A uniqueness theorem (or its proof) is, at least within the mathematics of differential equations, often combined with an existence theorem (or its proof) to a combined existence and uniqueness theorem (e.g., existence and uniqueness of solution to first-order differential equations with boundary condition).

本文补全了一些课程中用到的，但高等数学课程中并未出现（然而任课老师以为都知道？ ）的定理的叙述和证明. 1. 存在唯一性. 考虑带初值的一阶方程. \begin {cases} \frac { {\rm d}y} { {\rm d}t}=f (t,y)\\ y (t)\mid_ {t=t_0}=y_0 \end {cases} \\ 其中函数 f (t,y) 连续可微. 设在矩形区域 [t_0-a,t_0+a]\times [y_0-b,y_0+b] 上，若 |f (t,y)|\leq M ,则. 在区间 [t_0-h,t_0+h] 上方程的解存在且唯一 , 其中 h=\min\ {a,\frac bM\} . （a）存在性的证明. 构造Picard序列.

The uniqueness theorem is quite useful for it sets forth constraints on the boundary conditions that guarantee there is only one solution to Maxwell’s equations, which we can find as usual. To prove the uniqueness theorem we begin by considering a volume V enclosed by surface S and governed by Maxwell’s equations: ∇ ∙ →Di = ρ

Uniqueness Theorem rtant concept in mathematics. Under certain conditions, ordinary di erential equation partial di erential equation and matrix equatio s will have unique solutions. But uniqueness is not alwa s guaranteed as we shall see. This …

By the existence and uniqueness theorem there is a unique solution for the problem on the interval [ h, h] where h = minfL, b/2g. Since our bounds are independent of b we may choose b large, in particular we may choose b larger than 2L, so …

1 天前 · A theorem, also called a unicity theorem, stating the uniqueness of a mathematical object, which usually means that there is only one object fulfilling given properties, or that all objects of a given class are equivalent (i.e., they can be represented by the same model).

2020年7月16日 · The uniqueness theorem can be stated as the following: "To every boundary value condition there exists a unique solution to the Laplace equation. Two solutions of Laplace equation that satisfy the same boundary value condition are (i) same for Dirichlet Boundary Value Condition and (ii) differ by an additive constant for Neumann Boundary Value ...

We’ll prove existence in two different ways and will prove uniqueness in two different ways. The first existence proof is constructive: we’ll use a method of successive approximations — the Picard iterates — and we’ll prove they converge to a solution. The second existence proof uses a fixed-point argument.

(Uniquness theorem): Suppose that f and @f=@y are continuous function in R (de ned in the existence theorem). Hence, both the f and @f=@y are bounded in R, i.e., x0j . Comment: Condition (b) can be replaced by a weaker condition which is known as Lipschitz condition. Thus, instead of continuity of @f=@y, we require.

The theorem below shows that one can, under the right conditions, assert that a DE has a unique solution, even if the solution can’t be written down in closed form.