如果你也在 怎样代写微分方程differential equation这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。微分方程differential equation在数学中，是将一个或多个未知函数及其导数联系起来的方程。在应用中，函数通常代表物理量，导数代表其变化率，而微分方程则定义了两者之间的关系。这种关系很常见；因此，微分方程在许多学科，包括工程、物理学、经济学和生物学中发挥着突出作用。
微分方程differential equation研究主要包括研究其解（满足每个方程的函数集合），以及研究其解的性质。只有最简单的微分方程可以用明确的公式求解；然而，一个给定的微分方程的解的许多属性可以在不精确计算的情况下确定。
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数学代写|微分方程代写differential equation代考|Euler’s method
For the initial value problem of ordinary differential equations, Euler’s method is the most straightforward. Euler’s method is named after a Swiss mathematician Leonhard Euler (1707-1783), who proposed this algorithm in his book “Institutionum calculi integralis” in 1768. It is the oldest numerical algorithm for differential equations.
Although the algorithm appears to be very simple, it is helpful in understanding other complicated algorithms to be presented later. Therefore, Euler’s method is introduced, and its MATLAB implementation is completed.
Assume that at time $t_{0}$, the initial state vector $\boldsymbol{x}\left(t_{0}\right)$ is known. A very small stepsize $h$ can be selected, and from the definition of derivatives in (3.1.1),
$$
\lim {h \rightarrow 0} \frac{\boldsymbol{x}\left(t{0}+h\right)-\boldsymbol{x}\left(t_{0}\right)}{t_{0}+h-t_{0}}=\boldsymbol{f}(t, \boldsymbol{x}(t)) .
$$
It can be seen that in mathematics there is no step-size $h$ satisfying $h \rightarrow 0$. A relatively small step-size $h$ can be chosen instead, and the limit sign in (3.2.1) can be removed, such that the $\boldsymbol{x}(t)$ at time $t_{0}+h$ can be approximately expressed as
$$
\hat{\boldsymbol{x}}\left(t_{0}+h\right) \approx \boldsymbol{x}\left(t_{0}\right)+h \boldsymbol{f}\left(t_{0}, \boldsymbol{x}\left(t_{0}\right)\right) .
$$
Strictly speaking, such an approximation may yield errors. Therefore the state vector at time $t_{0}+h$ can be expressed as
$$
\boldsymbol{x}\left(t_{0}+h\right)=\hat{\boldsymbol{x}}\left(t_{0}+h\right)+\boldsymbol{R}{0}=\boldsymbol{x}\left(t{0}\right)+h \boldsymbol{f}\left(t, \boldsymbol{x}\left(t_{0}\right)\right)+\boldsymbol{R}_{0}
$$
数学代写|微分方程代写differential equation代考|Second-order Runge–Kutta algorithm
Since the accuracy of the Euler’s method described earlier is too low, better algorithms with higher accuracy are expected. Runge-Kutta methods are a family of widely used numerical methods. Runge-Kutta methods are named after two German mathematicians Carl David Tolmé Runge (1856-1927) and Martin Wilhelm Kutta (1867-1944), where Runge proposed the method in 1895, and Kutta proposed a series of computation formulas of orders less than 4 .
Euler’s method discussed earlier is also a special case of these formulas, known as the first-order Runge-Kutta method. Here, another simple Runge-Kutta method second-order Runge-Kutta algorithm – is illustrated.
数学代写|微分方程代写DIFFERENTIAL EQUATION代考|Fourth-order Runge–Kutta algorithm
The fourth-order fixed-step Runge-Kutta algorithm is often taught in numerical analysis and system simulation courses. It is regarded as an effective algorithm. The structure of the algorithm is simple and suitable for computer implementation.
Theorem 3.5. With the fourth-order Runge-Kutta algorithm, the states can be recursively computed from
$$
\boldsymbol{x}{k+1}=\boldsymbol{x}{k}+\frac{h}{6}\left(\boldsymbol{k}{1}+2 \boldsymbol{k}{2}+2 \boldsymbol{k}{3}+\boldsymbol{k}{4}\right)
$$
where the four intermediate vectors can be computed from
$$
\left{\begin{array}{l}
\boldsymbol{k}{1}=\boldsymbol{f}\left(t{k}, \boldsymbol{x}{k}\right), \ \boldsymbol{k}{2}=\boldsymbol{f}\left(t_{k}+h / 2, \boldsymbol{x}{k}+h \boldsymbol{k}{1} / 2\right), \
\boldsymbol{k}{3}=\boldsymbol{f}\left(t{k}+h / 2, \boldsymbol{x}{k}+h \boldsymbol{k}{2} / 2\right), \
\boldsymbol{k}{4}=\boldsymbol{f}\left(t{k}+h, \boldsymbol{x}{k}+h \boldsymbol{k}{3}\right)
\end{array}\right.
$$
where $h$ is the step-size, which can be a constant.
微分方程代写
数学代写|微分方程代写DIFFERENTIAL EQUATION代考|EULER’S METHOD
对于常微分方程的初值问题，欧拉法是最直接的。欧拉方法以瑞士数学家莱昂哈德欧拉命名1707−1783，他在 1768 年的《Institutionum calculi integralis》一书中提出了这个算法。它是最古老的微分方程数值算法。
尽管该算法看起来很简单，但它有助于理解稍后介绍的其他复杂算法。因此，引入欧拉法，完成其MATLAB实现。
假设当时吨0，初始状态向量X(吨0)是已知的。非常小的步长H可以选择，并从导数的定义中3.1.1,
$$
\lim {h \rightarrow 0} \frac{\boldsymbol{x}\left(t{0}+h\right)-\boldsymbol{x}\left(t_{0}\right)}{t_{0}+h-t_{0}}=\boldsymbol{f}(t, \boldsymbol{x}(t)) .
$$
It can be seen that in mathematics there is no step-size $h$ satisfying $h \rightarrow 0$. A relatively small step-size $h$ can be chosen instead, and the limit sign in (3.2.1) can be removed, such that the $\boldsymbol{x}(t)$ at time $t_{0}+h$ can be approximately expressed as
$$
\hat{\boldsymbol{x}}\left(t_{0}+h\right) \approx \boldsymbol{x}\left(t_{0}\right)+h \boldsymbol{f}\left(t_{0}, \boldsymbol{x}\left(t_{0}\right)\right) .
$$
Strictly speaking, such an approximation may yield errors. Therefore the state vector at time $t_{0}+h$ can be expressed as
$$
\boldsymbol{x}\left(t_{0}+h\right)=\hat{\boldsymbol{x}}\left(t_{0}+h\right)+\boldsymbol{R}{0}=\boldsymbol{x}\left(t{0}\right)+h \boldsymbol{f}\left(t, \boldsymbol{x}\left(t_{0}\right)\right)+\boldsymbol{R}_{0}
$$
数学代写|微分方程代写DIFFERENTIAL EQUATION代考|SECOND-ORDER RUNGE–KUTTA ALGORITHM
由于前面描述的 Euler 方法的精度太低，因此需要具有更高精度的更好算法。Runge-Kutta 方法是一系列广泛使用的数值方法。Runge-Kutta 方法以两位德国数学家 Carl David Tolmé Runge 的名字命名1856−1927和马丁·威廉·库塔1867−1944，其中龙格在 1895 年提出了该方法，库塔提出了一系列小于 4 阶的计算公式。
前面讨论的欧拉方法也是这些公式的一个特例，称为一阶龙格-库塔方法。在这里，说明了另一种简单的 Runge-Kutta 方法二阶 Runge-Kutta 算法。
数学代写|微分方程代写DIFFERENTIAL EQUATION代考|FOURTH-ORDER RUNGE–KUTTA ALGORITHM
四阶固定步长 Runge-Kutta 算法经常在数值分析和系统仿真课程中教授。它被认为是一种有效的算法。该算法结构简单，适合计算机实现。
定理 3.5。使用四阶 Runge-Kutta 算法，可以从
$$
\boldsymbol{x}{k+1}=\boldsymbol{x}{k}+\frac{h}{6}\left(\boldsymbol{k}{1}+2 \boldsymbol{k}{2}+2 \boldsymbol{k}{3}+\boldsymbol{k}{4}\right)
$$
where the four intermediate vectors can be computed from
$$
\left{\begin{array}{l}
\boldsymbol{k}{1}=\boldsymbol{f}\left(t{k}, \boldsymbol{x}{k}\right), \ \boldsymbol{k}{2}=\boldsymbol{f}\left(t_{k}+h / 2, \boldsymbol{x}{k}+h \boldsymbol{k}{1} / 2\right), \
\boldsymbol{k}{3}=\boldsymbol{f}\left(t{k}+h / 2, \boldsymbol{x}{k}+h \boldsymbol{k}{2} / 2\right), \
\boldsymbol{k}{4}=\boldsymbol{f}\left(t{k}+h, \boldsymbol{x}{k}+h \boldsymbol{k}{3}\right)
\end{array}\right.
$$
在哪里H是步长，可以是一个常数。
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