线性方程组解的存在唯一性

上的解。 在区间− ∞ < t < +∞ 上的解。 解
e 0 1 u ( 0) = 0 = − e − 1
是给定初值问题的解。 因此 u(t ) 是给定初值问题的解。
Existence & Uniqueness Theorems of Linear ODEs § 5.1 E
b11 (t ) b12 (t ) b (t ) b (t ) 22 21 B(t ) = L L bn1 (t ) bn 2 (t )
L b1n (t ) L b2 n (t ) L L L bnn (t )
u1 (t ) u (t ) u (t ) = 2 M u n (t )
Existence & Uniqueness Theorems of Linear ODEs § 5.1 E
x(t0 ) = η1 , x′(t0 ) = η 2
x′′ + p (t ) x′ + q (t ) x = f (t )
满足
x = ϕ1 (t )
ϕ1′′(t ) + p (t )ϕ1′(t ) + q (t )ϕ1 (t ) = f (t )
Existence & Uniqueness Theorems of Linear ODEs § 5.1 E
∫ B(t )dt = (∫ b (t )dt )
ij
n×n
u(t )dt = ( ∫ u1 (t ) dt , ∫ u 2 (t ) dt , L , ∫ u n (t ) dt )T ∫
Existence & Uniqueness Theorems of Linear ODEs § 5.1 E
5.1.1 记号与定义/Symbol and Definition/ 一阶微分方程组
′ x1 = f 1 ( t , x1 , x 2 ,⋅ ⋅ ⋅, x n ) x ′ = f ( t , x , x ,⋅ ⋅ ⋅, x ) 2 2 1 2 n L L L L x n = f n ( t , x1 , x 2 ,⋅ ⋅ ⋅, x n ) ′
f1 (t ) f (t ) f (t ) = 2 M f n (t )
L a1n (t ) L a2 n (t ) L L L ann (t )
……….(5.2)
x1 x x = 2 M xn
′ x1 x′ dx = x ′ = 2 ……(5.3) M dt n x′
………….(5.4)
dx = x ′ = A (t ) x + f (t ) dt
Existence & Uniqueness Theorems of Linear ODEs § 5.1 E
在某区间α ≤ t ≤ β ([α, β ] ⊂ [a, b]) 的解就是向量
u (t ) 在区间 α ≤ t ≤
β 上连续且满足
u′(t ) = A(t ) u(t ) + f (t )
Existence & Uniqueness Theorems of Linear ODEs § 5.1 E
定义2 初值问题(Cauchy Problem) 定义 初值问题
1) ( A(t ) + B (t ))′ = A′(t ) + B′(t ) (u(t ) + v (t ))′ = u′(t ) + v ′(t )
2) ( A(t ) ⋅ B (t ))′ = A′(t ) B (t ) + A(t ) B′(t ) 3) ( A(t ) ⋅ u(t ))′ = A′(t )u(t ) + A(t )u′(t )
解
x = ϕ (t )
构造 向量
ϕ ′′(t ) + p (t )ϕ ′(t ) + q (t )ϕ (t ) = f (t )
ϕ′(t) (t (t (t ϕ′(t) ϕ (t ) x= ϕ′′(t) = − p(t )ϕ′(t ) − q(t )ϕ(t ) + f (t ) ϕ′(t ) 1 ϕ (t ) 0 0 满足 = ϕ′(t ) + f (t ) − q(t ) − p(t ) 1 0 0 x1 (t0 ) = η1 , x2 (t0 ) = η 2 x′ = x + f (t ) − q(t ) − p(t )
aij (t ), f i (t ) i, j = 1 2， , n 在[a, b]上连续 ，L
…(5.1)
Existence & Uniqueness Theorems of Linear ODEs § 5.1 E
a11 (t ) a12 (t ) a (t ) a (t ) 22 A(t ) = 21 L L an1 (t ) an 2 (t )
初值条件 x1 (t0 ) = η1 , x2 (t0 ) = η 2 , ⋅ ⋅⋅, xn (t0 ) = η n
Existence & Uniqueness Theorems of Linear ODEs § 5.1 E
一阶线性微分方程组
′ x1 = a11 (t ) x1 + a12 (t ) x2 + ⋅ ⋅ ⋅ + a1n (t ) xn + f1 (t ) x′ = a (t ) x + a (t ) x + ⋅ ⋅ ⋅ + a (t ) x + f (t ) 2 21 1 22 2 2n 2 2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ xn = an1 (t ) x1 + an 2 (t ) x2 + ⋅ ⋅ ⋅ + ann (t ) xn + f n (t ) ′
在区间 a ≤ t ≤ b 可定义矩阵与向量函数
B (t ) = (bij (t )) n×n
u ( t ) = ( u 1 ( t ), u 2 ( t ), L , u n ( t )) T
连续: 连续 可微: 可微 可积: 可积
bij (t ) ui (t ) 在区间 a ≤ t ≤ b 连续。 连续。
′(t (t 1 ϕ1 (t ) 0 ϕ1(t) 0 ϕ′ (t) = − q(t ) − p(t ) ϕ (t ) + f (t ) 2 2 (t ϕ1 (t ) x= ϕ2 (t )
解
ϕ2 (t) = − q(t)ϕ1 (t) − p(t)ϕ2 (t ) + f (t )
e − t 例1 验证向量 u(t) = −t 是初值问题 − e
0 1 x′ = x 1 0
− e −t u′(t ) = −t , e
1 x (0) = − 1
0 1 e − t − e − t 1 0 −t = −t − e e
( n −1)
+ ⋅ ⋅ ⋅ + an −1 (t ) x′ + an (t ) x = f (t )
( n −1) 令 x1 = x, x2 = x′, x3 = x′′, ⋅ ⋅⋅, xn = x
′ x1 = x′ = x2
′ x2 = x′′ = x3 ⋅⋅⋅⋅⋅⋅⋅
′ xn −1 = x ( n −1) = xn
1 0 0 x′ = x + f (t ) − q(t ) − p(t )
Existence & Uniqueness Theorems of Linear ODEs § 5.1 E
x(t0 ) = η1 , x′(t0 ) = η 2
x′′ + p (t ) x′ + q (t ) x = f (t )
Existence & Uniqueness Theorems of Linear ODEs § 5.1 E
上的连续 n × n 矩阵， 定义1 矩阵， 定义1 设 A(t ) 是区间 a≤t ≤b
f (t ) 是区间 a≤t ≤b 上的连续
n
维向量， 维向量，方程组 ………….(5.4)
dx = x ′ = A (t ) x + f (t ) dt
dx = x ′ = A(t ) x + f (t ) dt x (t0 ) = η
………….(5.5)
的解就是方程组(5.4)在包含 t0的区间α ≤ t ≤ β 在包含 的解就是方程组
上的解u(t ), 使得 u(t0 ) = η
Existence & Uniqueness Theorems of Linear ODEs § 5.1 E
bij (t ) ui (t ) 在区间 a ≤ t ≤ b 可微。 可微。
′ B′(t ) = (bij (t )) n×n
′ ′ ′ u ′ ( t ) = ( u 1 ( t ), u 2 ( t ), L , u n ( t )) T
bij (t ) ui (t ) 在区间 a ≤ t ≤ b 可积。 可积。
5.1.2 n 阶线性微分方程与一阶线性微分方程组等价 例1 解
x′′ + p (t ) x′ + q (t ) x = f (t )
令
x1 = x,
x2 = x′,
′ x1 = x′ = x2
′ x2 = x′′ = − p (t ) x′ − q(t ) x + f (t )
′ x1 = x2 2 x′ = − q (t ) x1 − p (t ) x2 + f (t )
§ 5.1 线性微分方程组解的 存在唯一性定理
Existence & Uniqueness Theorems of Linear ODEs
源自文库节要求/Requirements/
掌握高阶线性微分方程与线性微分方程组的关系。 掌握高阶线性微分方程与线性微分方程组的关系。 理解线性微分方程组解的存在唯一性定理。 理解线性微分方程组解的存在唯一性定理。 熟练掌握解的逐次逼近序列的构造方法。 熟练掌握解的逐次逼近序列的构造方法。
′ = x ( n ) = − an (t ) x1 − an −1 (t ) x2 − ⋅ ⋅ ⋅ − a1 (t ) xn + f (t ) xn
Existence & Uniqueness Theorems of Linear ODEs § 5.1 E
1 0 0 L 0 0 0 0 0 1 0 L x′ = M M M M M x + M 0 L 0 1 0 0 − an (t ) − an−1 (t ) L − a2 (t ) − a1 (t ) f (t )
Existence & Uniqueness Theorems of Linear ODEs § 5.1 E
x ( n ) + a1 (t ) x ( n −1) + ⋅ ⋅ ⋅ + an −1 (t ) x′ + an (t ) x = f (t ) ………(5.6) ( n −1) x(t0 ) = η1 , x′(t0 ) = η 2 ,⋅ ⋅ ⋅, x (t0 ) = η n ψ (t ) ψ ′(t ) ψ (t )
x1 (t0 ) = η1 , x2 (t0 ) = η 2
1 0 0 x′ = x + f (t ) − q(t ) − p(t )
Existence & Uniqueness Theorems of Linear ODEs § 5.1 E
x
(n)
+ a1 (t ) x
等价
M ( n −1) ψ (t )
1 0 0 0 x′ = M M 0 0 − an (t) − an−1(t)