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# 数学代写|数值分析代写Numerical analysis代考|STAT7604 The Sturm–Liouville eigenvalue problem

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## 数学代写|数值分析代写Numerical analysis代考|The Sturm–Liouville eigenvalue problem

Suppose that $r$ is a real-valued function, defined and continuous on the closed interval $[a, b], p$ is a real-valued function, defined and continuously differentiable on $[a, b]$, and $r(x) \geq 0, p(x) \geq c_0>0$ for all $x \in[a, b]$. The differential equation
$$-\frac{\mathrm{d}}{\mathrm{d} x}\left(p(x) \frac{\mathrm{d} y}{\mathrm{~d} x}\right)+r(x) y=\lambda y, \quad a<x<b,$$
with homogeneous boundary conditions $y(a)=y(b)=0$, has only the trivial solution $y \equiv 0$, except for an infinite sequence of positive eigenvalues $\lambda=\lambda_m, m=1,2, \ldots$. We shall now consider a numerical method for finding these eigenvalues and the corresponding eigenfunctions, $y_{(m)}(x)$, $m=1,2, \ldots$

In the simple case where $p(x) \equiv 1$ and $r(x) \equiv 0$ the solution to this problem is, of course, $\lambda_m=[m \pi /(b-a)]^2, y_{(m)}(x)=A \sin m \pi t$, $m=1,2, \ldots$, where $A$ is a nonzero constant and $t=(x-a) /(b-a)$.
Using the same finite difference approximation as in the previous section, we obtain the equations
$$\begin{array}{r} -\frac{p_{j+1 / 2}\left(Y_{j+1}-Y_j\right)-p_{j-1 / 2}\left(Y_j-Y_{j-1}\right)}{h^2}+r_j Y_j=\Lambda Y_j, \ j=1,2, \ldots, n-1 . \end{array}$$

## 数学代写|数值分析代写Numerical analysis代考|The shooting method

The methods we have described for the linear boundary value problem may be extended to nonlinear differential equations. We shall not discuss how this is done; instead, we shall describe an alternative approach, called the shooting method. We shall consider the nonlinear model problem
$$y^{\prime \prime}=f(x, y), \quad a<x<b, \quad y(a)=A, \quad y(b)=B,$$
where we assume that the function $f(x, y)$ is continuous and differentiable, and that
$$\frac{\partial f}{\partial y}(x, y) \geq 0, \quad a<x<b, \quad y \in \mathbb{R} .$$
The central idea of the method is to replace the boundary value problem under consideration by an initial value problem of the form
$$y^{\prime \prime}=f(x, y), \quad a<x \leq b, \quad y(a)=A, \quad y^{\prime}(a)=t,$$
where $t$ is to be chosen in such a way that $y(b)=B$. This can be thought of as a problem of trying to determine the angle of inclination $\tan ^{-1} t$ of a loaded gun, so that, when shot from height $A$ at the point $x=a$, the bullet hits the target placed at height $B$ at the point $x=b$. Hence the name, shooting method.

## 数学代写|数值分析代写NUMERICAL ANALYSIS代考|THE STURM-LIOUVILLE EIGENVALUE PROBLEM

$$-\frac{\mathrm{d}}{\mathrm{d} x}\left(p(x) \frac{\mathrm{d} y}{\mathrm{~d} x}\right)+r(x) y=\lambda y, \quad a<x<b,$$

$$-\frac{p_{j+1 / 2}\left(Y_{j+1}-Y_j\right)-p_{j-1 / 2}\left(Y_j-Y_{j-1}\right)}{h^2}+r_j Y_j=\Lambda Y_j, j=1,2, \ldots, n-1 .$$

## 数学代写|数值分析代写NUMERICAL ANALYSIS代考|THE SHOOTING METHOD

$$y^{\prime \prime}=f(x, y), \quad a<x<b, \quad y(a)=A, \quad y(b)=B,$$

$$\frac{\partial f}{\partial y}(x, y) \geq 0, \quad a<x<b, \quad y \in \mathbb{R} .$$

$$y^{\prime \prime}=f(x, y), \quad a<x \leq b, \quad y(a)=A, \quad y^{\prime}(a)=t,$$

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