# 数学代写|数值分析代写numerical analysis代考|MATH3003

## 数学代写|数值分析代写numerical analysis代考|Numerical examples

In this section, we apply the Euler method to particular examples.
Example 4.4. As a first example, we consider the initial value problem associated with the logistic equation
$$x^{\Lambda}(t)=(\alpha \ominus(\alpha x(t))) x(t), \quad x(0)=2,$$
where $\mathbb{T}=\mathbb{N}_0$ and $\alpha$ is a real number. The logistic equation in both continuous and discrete cases is known to be one of the basic models of the population growth [1]. It is not as simple as the exponential growth model since it takes into account the carrying capacity of the system on which the population of a certain species is studied. However, this model ignores many features and external effects of the population growth. Nevertheless, it is still widely used in population related problems.

Notice that the problem can be written as
$$\chi^{\Delta}=\frac{\alpha(1-x)}{1+\alpha \mu(t) x} x, \quad \chi(0)=2 .$$
We apply the Euler method to the problem with three different step sizes explained below.

Case 1. Let $t \in[0,30]{\mathrm{T}}$ and let $t_0=0, t_i=\sigma^{l{i-1}}\left(t_{i-1}\right)=t_{i-1}+1$, where $i=1, \ldots, 30$. Then we have $l_i=1$ and $\sigma\left(t_{i-1}\right)-t_{i-1}=1$ for all $i=1, \ldots, 30$. Hence the computed sequence of values of the solution $x$ is defined as
$$x_i=x_{i-1}+\left(\sigma\left(x_{i-1}\right)-x_{i-1}\right) \frac{\alpha\left(1-x_{i-1}\right)}{1+\alpha \mu\left(t_{i-1}\right) x_{i-1}} x_{i-1}=\frac{(1+\alpha) x_{i-1}}{1+\alpha x_{i-1}}, \quad i=1, \ldots, 30 .$$
In fact, the exact solution of the problem is obtained as
$$x_i^{(e)}(t)=\frac{(1+\alpha) x_{i-1}^{(e)}(t)}{1+\alpha x_{i-1}^{(e)}(t)}$$

## 数学代写|数值分析代写numerical analysis代考|The trapezoid rule

The trapezoid rule is a method that can be deduced from the Taylor series method of order 2 . It is an implicit method and its application to nonlinear differential equations requires use of suitable numerical methods.

In this section, we will introduce the trapezoid rule for IVPs associated with the first order dynamic equations on time scales [11].
We start again with the Taylor formula for $x^{\Lambda}$ which gives
\begin{aligned} \chi^{\Delta}(t+r) &=\chi^{\Delta}(t)+r \chi^{\Delta^2}(t)+\int_t^{\rho(t+r)} h_1(t+r, \sigma(\tau)) \chi^{\Delta^3}(\tau) \Delta \tau \ &=x^{\Delta}(t)+r \chi^{\Delta^2}(t)+R_1(r), \end{aligned}
whereupon
$$r x^{\Delta^2}(t)=\chi^{\Delta}(t+r)-\chi^{\Delta}(t)-R_1(r), \quad t, t+r \in\left[t_0, t_f\right], \quad r>0 .$$
We substitute the latter relation into equation (5.2) and find
\begin{aligned} \chi(t+r) &=\chi(t)+r x^{\Delta}(t)+h_2(t+r, t) \chi^{\Delta^2}(t)+R_2(r) \ &=\chi(t)+r x^{\Delta}(t)+\frac{h_2(t+r, t)}{r}\left(r x^{\Delta^2}(t)\right)+R_2(r) \ &=\chi(t)+r x^{\Delta}(t)+\frac{h_2(t+r, t)}{r}\left(x^{\Delta}(t+r)-\chi^{\Lambda}(t)-R_1(r)\right)+R_2(r) \end{aligned}

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Numerical examples

$$x^{\Lambda}(t)=(\alpha \ominus(\alpha x(t))) x(t), \quad x(0)=2,$$

$$x_i=x_{i-1}+\left(\sigma\left(x_{i-1}\right)-x_{i-1}\right) \frac{\alpha\left(1-x_{i-1}\right)}{1+\alpha \mu\left(t_{i-1}\right) x_{i-1}} x_{i-1}=\frac{(1+\alpha) x_{i-1}}{1+\alpha x_{i-1}}, \quad i=1, \ldots, 30 .$$

$$x_i^{(e)}(t)=\frac{(1+\alpha) x_{i-1}^{(e)}(t)}{1+\alpha x_{i-1}^{(e)}(t)}$$

## 数学代写|数值分析代写numerical analysis代考|The trapezoid rule

$$\chi^{\Delta}(t+r)=\chi^{\Delta}(t)+r \chi^{\Delta^2}(t)+\int_t^{\rho(t+r)} h_1(t+r, \sigma(\tau)) \chi^{\Delta^3}(\tau) \Delta \tau \quad=x^{\Delta}(t)+r \chi^{\Delta^2}(t)+R_1(r)$$

$$r x^{\Delta^2}(t)=\chi^{\Delta}(t+r)-\chi^{\Delta}(t)-R_1(r), \quad t, t+r \in\left[t_0, t_f\right], \quad r>0 .$$

$$\chi(t+r)=\chi(t)+r x^{\Delta}(t)+h_2(t+r, t) \chi^{\Delta^2}(t)+R_2(r) \quad=\chi(t)+r x^{\Delta}(t)+\frac{h_2(t+r, t)}{r}\left(r x^{\Delta^2}(t)\right)$$

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