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Up: 10.001: Numerical Solution of
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It was mentioned in the introduction (see Eq. 4) that systems of first order ODEs can arise from
a single ODE of order larger than unity. Systems of ODEs also arises naturally from physical
modeling. For instance, the following IVP describes the concentrations yi,
of
n chemicals in a reactor as a function of time.
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![$\displaystyle \frac{dy_1}{dt} = f_1(y_1,y_2,\cdots , y_n, t)$](img53.gif) |
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![$\displaystyle \frac{dy_2}{dt} = f_2(y_2,y_2,\cdots , y_n, t)$](img54.gif) |
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.................................... |
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![$\displaystyle \frac{dy_n}{dt} = f_n(y_2,y_2,\cdots , y_n, t)$](img55.gif) |
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![$\displaystyle y_1(0) = y_{10}, \:\: y_2(0) =y_{20},\cdots , y_n(0) = y_{n0}.$](img56.gif) |
(26) |
The methods we discussed for the solution of a single first order ODE can be extended to the system of ODEs.
Let's define the n-dimensional vector of solutions
and the
n dimensional vector of the right hand side functions
.
Similarly, the initial conditions can be arranged in a vector
.
This allows us to write the IVP in the vector notation as
![\begin{displaymath}\frac{d{\bf {y}}}{dt} = {\bf {f}}({\bf {y}},t), \:\:\: {\bf {y}}(0) = {\bf {y_0}}.
\end{displaymath}](img60.gif) |
(27) |
Now, if we apply the forward Euler method for the solution of Eq. 28, with constant time step size h, we get
the following explicit equation for the solution at the (k+1)th step given the solution at the k the step:
![\begin{displaymath}{\bf {y}}_{k+1} = {\bf {y}}_{k} + h {\bf {f}}({\bf {y}}_{k}, t_k).
\end{displaymath}](img61.gif) |
(28) |
However, the implicit backward Euler method gives
![\begin{displaymath}{\bf {y}}_{k+1} = {\bf {y}}_{k} + h {\bf {f}}({\bf {y}}_{k+1}, t_{k+1}),
\end{displaymath}](img62.gif) |
(29) |
which necessitates in general the solution of a system of non-linear algebraic equations at every time step.
We can employ Newton's method for a system of non-linear equations to achieve this.
Next: Boundary Value Problems: The
Up: 10.001: Numerical Solution of
Previous: Predictor-Corrector Methods
Michael Zeltkevic
1998-04-15