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数学代写|数值分析代写Numerical analysis代考|MATH2722 Difficulties in the Execution of the Simple Shooting Method

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数学代写数值分析代写Numerical analysis代考|Difficulties in the Execution of the Simple Shooting Method

To be useful in practical applications, methods for solving the boundaryvalue problem
$$y^{\prime}=f(x, y), \quad r(y(a), y(b))=0,$$
should yield an approximation to $y\left(x_0\right)$ for every choice of $x_0$ in the region of definition of a solution $y$. In the shooting method discussed so far, an approximate value $\bar{s}$ for the solution $y(a)$ is computed only at one point, the point $x_0=a$. This seems to suggest that the boundary value problem has thus been solved, since the value $y\left(x_0\right)$ of the solution at every other point $x_0$ can be (approximately) determined by solving the initial-value problem
$(7.3 .4 .1) \quad y^{\prime}=f(x, y), \quad y(a)=\bar{s}$,
say, with the methods of Section 7.2. This, however, is true only in principle. In practice, there often accrue considerable inaccuracies if the solution $y(x)=y(x ; \bar{s})$ of $(7.3 .4 .1)$ depends very sensitively on $\bar{s}$, as shown in the following example.
EXAMPLE 1 . The linear system of differential equations
$$\left[\begin{array}{l} y_1 \ y_2 \end{array}\right]^{\prime}=\left[\begin{array}{cc} 0 & 1 \ 100 & 0 \end{array}\right]\left[\begin{array}{l} y_1 \ y_2 \end{array}\right]$$
has the general solution
(7.3.4.3) $y(x)=\left[\begin{array}{l}y_1(x) \ y_2(x)\end{array}\right]=c_1 e^{-10 x}\left[\begin{array}{r}1 \ -10\end{array}\right]+c_2 e^{10 x}\left[\begin{array}{r}1 \ 10\end{array}\right], \quad c_1, c_2$ arbitrary.
Let $y(x ; s)$ be the solution of (7.3.4.2) satisfying the initial condition
$$y(-5)=s=\left[\begin{array}{l} s_1 \ s_2 \end{array}\right] .$$

数学代写|数值分析代写Numerical analysis代考|The Multiple Shooting Method

The multiple shooting method has been described repeatedly in the literature, for example in Keller (1968), Osborne (1969), and Bulirsch (1971). A FORTRAN program can be found in Oberle and Grimm (1989).
In a multiple shooting method, the values
$$\bar{s}_k=y\left(x_k\right), \quad k=1,2, \ldots, m,$$
of the exact solution $y(x)$ of a boundary-value problem
$$y^{\prime}=f(x, y), \quad r(y(a), y(b))=0,$$
at several points
$$a=x_1<x_2<\cdots<x_m=b$$
are computed simultaneously by iteration. To this end, let $y\left(x ; x_k, s_k\right)$ be the solution of the initial-value problem
$$y^{\prime}=f(x, y), \quad y\left(x_k\right)=s_k .$$

The problem now consists in determining the vectors $s_k, k=1,2, \ldots, m$, in such a way that the function
\begin{aligned} y(x) &:=y\left(x ; x_k, s_k\right) \quad \text { for } x \in\left[x_k, x_{k+1}\right), \quad k=1,2, \ldots, m-1, \ y(b) &:=s_m, \end{aligned}
pieced together by the $y\left(x ; x_k, s_k\right)$, is continuous, and thus a solution of the differential equation $y^{\prime}=f(x, y)$, and in addition satisfies the boundary conditions $r(y(a), y(b))=0$ (see Figure 20).

数学代写数值分析代写Numerical analysis代考|Difficulties in the Execution of the Simple Shooting Method

$$y^{\prime}=f(x, y), \quad r(y(a), y(b))=0,$$

$$\text { (7.3.4.1) } y^{\prime}=f(x, y), \quad y(a)=\bar{s}$$

$$\left[\begin{array}{ll} y_1 & y_2 \end{array}\right]^{\prime}=\left[\begin{array}{llll} 0 & 1 & 100 & 0 \end{array}\right]\left[\begin{array}{ll} y_1 & y_2 \end{array}\right]$$

(7.3.4.3) $y(x)=\left[y_1(x) y_2(x)\right]=c_1 e^{-10 x}[1-10]+c_2 e^{10 x}[110], \quad c_1, c_2$ 随意的。

$$y(-5)=s=\left[\begin{array}{ll} s_1 & s_2 \end{array}\right] .$$

数学代写|数值分析代写Numerical analysis代考|The Multiple Shooting Method

$$\bar{s}k=y\left(x_k\right), \quad k=1,2, \ldots, m,$$ 的精确解 $y(x)$ 边值问题 $$y^{\prime}=f(x, y), \quad r(y(a), y(b))=0,$$ 在几个点 $$a=x_1{k+1}\right), \quad k=1,2, \ldots, m-1, y(b) \quad:=s_m,$$

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