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# 数学代写|最优化作业代写optimization theory代考|CSC591611 Interlination of Functions and Rvachov Structural Method

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## 数学代写|最优化作业代写optimization theory代考|Interlination of Functions and Rvachov Structural Method

In the works of V. L. Rvachov, the method of functions was proposed which allowed to create a general approach to the construction of complete systems of coordinate functions that completely satisfy the set boundary conditions. The fundamental principles of the structural method for solving boundary value problems for differential equations with partial derivatives in the case of domains of complex form are formulated. The problem that was not solved within the structural method is analyzed. The basic idea of this method is explained by V. L. Rvachov, and then it was developed in the works of his students, foremost in the direction of constructing the structures of approximate solutions of the boundary value problems for different domains and different types of boundary conditions.

According to the basic statements of the theory of interlineation and interflatation of functions, the structures of approximate solutions of boundary-value problems for domains of complex shape can be constructed by operators of interlineation and interflatation of functions which allow closely satisfy the boundary conditions of different types without using $R$ functions. These structures lead to new methods of solving boundary-value problems for partial differential equations – the methods of reduction to the systems of ordinary linear (LIDE) and nonlinear (NIDE) integrodifferential equations. They are the generalizations of the Kantorovich method.

## 数学代写|最优化作业代写optimization theory代考|Basic Principles of the Structural Method of Solving Mathematical Physics Problems

We will explore the methods of constructing the structures of approximate solutions of the boundary value problems with the partial derivatives:
$$A u \equiv \sum_{|s|=0}^m(-1)^{|s|} D^s\left(a_s(x) D^s u\right)=f(x), x=\left(x_1, \ldots, x_n\right) \in G \subset \mathbb{R}^n, \quad \text { (4.106) }$$
$$B_j u(x)=\varphi_j(x), x \in \partial G, j=0,1, \ldots, n-1,$$

where $m, n$ are given natural numbers; $a_s(x), 0 \leq|s| \leq m ; f(x)$ are given functions; $s=\left(s_1, \ldots, s_n\right),|s|=\sum_{k=1}^n s_k ; D^s=\frac{\partial^{|s|}}{\partial x_{x_1}^{s_1} \ldots \partial x_{x_n^s}^{s i n}} ; \quad B_j u=\sum_{|s|=0}^{m_j} b_{j, s}(x) D^s u(x)$ are given differential operators of the order $m_j$; $G$ is a given domain. Widely used methods of solving time independent boundary value problems (4.106) and (4.107) and and unstable boundary-value problems:
$$\frac{\partial u(x, t)}{\partial t}-A u(x, t)=f(x, t), x \in G, 0<t<T,$$
where the required function $u(x, t)$ satisfies the boundary conditions:
$$B_j u(x, t)=\varphi_j(x, t), x \in \partial D, j=0,1, \ldots, n-1,$$
and the initial condition:
$$u(x, 0)=\psi(x), x \in G,$$
According to the theory of the structural Rvachov method, to find an approximate solution of the above problems, the following steps must be performed:

• Construct the structure of the approximate solution of problems (4.106), (4.107), or $(4.108,4.109$, and $4.110)$.
• Find unknown components of the structure of the approximate solution (parameters, unknown functions of one variable $t$, etc.) by solving appropriate systems of equations (linear or nonlinear, algebraic, or transcendental) which can also be the systems of differential equations that must be solved together with the appropriate initial conditions – Cauchy conditions.

## 数学代写|最优化作业代写优化理论代考|解决数学物理问题的结构方法基本原理

$$A u \equiv \sum_{|s|=0}^m(-1)^{|s|} D^s\left(a_s(x) D^s u\right)=f(x), x=\left(x_1, \ldots, x_n\right) \in G \subset \mathbb{R}^n, \quad \text { (4.106) }$$
$$B_j u(x)=\varphi_j(x), x \in \partial G, j=0,1, \ldots, n-1,$$

$$\frac{\partial u(x, t)}{\partial t}-A u(x, t)=f(x, t), x \in G, 0<t<T,$$

$$B_j u(x, t)=\varphi_j(x, t), x \in \partial D, j=0,1, \ldots, n-1,$$

$$u(x, 0)=\psi(x), x \in G,$$

• 构造问题(4.106)、(4.107)或近似解的结构 $(4.108,4.109$，以及 $4.110)$.
• 求近似解结构的未知分量(参数，一个变量的未知函数 $t$等)通过求解适当的方程组(线性或非线性、代数或超越的)，这些方程组也可以是微分方程组，必须与适当的初始条件-柯西条件一起求解

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