## 数学代写|有限元方法代写Finite Element Method代考|ENEM28001

2022年9月27日

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Consider a differential operator $\mathcal{L}$ and two arbitrary functions $u(x)$ and $v(x)$ with homogeneous end conditions, $u\left(x_0\right)=u\left(x_L\right)=0$ and $v\left(x_0\right)=v\left(x_L\right)=0$. The inner product of $u(x)$ with $\mathcal{L}(v)$ is defined as follows:
$$\langle u, \mathcal{L}(v)\rangle=\int_{x_0}^{x_L} u \mathcal{L}(v) d x$$
An adjoint differential operator $\mathcal{L}^$ satisfies the following relationship: $$\langle u, \mathcal{L}(v)\rangle=\left\langle v, \mathcal{L}^(u)\right\rangle$$

Consider the following second order differential equation with variable coefficients $a=a(x), b=b(x)$ and $c=c(x)$,
$$\mathcal{L}(u)=a \frac{d^2 u}{d x^2}+b \frac{d u}{d x}+c u=0 \text { in } x_0 \leq x \leq x_L$$
By using integration by parts and the end conditions stated above, it can be shown that the adjoint differential operator $\mathcal{L}^$ associated with $\mathcal{L}$ for Eq. (3.15) is given as follows [2], $$\mathcal{L}^(v)=\frac{d^2}{d x^2}(a v)-\frac{d}{d x}(b v)+c v$$
Differential operators for which the following condition holds,
$$\mathcal{L}^*=\mathcal{L}$$
By using Eqs. (3.15) and (3.16) it can be shown that a second order, linear differential equation is self-adjoint if and only if it is of the form [2].
$$\mathcal{L}(\cdot)=\frac{d}{d x}\left(a \frac{d(\cdot)}{d x}\right)+c(\cdot)$$
where (.) represents any arbitrary function of the type defined in this section. This differential operator is called the Sturm-Liouville differential operator.
Similarly, the following fourth order differential operator,
$$\mathcal{L}(\cdot)=\frac{d}{d x^2}\left(s \frac{d(\cdot)}{d x^2}\right)+\frac{d}{d x}\left(a \frac{d(\cdot)}{d x}\right)+c(\cdot)$$
where $a=a(x), c=c(x)$ and $s=s(x)$, is self-adjoint, if the boundary conditions are homogenous and of the form [2],
$$u=\frac{d u}{d x}=0 \text { or } u=s \frac{d^2 u}{d x^2}=0 \text { or } \frac{d u}{d x}=\frac{d}{d x}\left(s \frac{d^2 u}{d x^2}\right)=0$$
on the boundaries.

## 数学代写|有限元方法代写Finite Element Method代考|Variation of a functional

Consider the functional given in Eq. (3.21). Let’s assume first, that the values of the independent variable $u$ on the boundaries are given as,
$$u\left(x_0\right)=u_0 \text { and } u\left(x_L\right)=u_L$$
where $u_0$ and $u_L$ are prescribed values. Thus, we see that essential boundary conditions are specified on both ends of the domain. Recall that $I(u)$ is a scalar therefore its value between these points will depend on the function $u(x)$ or the chosen path between the end points (Fig. 3.2).

Let’s assume that a path $u(x)$ which extremizes the functional $I$ exists. We will see that this path, called the extremal path, is the function that represents the solution of the problem. Let’s also call all of the other paths between these points, $\tilde{u}(x)$, varied paths. In Fig. 3.2, we can see that the varied paths can be defined as follows:
$$\tilde{u}=u+\varepsilon v$$
where $\varepsilon$ is an arbitrarily small parameter and $v(x)$ is any differentiable function. The varied function $\tilde{u}$ and $v(x)$, should have the same values as the function $u$ at the end points. Thus we see that the varied function $v(x)$ should have the following property,
$$v\left(x_0\right)=0 \text { and } v\left(x_L\right)=0$$
In other words, $v(x)$ should satisfy the homogenous form of the essential boundary conditions.

The difference between the extremal path and one of the varied paths is defined as follows:
$$\delta u=\tilde{u}-u$$
By using Eq. (3.26), the variation of $u, \delta u$ can be found as follows:
$$\delta u=\tilde{u}-u=\varepsilon v$$
The function $\delta u=\delta u(x)$ represents the variation of $\tilde{u}(x)$ from $u(x)$. The symbol $\delta$ is called the del-operator or the variational operator. The variational operator represents variation of the function rather than a pointwise difference between the functions. Comparing Eqs. (3.29) with (3.27), we see that $\delta u(x)$ should have the same property (3.27) as $v(x)$. Thus the boundary conditions for the deloperator can be expressed as follows.

# 有限元方法代考

## 有限元方法代写

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## MATLAB代写

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