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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|有限元方法代写Finite Element Method代考|Elements of calculus of variations

In this section we introduce the concept of space-time functionals. In the abstract sense, we mean a mapping or a correspondence that assigns a definite real number to each function belonging to some class or space. Space-time functionals are naturally functions of spatial coordinates as well as time. Such functionals play an important role in obtaining approximations of the theoretical solutions of IVPs encountered in mathematical physics, science, and engineering. In this section we study elements of the calculus of variations, a branch of mathematics that deals with extremums of spacetime functionals, i.e. maximums, saddle points, and minimums. We establish a correspondence between the solutions of IVPs and the extremums of the space-time functionals that are constructed using the IVPs. Therein lies our interest in studying the elements of the calculus of variations associated with space-time functionals. There are four basic lemmas that are important in this regard. We state these and provide their proofs.

Lemma 2.1 (Fundamental Lemma). If $\alpha(x, t)$ is continuous on $\bar{\Omega}{x t}=$ $\Omega{x t} \cup \Gamma$ and if
$$\int_{\Omega_{x t}} \alpha(x, t) h(x, t) d \Omega_{x t}=0 \quad \forall h(x, t) \in H^{(1)}: \quad h(\Gamma)=0$$
Then
$$\alpha(x, t)=0 \quad \forall(x, t) \in \bar{\Omega}{x t}$$ Proof. We construct the proof of this lemma by contradiction. Suppose $\alpha(x, t)$ is nonzero, say positive at some point in $\bar{\Omega}{x t}$. If we let
$$h(x, t)=\left(x-x_{1}\right)\left(x_{2}-x\right)\left(t-t_{1}\right)\left(t_{2}-t\right)$$
for some $(x, t) \in\left[x_{1}, x_{2}\right] \times\left[t_{1}, t_{2}\right]$ and $h(x, t)=0$ otherwise, then $h(x, t)$ satisfies the conditions of the lemma. However, we have
$$\int_{\bar{\Omega}{x t}} \alpha(x, t) h(x, t) d \Omega{x t}=\int_{t_{1} x_{1}}^{t_{2} x_{2}} \int_{x_{1}} \alpha(x)\left(x-x_{1}\right)\left(x_{2}-x\right)\left(t-t_{1}\right)\left(t_{2}-t\right) d x d t>0$$
Since the integrand is positive (except at $x_{1}, x_{2}, t_{1}, t_{2}$ ), this contradiction proves the lemma.

数学代写|有限元方法代写Finite Element Method代考|Concept of variation of a space-time functional

Variation means change or, in the sense of calculus, differential. Let $I(y)$ with $y=y(x, t)$ be a functional defined over some normed linear space and let
$$\Delta I(h)=I(y+h)-I(y)$$
be increment in $I$ corresponding to an increment $h=h(x, t)$ of the dependent variable $y=y(x, t)$. If $y$ is fixed, then $\Delta I(h)$ is a function of $h$, in general a non-linear functional. Suppose that
$$\Delta I(h)=\Phi(h)+\epsilon|h|$$
where $\Phi(h)$ is a linear functional and $\epsilon|h| \rightarrow 0$ as $|h| \rightarrow 0$. Then the functional $I$ is said to be differentiable and the principal linear part of the increment $\Delta I(h)$, i.e. the linear functional $\Phi(h)$ which differs from $\Delta I(h)$ by an infinitesimal of order higher than one relative to $|h|$, is called the variation of $I(y)$ denoted by $\delta I$ or $\delta I(h)$.
Theorem 2.7. The variation of a space-time functional is unique.
Theorem 2.8. A necessary condition for a space-time functional $I(y(x, t))$ to have an extremum for $y=y^{}$ is that its variation vanishes for $y=y^{}$, i.e. $\delta I(h)=0$ for $y=y *$ for all admissible $h=h(x, t)$.

有限元方法代考

数学代写|有限元方法代写Finite Element Method代考|Elements of calculus of variations

$$\int_{\Omega_{\boldsymbol{A} t}} \alpha(x, t) h(x, t) d \Omega_{x t}=0 \quad \forall h(x, t) \in H^{(1)}: \quad h(\Gamma)=0$$
$$\alpha(x, t)=0 \quad \forall(x, t) \in \bar{\Omega} x t$$

$$h(x, t)=\left(x-x_{1}\right)\left(x_{2}-x\right)\left(t-t_{1}\right)\left(t_{2}-t\right)$$

$$\int_{\bar{\Omega} x t} \alpha(x, t) h(x, t) d \Omega x t=\int_{t_{1} x_{1}}^{t_{2} x_{2}} \int_{x_{1}} \alpha(x)\left(x-x_{1}\right)\left(x_{2}-x\right)\left(t-t_{1}\right)\left(t_{2}-t\right) d x d t>0$$

数学代写|有限元方法代写Finite Element Method代考|Concept of variation of a space-time functional

$$\Delta I(h)=I(y+h)-I(y)$$

$$\Delta I(h)=\Phi(h)+\epsilon|h|$$

有限元方法代写

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MATLAB代写

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