Пример. Увеличенный якобиан для Stiffb и Stiffr

Функции > Решение и оптимизация > Решатели дифференциальных уравнений > Пример. Увеличенный якобиан для Stiffb и Stiffr

Используйте функцию Jacob, чтобы вычислить расширенный якобиан для обыкновенного дифференциального уравнения (ОДУ), а затем предоставить ее как входные данные для решателей Stiffb и Stiffr.

1. Определите систему четырех неизвестных:

Щелкните для копирования этого выражения

Переменные yi являются функциями x.

2. Определите константы системы.

3. Определите начальные значения.

4. Определите векторную функцию D(x,y), соответствующую правой части системы.

5. Используйте функцию augment, чтобы создать расширенный якобиан:

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 <define xmlns="http://schemas.mathsoft.com/math50">
 <function>
 <id xml:space="preserve" labels="VARIABLE">AJ</id>
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 <id xml:space="preserve" labels="VARIABLE">y</id>
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 <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">augment</id>
 <sequence>
 <apply>
 <id xml:space="preserve">Jacob</id>
 <sequence>
 <apply>
 <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">D</id>
 <sequence>
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 <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">y</id>
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 <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
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 <sequence>
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 <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">D</id>
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 <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
 <id xml:space="preserve">y</id>
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 <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">y</id>
 </sequence>
 </apply>
 </sequence>
 </apply>
 <command>
 <placeholder />
 </command>
 <symResult>
 <matrix rows="4" cols="5">
 <real>0</real>
 <real>0</real>
 <real>0</real>
 <real>0</real>
 <apply>
 <minus />
 <apply>
 <mult />
 <real>-1.34</real>
 <apply>
 <indexer />
 <id xml:space="preserve" labels="*">y</id>
 <real>2</real>
 </apply>
 </apply>
 <apply>
 <mult />
 <real>8000</real>
 <apply>
 <indexer />
 <id xml:space="preserve" labels="*">y</id>
 <real>1</real>
 </apply>
 </apply>
 </apply>
 <apply>
 <plus />
 <apply>
 <mult />
 <real>1.34</real>
 <apply>
 <indexer />
 <id xml:space="preserve" labels="*">y</id>
 <real>2</real>
 </apply>
 </apply>
 <apply>
 <mult />
 <real>8000</real>
 <apply>
 <indexer />
 <id xml:space="preserve" labels="*">y</id>
 <real>1</real>
 </apply>
 </apply>
 </apply>
 <apply>
 <mult />
 <real>-1.34</real>
 <apply>
 <indexer />
 <id xml:space="preserve" labels="*">y</id>
 <real>2</real>
 </apply>
 </apply>
 <apply>
 <mult />
 <real>8000</real>
 <apply>
 <indexer />
 <id xml:space="preserve" labels="*">y</id>
 <real>1</real>
 </apply>
 </apply>
 <apply>
 <mult />
 <real>-8000</real>
 <apply>
 <indexer />
 <id xml:space="preserve" labels="*">y</id>
 <real>0</real>
 </apply>
 </apply>
 <apply>
 <minus />
 <apply>
 <plus />
 <apply>
 <mult />
 <apply>
 <neg />
 <parens>
 <apply>
 <mult />
 <real>1.6</real>
 <apply>
 <pow />
 <real>10</real>
 <real>9</real>
 </apply>
 </apply>
 </parens>
 </apply>
 <apply>
 <indexer />
 <id xml:space="preserve" labels="*">y</id>
 <real>2</real>
 </apply>
 </apply>
 <apply>
 <mult />
 <real>8000</real>
 <apply>
 <indexer />
 <id xml:space="preserve" labels="*">y</id>
 <real>0</real>
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 </apply>
 </apply>
 <apply>
 <mult />
 <real>160000000</real>
 <apply>
 <indexer />
 <id xml:space="preserve" labels="*">y</id>
 <real>1</real>
 </apply>
 </apply>
 </apply>
 <apply>
 <mult />
 <apply>
 <neg />
 <parens>
 <apply>
 <mult />
 <real>1.6</real>
 <apply>
 <pow />
 <real>10</real>
 <real>9</real>
 </apply>
 </apply>
 </parens>
 </apply>
 <apply>
 <indexer />
 <id xml:space="preserve" labels="*">y</id>
 <real>2</real>
 </apply>
 </apply>
 <apply>
 <mult />
 <real>8000</real>
 <apply>
 <indexer />
 <id xml:space="preserve" labels="*">y</id>
 <real>0</real>
 </apply>
 </apply>
 <apply>
 <mult />
 <real>-1.34</real>
 <apply>
 <indexer />
 <id xml:space="preserve" labels="*">y</id>
 <real>0</real>
 </apply>
 </apply>
 <apply>
 <plus />
 <apply>
 <mult />
 <real>1.34</real>
 <apply>
 <indexer />
 <id xml:space="preserve" labels="*">y</id>
 <real>0</real>
 </apply>
 </apply>
 <apply>
 <mult />
 <apply>
 <neg />
 <parens>
 <apply>
 <mult />
 <real>1.6</real>
 <apply>
 <pow />
 <real>10</real>
 <real>9</real>
 </apply>
 </apply>
 </parens>
 </apply>
 <apply>
 <indexer />
 <id xml:space="preserve" labels="*">y</id>
 <real>1</real>
 </apply>
 </apply>
 </apply>
 <apply>
 <plus />
 <apply>
 <mult />
 <real>-1.34</real>
 <apply>
 <indexer />
 <id xml:space="preserve" labels="*">y</id>
 <real>0</real>
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 <mult />
 <apply>
 <neg />
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 <apply>
 <mult />
 <real>1.6</real>
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 <pow />
 <real>10</real>
 <real>9</real>
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 <indexer />
 <id xml:space="preserve" labels="*">y</id>
 <real>1</real>
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 <real>0</real>
 <real>0</real>
 <real>1</real>
 <real>-1</real>
 </matrix>
 </symResult>
 </symEval>
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6. Вызовите функции Stiffb и Stiffr:

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Возвращенные матрицы содержат 5 столбцов, соответствующих числу точек и решениям для четырех неизвестных.

7. Извлеките решения для четырех неизвестных из возвращенных матриц Stiffb и Stiffr:

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8. Постройте графики и сравните возвращенные решения для каждого неизвестного из двух функций:

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Графики показывают, что два решателя ОДУ возвращают идентичные решения.

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Решатели ОДУ

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