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## 数学代写|多变量微积分和常微分方程代考Multivariate Calculus & Ordinary Differential Equations代写|Systems with Constant Coeﬃﬃﬃcients

Our discussion in Lecture 18 has restricted usage of obtaining explicit solutions of homogeneous and, in general, of nonhomogeneous differential systems. This is so because the solution (18.4) involves an infinite series with repeated integrations and (18.14) involves its inversion. In fact, even if the matrix $A(x)$ is of second order, no general method of finding the explicit form of the fundamental matrix is available. Further, if the matrix $A$ is constant, then the computation of the elements of the fundamental matrix $e^{A x}$ from the series (18.4) may turn out to be difficult, if not impossible. However, in this case the notion of eigenvalues and eigenvectors of the matrix $A$ can be used to avoid unnecessary computation. For this, the first result we prove is the following theorem.

Theorem 19.1. Let $\lambda_1, \ldots, \lambda_n$ be the distinct eigenvalues of the matrix $A$ and $v^1, \ldots, v^n$ be the corresponding eigenvectors. Then the set
$$u^1(x)=v^1 e^{\lambda_1 x}, \quad \cdots \quad, u^n(x)=v^n e^{\lambda_n x}$$
is a fundamental set of solutions of (18.6).
Proof. Since $v^i$ is an eigenvector of $A$ corresponding to the eigenvalue $\lambda_i$, we find
$$\left(u^i(x)\right)^{\prime}=\left(v^i e^{\lambda_i x}\right)^{\prime}=\lambda_i v^i e^{\lambda_i x}=A v^i e^{\lambda_i x}=A u^i(x)$$
and hence $u^i(x)$ is a solution of (18.6). To show that (19.1) is a fundamental set, we note that $W(0)=\operatorname{det}\left[v^1, \ldots, v^n\right] \neq 0$, since $v^1, \ldots, v^n$ are linearly independent from Theorem 14.1. Thus, the result follows from Theorem 17.1.
Obviously, from Theorem $19.1$ it follows that
$$e^{A x}=\left[v^1 e^{\lambda_1 x}, \ldots, v^n e^{\lambda_n x}\right]\left[v^1, \ldots, v^n\right]^{-1}$$
and the general solution of $(18.6)$ can be written as
$$u(x)=\sum_{i=1}^n c_i v^i e^{\lambda_i x} .$$

## 数学代写|多变量微积分和常微分方程代考Multivariate Calculus & Ordinary Differential Equations代写|Periodic Linear Systems

A function $y(x)$ is called periodic of period $\omega>0$ if for all $x$ in the domain of the function
$$y(x+\omega)=y(x) .$$
Geometrically, this means that the graph of $y(x)$ repeats itself in successive intervals of length $\omega$. For example, the functions $\sin x$ and $\cos x$ are periodic of period $2 \pi$. For convenience, we shall assume that $\omega$ is the smallest positive number for which (20.1) holds. If each component $u_i(x), 1 \leq i \leq n$ of $u(x)$ and each element $a_{i j}(x), 1 \leq i, j \leq n$ of $A(x)$ are periodic of period $\omega$, then $u(x)$ and $A(x)$ are said to be periodic of period $\omega$. Periodicity of solutions of differential systems is an interesting and important aspect of qualitative study. Here we shall provide certain characterizations for the existence of such solutions of linear differential systems.

To begin with we shall provide necessary and sufficient conditions for the differential system (17.1) to have a periodic solution of period $\omega$.

Theorem 20.1. Let the matrix $A(x)$ and the function $b(x)$ be continuous and periodic of period $\omega$ in $\mathbb{R}$. Then the differential system (17.1) has a periodic solution $u(x)$ of period $\omega$ if and only if $u(0)=u(\omega)$.

Proof. Let $u(x)$ be a periodic solution of period $\omega$, then by definition it is necessary that $u(0)=u(\omega)$. To show sufficiency, let $u(x)$ be a solution of (17.1) satisfying $u(0)=u(\omega)$. If $v(x)=u(x+\omega)$, then it follows that $v^{\prime}(x)=u^{\prime}(x+\omega)=A(x+\omega) u(x+\omega)+b(x+\omega)=A(x) v(x)+b(x)$; i.e., $v(x)$ is a solution of (17.1). However, since $v(0)=u(\omega)=u(0)$, the uniqueness of the initial value problems implies that $u(x)=v(x)=u(x+\omega)$, and hence $u(x)$ is periodic of period $\omega$.

Corollary 20.2. Let the matrix $A(x)$ be continuous and periodic of period $\omega$ in $\mathbb{R}$. Further, let $\Psi(x)$ be a fundamental matrix of the differential system (17.3). Then the differential system (17.3) has a nontrivial periodic solution $u(x)$ of period $\omega$ if and only if $\operatorname{det}(\Psi(0)-\Psi(\omega))=0$.

Proof. We know that the general solution of the differential system (17.3) is $u(x)=\Psi(x) c$, where $c$ is an arbitrary constant vector. This $u(x)$ is periodic of period $\omega$ if and only if $\Psi(0) c=\Psi(\omega) c$, i.e., the system $(\Psi(0)-$ $\Psi(\omega)) c=0$ has a nontrivial solution vector $c$. But, from Theorem $13.2$ this system has a nontrivial solution if and only if $\operatorname{det}(\Psi(0)-\Psi(\omega))=0$.

# 多变量微积分和常微分方程代考

## 数学代写多变量微积分和常微分方程代考Multivariate Calculus \& Ordinary Differential Equations代写|Systems with Constant Coeffiffifficients

$$u^1(x)=v^1 e^{\lambda_1 x}, \quad \cdots \quad, u^n(x)=v^n e^{\lambda_n x}$$

$$\left(u^i(x)\right)^{\prime}=\left(v^i e^{\lambda_i x}\right)^{\prime}=\lambda_i v^i e^{\lambda_{i x}}=A v^i e^{\lambda_{i x}}=A u^i(x)$$

$$e^{A x}=\left[v^1 e^{\lambda_1 x}, \ldots, v^n e^{\lambda_n x}\right]\left[v^1, \ldots, v^n\right]^{-1}$$

$$u(x)=\sum_{i=1}^n c_i v^i e^{\lambda_i x} .$$

## 数学代写多变量凱积分和常微分方程代坒Multivariate Calculus \& Ordinary Differential Equations代胃|Periodic Linear Systems

$$y(x+\omega)=y(x) .$$

$v^{\prime}(x)=u^{\prime}(x+\omega)=A(x+\omega) u(x+\omega)+b(x+\omega)=A(x) v(x)+b(x) ; \mathbb{I}$ 。 $v(x)$ 是 (17.1) 的解。然而，由于 $v(0)=u(\omega)=u(0)$ ，初始值问题的唯一性意味着 $u(x)=v(x)=u(x+\omega)$ ，因此 $u(x)$ 是周期的周期 $\omega$.

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Ordinary Differential Equations, 多变量微积分和常微分方程, 常微分方程, 数学代写

## avatest™帮您通过考试

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## 数学代写|多变量微积分和常微分方程代考Multivariate Calculus & Ordinary Differential Equations代写|Preliminary Results from Algebra and Analysis

For future reference we collect here several fundamental concepts and results from algebra and analysis.
A function $P_n(x)$ defined by
$$P_n(x)=a_0+a_1 x+\cdots+a_n x^n=\sum_{i=0}^n a_i x^i, \quad a_n \neq 0$$
where $a_i \in \mathbb{R}, 0 \leq i \leq n$, is called a polynomial of degree $n$ in $x$. If $P_n\left(x_1\right)=0$, then the number $x=x_1$ is called a zero of $P_n(x)$. The following fundamental theorem of algebra of complex numbers is valid.

Theorem 13.1. Every polynomial of degree $n \geq 1$ has at least one zero.

Thus, $P_n(x)$ has exactly $n$ zeros; however, some of these may be the same, i.e., $P_n(x)$ can be written as
$$P_n(x)=a_n\left(x-x_1\right)^{r_1}\left(x-x_2\right)^{r_2} \cdots\left(x-x_k\right)^{r_k}, \quad r_i \geq 1, \quad 1 \leq i \leq k,$$
where $\sum_{i=1}^k r_i=n$. If $r_i=1$, then $x_i$ is called a simple zero, and if $r_i>$ 1 , then multiple zero of multiplicity $r_i$. Thus, if $x_i$ is a multiple zero of multiplicity $r_i$, then $P^{(j)}\left(x_i\right)=0,0 \leq j \leq r_i-1$ and $P^{\left(r_i\right)}\left(x_i\right) \neq 0$.

A rectangular table of $m \times n$ elements arranged in $m$ rows and $n$ columns
$$\left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \ a_{21} & a_{22} & \cdots & a_{2 n} \ \cdots & & & \ a_{m 1} & a_{m 2} & \cdots & a_{m n} \end{array}\right]$$
is called an $m \times n$ matrix and in short represented as $A=\left(a_{i j}\right)$. We shall mainly deal with square matrices $(m=n)$, row matrices or row vectors $(m=1)$, and column matrices or column vectors $(n=1)$.

## 数学代写|多变量微积分和常微分方程代考Multivariate Calculus & Ordinary Differential Equations代写|Preliminary Results from Algebra and Analysis (Contd.)

The number $\lambda$, real or complex, is called an eigenvalue of the matrix $A$ if there exists a nonzero real or complex vector $v$ such that $A v=\lambda v$. The vector $v$ is called an eigenvector corresponding to the eigenvalue $\lambda$. From Theorem 13.2, $\lambda$ is an eigenvalue of $A$ if and only if it is a solution of the characteristic equation $p(\lambda)=\operatorname{det}(A-\lambda I)=0$. Since the matrix $A$ is $n \times n, p(\lambda)$ is a polynomial of degree exactly $n$, and it is called the characteristic polynomial of $A$. Hence, from Theorem $13.1$ it follows that $A$ has exactly $n$ eigenvalues counting their multiplicities.

In the case when the eigenvalues $\lambda_1, \ldots, \lambda_n$ of $A$ are distinct it is easy to find the corresponding eigenvectors $v^1, \ldots, v^n$. For this, first we note that for the fixed eigenvalue $\lambda_j$ of $A$ at least one of the cofactors of $\left(a_{i i}-\lambda_j\right)$ in the matrix $\left(A-\lambda_j I\right)$ is nonzero. If not, then from (13.5) it follows that $p^{\prime}(\lambda)=-\left[\right.$ cofactor of $\left.\left(a_{11}-\lambda\right)\right]-\cdots-\left[\operatorname{cofactor}\right.$ of $\left.\left(a_{n n}-\lambda\right)\right]$, and hence $p^{\prime}\left(\lambda_j\right)=0$; i.e., $\lambda_j$ was a multiple root, which is a contradiction to our assumption that $\lambda_j$ is simple. Now let the cofactor of $\left(a_{k k}-\lambda_j\right)$ be different from zero, then one of the possible nonzero solution of the system $\left(A-\lambda_j I\right) v^j=0$ is $v_i^j=$ cofactor of $a_{k i}$ in $\left(A-\lambda_j I\right), 1 \leq i \leq n, i \neq k$, $v_k^j=$ cofactor of $\left(a_{k k}-\lambda_j\right)$ in $\left(A-\lambda_j I\right)$. Since for this choice of $v^j$, it follows from (13.2) that every equation, except the $k$ th one, of the system $\left(A-\lambda_j I\right) v^j=0$ is satisfied, and for the $k$ th equation from (13.1), we have $\sum_{i \neq \hbar}^n a_{k i}\left[\right.$ cofactor of $\left.a_{k i}\right]+\left(a_{k k}-\lambda_j\right)\left[\operatorname{cofactor}\right.$ of $\left.\left(a_{k k}-\lambda_j\right)\right]=\operatorname{det}\left(A-\lambda_j I\right)$,
which is also zero. In conclusion this $v^j$ is the eigenvector corresponding to the eigenvalue $\lambda_j$.

# 多变量微积分和常微分方程代考

## 数学代写|多变量微积分和常微分方程代考Multivariate Calculus \& Ordinary Differential Equations代写|Preliminary Results from Algebra and Analysis

$$P_n(x)=a_0+a_1 x+\cdots+a_n x^n=\sum_{i=0}^n a_i x^i, \quad a_n \neq 0$$

$$P_n(x)=a_n\left(x-x_1\right)^{r_1}\left(x-x_2\right)^{r_2} \cdots\left(x-x_k\right)^{\tau_k}, \quad r_i \geq 1, \quad 1 \leq i \leq k,$$

## 数学代写|多变量微积分和常微分方程代考Multivariate Calculus \& Ordinary Differential Equations代写|Preliminary Results from Algebra and Analysis (Contd.)

$\left.\left(a_{11}-\lambda\right)\right]-\cdots-\left[\right.$ cofactor的 $\left(a_{n n}-\lambda\right)$ ， 因此 $p^{\prime}\left(\lambda_j\right)=0 ; \mathrm{E}^2$ 。 $\lambda_j$ 是 个多重根，这与我们的假设相矛盾 $\lambda_j$ 很简单。现在 让 $\left(a_{k k}-\lambda_j\right)$ 不为零，则系统可能的非零解之一 $\left(A-\lambda_j I\right) v^j=0$ 是 $v_i^j=$ 的辅因子 $a_{k i}$ 在 $\left(A-\lambda_j I\right), 1 \leq i \leq n, i \neq k ， v_k^j=$ 的辅因子 $\left(a_{k k}-\lambda_j\right)$ 在 $\left(A-\lambda_j I\right)$. 因为对于这个选择 $v^j$ ，从 (13.2) 可以得出，每个方程，除了 $k$ 一系统的 $\left(A-\lambda_j I\right) v^j=0$ 是满意的，并且对于 $k(13.1)$ 中的方程，我们有 $\sum_{i \neq h}^n a_{k i}\left[\right.$ 的辅因子 $\left.a_{k i}\right]+\left(a_{k k}-\lambda_j\right)[$ cofactor的 $\left.\left(a_{k k}-\lambda_j\right)\right]=\operatorname{det}\left(A-\lambda_j I\right)$

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Ordinary Differential Equations, 多变量微积分和常微分方程, 常微分方程, 数学代写

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 数学代写|多变量微积分和常微分方程代考Multivariate Calculus & Ordinary Differential Equations代写|Exact Equations

Let, in the DE of first order and first degree (1.9), the function $f(x, y)=$ $-M(x, y) / N(x, y)$, so that it can be written as
$$M(x, y)+N(x, y) y^{\prime}=0,$$
where $M$ and $N$ are continuous functions having continuous partial derivatives $M_y$ and $N_x$ in the rectangle $S:\left|x-x_0\right|<a,\left|y-y_0\right|<b(0<a, b<$ $\infty)$

Equation (3.1) is said to be exact if there exists a function $u(x, y)$ such that
$$u_x(x, y)=M(x, y) \text { and } u_y(x, y)=N(x, y) .$$
The nomenclature comes from the fact that
$$M+N y^{\prime}=u_x+u_y y^{\prime}$$
is exactly the derivative $d u / d x$.
Once the DE (3.1) is exact its implicit solution is
$$u(x, y)=c .$$
If (3.1) is exact, then from (3.2) we have $u_{x y}=M_y$ and $u_{y x}=N_x$. Since $M_y$ and $N_x$ are continuous, we must have $u_{x y}=u_{y x}$; i.e., for (3.1) to be exact it is necessary that
$$M_y=N_x .$$
Conversely, if $M$ and $N$ satisfy (3.4) then the equation (3.1) is exact. To establish this we shall exhibit a function $u$ satisfying (3.2). We integrate both sides of $u_x=M$ with respect to $x$, to obtain
$$u(x, y)=\int_{x_0}^x M(s, y) d s+g(y) .$$

## 数学代写|多变量微积分和常微分方程代考Multivariate Calculus & Ordinary Differential Equations代写|Elementary First-Order Equations

Suppose in the DE of first order (3.1), M(x,y) $=X_1(x) Y_1(y)$ and $N(x, y)=X_2(x) Y_2(y)$, so that it takes the form
$$X_1(x) Y_1(y)+X_2(x) Y_2(y) y^{\prime}=0 .$$
If $Y_1(y) X_2(x) \neq 0$ for all $(x, y) \in S$, then (4.1) can be written as an exact DE
$$\frac{X_1(x)}{X_2(x)}+\frac{Y_2(y)}{Y_1(y)} y^{\prime}=0$$
in which the variables are separated. Such a DE (4.2) is said to be separable. The solution of this exact equation is given by
$$\int \frac{X_1(x)}{X_2(x)} d x+\int \frac{Y_2(y)}{Y_1(y)} d y=c .$$
Here both the integrals are indefinite and constants of integration have been absorbed in $c$.

Equation (4.3) contains all the solutions of (4.1) for which $Y_1(y) X_2(x) \neq$ 0 . In fact, when we divide (4.1) by $Y_1 X_2$ we might have lost some solutions, and the ones which are not in (4.3) for some $c$ must be coupled with (4.3) to obtain all solutions of (4.1).
Example 4.1. The DE in Example $3.2$ may be written as
$$\frac{1}{x}+\frac{1}{y(1-y)} y^{\prime}=0, \quad x y(1-y) \neq 0$$
for which (4.3) gives the solution $y=(1-c x)^{-1}$. Other possible solutions for which $x\left(y-y^2\right)=0$ are $x=0, y=0$, and $y=1$. However, the solution $y=1$ is already included in $y=(1-c x)^{-1}$ for $c=0$, and $x=0$ is not a solution, and hence all solutions of this DE are given by $y=0, y=(1-c x)^{-1}$.

# 多变量微积分和常微分方程代考

## 数学代写|多变量微积分和常微分方程代考Multivariate Calculus \& Ordinary Differential Equations代写|Exact Equations

$$M(x, y)+N(x, y) y^{\prime}=0,$$

$$u_x(x, y)=M(x, y) \text { and } u_y(x, y)=N(x, y) .$$

$$M+N y^{\prime}=u_x+u_y y^{\prime}$$

$$u(x, y)=c .$$

$$M_y=N_x .$$

$$u(x, y)=\int_{x_0}^x M(s, y) d s+g(y) .$$

## 数学代写|多变量微积分和常微分方程代考Multivariate Calculus \& Ordinary Differential Equations代写|Elementary First-Order Equations

$$X_1(x) Y_1(y)+X_2(x) Y_2(y) y^{\prime}=0 .$$

$$\frac{X_1(x)}{X_2(x)}+\frac{Y_2(y)}{Y_1(y)} y^{\prime}=0$$

$$\int \frac{X_1(x)}{X_2(x)} d x+\int \frac{Y_2(y)}{Y_1(y)} d y=c .$$

$$\frac{1}{x}+\frac{1}{y(1-y)} y^{\prime}=0, \quad x y(1-y) \neq 0$$
$y=1$ 经包含在 $y=(1-c x)^{-1}$ 为了c $=0$ ，和 $x=0$ 不是解，因此这个 $\mathrm{DE}$ 的所有解都由下式給出 $y=0, y=(1-c x)^{-1}$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。