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# 数学代写|数值分析代写Numerical analysis代考|MATH/CS514 Numerical Solution of Ordinary Differential Equations

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## 数学代写数值分析代写Numerical analysis代考|First Order Systems of Ordinary Differential Equations

Let us begin by reviewing the theory of ordinary differential equations. Many physical applications lead to higher order systems of ordinary differential equations, but there is a simple reformulation that will convert them into equivalent first order systems. Thus, we do not lose any generality by restricting our attention to the first order case throughout. Moreover, numerical solution schemes for higher order initial value problems are entirely based on their reformulation as first order systems.
First Order Systems
A first order system of ordinary differential equations has the general form
$$\frac{d u_1}{d t}=F_1\left(t, u_1, \ldots, u_n\right), \quad \ldots \quad \frac{d u_n}{d t}=F_n\left(t, u_1, \ldots, u_n\right) .$$
The unknowns $u_1(t), \ldots, u_n(t)$ are scalar functions of the real variable $t$, which usually represents time. We shall write the system more compactly in vector form
$$\frac{d \mathbf{u}}{d t}=\mathbf{F}(t, \mathbf{u}),$$
where $\mathbf{u}(t)=\left(u_1(t), \ldots, u_n(t)\right)^T$, and $\mathbf{F}(t, \mathbf{u})=\left(F_1\left(t, u_1, \ldots, u_n\right), \ldots, F_n\left(t, u_1, \ldots, u_n\right)\right)^T$ is a vector-valued function of $n+1$ variables. By a solution to the differential equation, we mean a vector-valued function $\mathbf{u}(t)$ that is defined and continuously differentiable on an interval $a<t<b$, and, moreover, satisfies the differential equation on its interval of definition. Each solution $\mathbf{u}(t)$ serves to parametrize a curve $C \subset \mathbb{R}^n$, also known as a trajectory or orbit of the system.

## 数学代写|数值分析代写Numerical analysis代考|Existence, Uniqueness, and Continuous Dependence

It goes without saying that there is no general analytical method that will solve all differential equations. Indeed, even relatively simple first order, scalar, non-autonomous ordinary differential equations cannot be solved in closed form. For example, the solution to the particular Riccati equation
$$\frac{d u}{d t}=u^2+t$$
cannot be written in terms of elementary functions, although there is a solution formula that relies on Airy functions. The Abel equation
$$\frac{d u}{d t}=u^3+t$$
fares even worse, since its general solution cannot be written in terms of even standard special functions – although power series solutions can be tediously ground out term by term. Understanding when a given differential equation can be solved in terms of elementary functions or known special functions is an active area of contemporary research, [6]. In this vein, we cannot resist mentioning that the most important class of exact solution techniques for differential equations are those based on symmetry. An introduction can be found in the author’s graduate level monograph $[\mathbf{4 1}]$; see also $[\mathbf{8}, \mathbf{2 7}]$.

Before worrying about how to solve a differential equation, either analytically, qualitatively, or numerically, it behooves us to try to resolve the core mathematical issues of existence and uniqueness. First, does a solution exist? If, not, it makes no sense trying to find one. Second, is the solution uniquely determined? Otherwise, the differential equation probably has scant relevance for physical applications since we cannot use it as a predictive tool. Since differential equations inevitably have lots of solutions, the only way in which we can deduce uniqueness is by imposing suitable initial (or boundary) conditions.

Unlike partial differential equations, which must be treated on a case-by-case basis, there are complete general answers to both the existence and uniqueness questions for initial value problems for systems of ordinary differential equations. (Boundary value problems are more subtle.) While obviously important, we will not take the time to present the proofs of these fundamental results, which can be found in most advanced textbooks on the subject, including $[\mathbf{4}, \mathbf{2 3}, \mathbf{2 6}, \mathbf{2 8}]$.

Let us begin by stating the Fundamental Existence Theorem for initial value problems associated with first order systems of ordinary differential equations.

## 数学代写数值分析代写Numerical analysis代考|First Order Systems of Ordinary Differential Equations

$$\frac{d u_1}{d t}=F_1\left(t, u_1, \ldots, u_n\right), \quad \ldots \quad \frac{d u_n}{d t}=F_n\left(t, u_1, \ldots, u_n\right) .$$
$$\frac{d \mathbf{u}}{d t}=\mathbf{F}(t, \mathbf{u}),$$

## 数学代写|数值分析代写Numerical analysis代考|Existence, Uniqueness, and Continuous Dependence

$$\frac{d u}{d t}=u^2+t$$

$$\frac{d u}{d t}=u^3+t$$

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