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# 数学代写|有限元方法作业代写finite differences method代考|Weighted residual and variational methods

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## 数学代写|有限元方法作业代写finite differences method代考|Classification of differential operators

The quantities of interest in many areas of applied mathematics are often to be found as the solution of certain partial differential equations, together with prescribed boundary and/or initial conditions.

The nature of the solution of a partial differential equation depends on the form that the equation takes. All linear, and quasi-linear, second-order equations are classified as elliptic, hyperbolic or parabolic. In each of these categories there are equations which model certain physical phenomena. The classification is determined by the coefficients of the highest partial derivatives which occur in the equation.

In this chapter, we shall consider functions which depend on two independent variables only, so that the resulting algebra does not obscure the underlying ideas.
Consider the second-order partial differential equation
$$\mathcal{L} u=f$$
where $\mathcal{L}$ is the operator defined by
$$\mathcal{L} u \equiv a \frac{\partial^{2} u}{\partial x^{2}}+b \frac{\partial^{2} u}{\partial x \partial y}+c \frac{\partial^{2} u}{\partial^{2} y^{2}}+F\left(x, y, u, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}\right)$$

## 数学代写|有限元方法作业代写finite differences method代考|Self-adjoint positive definite operators

Suppose that the function $u$ satisfies eqn (2.1) in a two-dimensional region $D$ bounded by a closed curve $C$, i.e.
$$\mathcal{L} u=f$$
where $f(x, y)$ is a given function of position. Suppose also that $u$ satisfies certain given homogeneous conditions on the boundary $C$. Usually these conditions are of the following types:
Dirichlet boundary condition: $u=0$;
$$\text { Neumann condition: } \frac{\partial u}{\partial n}=0 \text {; }$$
Robin condition: $\frac{\partial u}{\partial n}+\sigma(s) u=0$.
Here $s$ is the arc length measured along $C$ from some fixed point on $C$, and $\partial / \partial n$ represents differentiation along the outward normal to the boundary. Note that the Neumann condition may be obtained from the Robin condition by setting $\sigma \equiv 0$.

A problem is said to be properly posed, in the sense of Hadamard (1923), if and only if the following conditions hold:

1. A solution exists.
2. The solution is unique.
3 . The solution depends continuously on the data.

## 数学代写|有限元方法作业代写FINITE DIFFERENCES METHOD代考|Weighted residual methods

Consider the boundary-value problem
$$\mathcal{L} u=f \quad \text { in } \quad D$$
subject to the non-homogeneous Dirichlet boundary condition
$$u=g(s)$$
on some part $C_{1}$ of the boundary, and the non-homogeneous Robin condition
$$\frac{\partial u}{\partial n}+\sigma(s) u=h(s)$$
on the remainder $C_{2}$.
An approximate solution $\tilde{u}$ will not, in general, satisfy eqn (2.6) exactly, and associated with such an approximate solution is the residual defined by
$$r(\tilde{u})=\mathcal{L} \tilde{u}-f$$
If the exact solution is $u_{0}$, then
$$r\left(u_{0}\right) \equiv 0 .$$

## 数学代写|有限元方法作业代写FINITE DIFFERENCES METHOD代考|SELF-ADJOINT POSITIVE DEFINITE OPERATORS

Dirichlet 边界条件：在=0;
诺依曼条件： ∂在∂n=0;

1. 存在解决方案。
2. 解决方案是独一无二的。
3. 解决方案持续依赖于数据。

∂在∂n+σ(s)在=H(s)

r(在~)=大号在~−F

r(在0)≡0.