Чегаравий масалаларни ечиш усуллари.
Режа .
Системаларни интеграллаш .
Чегаравий масалаларни ечиш усуллари .
Отишма методи .
Айирмали методлар .
Рунге-Кутт методлари ЭҲМ ёрдамида ҳисоблаш учун қулай ҳисобланади. Чунки бу методнинг қуйидаги яхши хусусиятлари бор.
1) аниқликлари яхши (биринчи тартиблисидан бошқаларининг).
2) улар ошкор методлардирлар, яъни yn+1 , олдин аниқланган қийматлар орқали маълум формулалар орқали ошкор ифодаланади.
3) барча методларда қадамлар ўзгарувчи бўлиши мумкин: бу, ечим тез ўзгарадиган жойларда қадамни кичрайтириб, акс ҳолда қадамни катта қилиб ҳисоблашга имкон беради.
4) дастлабки ҳисоблашни турни танлаб, ![](data:image/png;base64,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) ни бериб, ҳисоблашни олдиндан маълум бўлган формулалар ёрдамида бажариш мумкин.
Бу хоссалар Рунге-Кутт методини ЭҲМ ёрдамида ҳисоблаш учун қулай эканлигини кўрсатди. Шу сабабли бу метод ёрдамида системани ечишни қараймиз. Умуман айтганда y ва f(x,y) ларни формал У ва F(X,Y) ларга алмаштириш кифоя.
Масалан,
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