### 数学代写|数学生态学作业代写Mathematical Ecology代考| Integral Equations

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Integral Equations

Integral equations contain integrals with unknown functions in their integrands. This textbook uses the following types of the integral equations:

• Volterra integral equation of the first kind with respect to the unknown function $x(t)$ of the one-dimensional independent variable $t \in[a, b]$ :
$$\int_{a}^{t} K(t, \tau) x(\tau) \mathrm{d} \tau=f(t), \quad t \in[a, b]$$
where the function $f(t)$ and the kernel $K(t, \tau)$ are given functions.
• Volterra integral equation of the second kind
$$x(t)=\int_{a}^{t} K(t, \tau) x(\tau) \mathrm{d} \tau+f(t), \quad t \in[a, b],$$
is more common as compared to (1.30) of the first kind.
Equations $(1.30)$ and (1.31) are named after Vito Volterra (1860-1940), a famous Italian mathematician and physicist, who introduced them and developed their theory and applications. The Volterra integral equations are widely used in population biology, physics, engineering, economics, and demography [3].

These equations are well suited for the description of dynamic processes. Indeed, if the variable $t$ is time, then the current state of a dynamic system (process) depends on the past states and cannot depend on the future. Hence, $K(t, \tau) \equiv 0$ at $\tau>t$ for dynamic systems, which is reflected in the integrals in (1.30) and (1.31).

Another major type of the linear integral equations is the Fredholm integral equations, which are described by the same expressions (1.31) and (1.31) where the upper integration limit $t$ is replaced with $b$. This small change causes significant differences in the qualitative dynamics of their solutions.

Despite the similarity of the Fredholm and Volterra integral equations, their properties are quite different. The Volterra integral equations (1.30) and (1.31) generalize the initial value problems for differential equations considered in Sect. 1.3.3, whereas the Fredholm integral equations correspond to boundary problems (not considered in this textbook).

Linear integral equations have well-developed theories. There is a variety of theorems about the existence and uniqueness of solutions for these equations [2]. The theorems differ in smoothness requirements and forms of the equation. In particular, the Volterra integral equation of the second kind has a unique solution under natural assumptions. A classic existence result is as follows:

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Existence and Uniqueness Theorem for Volterra Integral

If $K(t, \tau)$ is measurable on $[a, b] \otimes[a, b]$ and $f(t)$ is continuous on $[a, b]$, then a unique continuous solution $x(t)$ of the Volterra integral equation (1.31) of the second kind exists on $[a, b]$ and can be determined as
$$x(t)=\int_{a}^{t} R(t, \tau) f(\tau) d \tau+f(t)$$
where the resolvent kernel $R(t, \tau), \tau \in[a, b], t \in[a, b]$, is a solution of the following linear Volterra integral equation:
$$R(t, \tau)=\int_{\tau}^{t} R(t, u) K(u, \tau) \mathrm{d} u+K(t, \tau) .$$
In particular, if $K(t, \tau)=K=$ const, then
$$R(t, \tau)=K \mathrm{e}^{K(t-\tau)}$$
After discretization by the variable $t$, the linear integral equations (1.30) and (1.31) are reduced to systems of linear algebraic equations (1.4). The analogy between continuous integral models and their discrete analogues (1.4) is useful for better understanding and interpretation of the linear integral equations.

However, theory of linear continuous models is much more complex as compared to linear discrete models. In particular, a significant difference exists between the integral equations of the first and the second kind.
Nonlinear Volterra integral equation of the second kind
$$x(t)=\int_{a}^{t} F(t, \tau, x(\tau)) \mathrm{d} \tau+f(t), \quad t \in[a, b],$$
is the generalization of the linear integral equation (1.31) with the integrand $K(t, \tau) x$ replaced with a nonlinear function $F(t, \tau, x)$ of $x$.

Hammerstein-Volterra integral equation is the special case of the nonlinear equation (1.35) with the nonlinearity $F(t, \tau, x)=K(t, \tau) G(x)$ :
$$x(t)=\int_{a}^{t} K(t, \tau) G(x(\tau)) \mathrm{d} \tau+f(t), \quad t \in[a, b] .$$
A key condition for the existence and uniqueness of the solution $x$ to the nonlinear integral equation (1.35) is the Lipschitz condition for the function $F(t, s, x)$ with respect to $x:|F(t, s, x)-F(t, s, y)| \leq L(t, s)|x-y|$. If it holds, then (1.35) possesses a unique solution $x$, at least, for continuous $f$ and $L$.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Volterra Integral Equations with Variable Delay

The classic integral equations ( $1.30),(1.31),(1.35)$ take into account the distributed delay (after-effect, hereditary effects) on the interval $[a, t]$. The distributed delay means that a continuous sequence of the past states of a dynamical system affects the future evolution of the system. The integral equations with variable delay occur if the distributed delay exists at the initial time $t=a$ as well. Specifically, the linear integral equation with variable delay
$$x(t)=\int_{a(t)}^{t} K(t, \tau) x(\tau) \mathrm{d} \tau+f(t), \quad t \in\left[t_{0}, T\right]$$
with the initial condition
$$x(\tau)=x_{0}(\tau), \quad \tau \in\left[\tau_{0}, \quad t_{0}\right]$$
means that the solution $x$ depends on its known behavior $x_{0}$ over a certain prehistory interval $\left[\tau_{0}, t_{0}\right]$. The lower integration limit $a(t)$ in $(1.37)$ is a given function such as $\tau_{0} \leq a(t)<t .$

Equation (1.37) can be solved by reducing it to the standard Volterra integral equation

$1.3$ Review of Selected Mathematical Tools
19
$$x(t)=\int_{t_{0}}^{t} K(t, \tau) x(\tau) \mathrm{d} \tau+f_{1}(t), \quad t \in\left[t_{0}, t_{1}\right],$$
on some interval $\left[t_{0}, t_{1}\right]$ such that $a(t) \leq t_{0}, t \in\left[t_{0}, t_{1}\right]$. Then,
$$f_{1}(t)=\int_{a(t)}^{t_{0}} K(t, \tau) x_{0}(\tau) \mathrm{d} \tau+f(t)$$
is a given function on $\left[t_{0}, t_{1}\right]$. Next, this solution process is repeated on a new interval $\left[t_{1}, t_{2}\right]$ with the updated initial condition $x(\tau)=x_{0}(\tau), \tau \in\left[\tau_{0}, t_{1}\right]$, and so on.

The integral equations with variable delay are used for the modeling of economic development in Chaps. 4 and 5. Moreover, in some economic applications, the lower integration limit $a(t)$ can be an unknown control. Then, the model (1.39) leads to nonlinear integral equations with controllable delay (see Chap. 5).

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Integral Equations

• 关于未知函数的第一类沃尔泰拉积分方程X(吨)一维自变量的吨∈[一种,b] :
∫一种吨ķ(吨,τ)X(τ)dτ=F(吨),吨∈[一种,b]
函数在哪里F(吨)和内核ķ(吨,τ)被赋予功能。
• 第二类沃尔泰拉积分方程
X(吨)=∫一种吨ķ(吨,τ)X(τ)dτ+F(吨),吨∈[一种,b],
与第一种（1.30）相比更常见。
方程(1.30)和 (1.31) 以意大利著名数学家和物理学家 Vito Volterra (1860-1940) 的名字命名，他介绍了它们并发展了它们的理论和应用。Volterra 积分方程广泛用于人口生物学、物理学、工程学、经济学和人口学 [3]。

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Existence and Uniqueness Theorem for Volterra Integral

X(吨)=∫一种吨R(吨,τ)F(τ)dτ+F(吨)

R(吨,τ)=∫τ吨R(吨,在)ķ(在,τ)d在+ķ(吨,τ).

R(吨,τ)=ķ和ķ(吨−τ)

X(吨)=∫一种吨F(吨,τ,X(τ))dτ+F(吨),吨∈[一种,b],

Hammerstein-Volterra 积分方程是具有非线性的非线性方程 (1.35) 的特例F(吨,τ,X)=ķ(吨,τ)G(X) :
X(吨)=∫一种吨ķ(吨,τ)G(X(τ))dτ+F(吨),吨∈[一种,b].

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Volterra Integral Equations with Variable Delay

X(吨)=∫一种(吨)吨ķ(吨,τ)X(τ)dτ+F(吨),吨∈[吨0,吨]

X(τ)=X0(τ),τ∈[τ0,吨0]

1.3回顾选定的数学工具
19
X(吨)=∫吨0吨ķ(吨,τ)X(τ)dτ+F1(吨),吨∈[吨0,吨1],

F1(吨)=∫一种(吨)吨0ķ(吨,τ)X0(τ)dτ+F(吨)

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## MATLAB代写

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