 amp 4 end pmatrix begin pmatrix U 11 U 21 U 31 U 12 U 22 U 32 end pmatrix begin pmatrix h 11 2 U 01 U 10 h 2 f 21 U 20 h 2 f 31 U 41 U 30 h 2 f 12 U 02 U 13 h 2 f 22 U 23 h 2 f 32 U 42 U 33 end pmatrix nbsp Dlya rishennya takih sistem zastosovuyut iteracijni metodi hocha mozhut zastosovuvatisya metodi yaki vikoristovuyut blokovu strukturu Chiselni Redaguvati Metod strilbiDiv takozh RedaguvatiFunkciyi KurantaDzherela Redaguvati E A Volkov O reshenii kraevyh zadach dlya uravneniya Puassona v pryamougolnike Dokl AN SSSR 1962 tom 147 nomer 1 13 16 Otrimano z https uk wikipedia org w index php title Krajova zadacha amp oldid 31007253
