# 网课代修|Matlab代写|NUMERICAL METHODS

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## 网课代修|Matlab代写|NUMERICAL METHODS

By now you have seen most of the exact methods for finding solutions to first-order ordinary differential equations. The methods have also given you a view of the general behavior and properties of solutions to differential equations. However, it must be admitted that in many instances exact solutions cannot be found and we must resort to numerical solutions.

In this section we present the two most commonly used methods for solving differential equations: Euler and Runge-Kutta methods. There are many more methods and the interested student is referred to one of countless numerical methods books. A straightforward extension of these techniques can be applied to systems of first-order and higher-order differential equations.

• Euler and modified Euler methods
Consider the following first-order differential equation and initial condition:
$$\frac{d y}{d x}=f(x, y), \quad y\left(x_{0}\right)=y_{0}$$
Euler’s method is based on a Taylor series expansion of the solution about $x_{0}$ or
$$y\left(x_{0}+h\right)=y\left(x_{0}\right)+h y^{\prime}\left(x_{0}\right)+\frac{1}{2} y^{\prime \prime}(\xi) h^{2}, \quad x_{0}<\xi<x_{0}+h,$$
where $h$ is the step size. Euler’s method consists of taking a sufficiently small $h$ so that only the first two terms of this Taylor expansion are significant.
Let us now replace $y^{\prime}\left(x_{0}\right)$ by $f\left(x_{0}, y_{0}\right)$. Using subscript notation, we have that
$$y_{i+1}=y_{i}+h f\left(x_{i}, y_{i}\right)+O\left(h^{2}\right) .$$

## 网课代修|Matlab代写|Runge-Kutta method

As we have just shown, the accuracy of numerical solutions of ordinary differential equations can be improved by adding more terms to the Taylor expansion. The RungeKutta method ${ }^{13}$ builds upon this idea, just as the modified Euler method did.
Let us assume that the numerical solution can be approximated by
$$y_{i+1}=y_{i}+a k_{1}+b k_{2},$$
where
$$k_{1}=h f\left(x_{i}, y_{i}\right) \quad \text { and } \quad k_{2}=h f\left(x_{i}+A_{1} h, y_{i}+B_{1} k_{1}\right) .$$
Here $a, b, A_{1}$, and $B_{1}$ are four unknowns. Equation $1.7 .11$ was suggested by the modified Euler method that we just presented. In that case, the truncated Taylor series had an error of $O\left(h^{3}\right)$. We anticipate such an error in the present case.
Because the Taylor series expansion of $f(x+h, y+k)$ about $(x, y)$ is
\begin{aligned} f(x+h, y+k) &=f(x, y)+\left(h f_{x}+k f_{y}\right)+\frac{1}{2}\left(h^{2} f_{x x}+2 h k f_{x y}+k^{2} f_{y y}\right) \ &+\frac{1}{6}\left(h^{3} f_{x x x}+3 h^{2} k f_{x x y}+3 h k^{2} f_{x y y}+k^{3} f_{y y y}\right)+\cdots \end{aligned}
$k_{2}$ can be rewritten
\begin{aligned} k_{2} &=h f\left[x_{i}+A_{1} h, y_{i}+B h f\left(x_{i}, y_{i}\right)\right] \ &=h\left[f\left(x_{i}, y_{i}\right)+\left(A_{1} h f_{x}+B_{1} h f f_{y}\right)\right] \ &=h f+A_{1} h^{2} f_{x}+B_{1} h^{2} f f_{y} \end{aligned}
where we have retained only terms up to $O\left(h^{2}\right)$ and neglected all higher-order terms. Finally, substituting Equation 1.7.16 into Equation 1.7.11 gives
$$y_{i+1}=y_{i}+(a+b) h f+\left(A_{1} b f_{x}+B_{1} b f f_{y}\right) h^{2} .$$

## 网课代修|MATLAB代写|NUMERICAL METHODS

• Euler 和修正的 Euler 方法
考虑以下一阶微分方程和初始条件：
d是dX=F(X,是),是(X0)=是0
欧拉方法是基于关于解的泰勒级数展开X0或者
是(X0+H)=是(X0)+H是′(X0)+12是′′(X)H2,X0<X<X0+H,
在哪里H是步长。欧拉方法包括取一个足够小的H所以只有这个泰勒展开式的前两项是有意义的。
现在让我们替换是′(X0)经过F(X0,是0). 使用下标符号，我们有
是一世+1=是一世+HF(X一世,是一世)+○(H2).

## 网课代修|MATLAB代写|RUNGE-KUTTA METHOD

ķ1=HF(X一世,是一世) 和 ķ2=HF(X一世+一个1H,是一世+乙1ķ1).

F(X+H,是+ķ)=F(X,是)+(HFX+ķF是)+12(H2FXX+2HķFX是+ķ2F是是) +16(H3FXXX+3H2ķFXX是+3Hķ2FX是是+ķ3F是是是)+⋯
ķ2可以改写
ķ2=HF[X一世+一个1H,是一世+乙HF(X一世,是一世)] =H[F(X一世,是一世)+(一个1HFX+乙1HFF是)] =HF+一个1H2FX+乙1H2FF是

## Matlab代写

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