Mathematical Physics x14 Laplace...
Transcript of Mathematical Physics x14 Laplace...
![Page 1: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/1.jpg)
Introduction Laplace Transform Homework Appendix
Methods of Mathematical Physics
§14 Laplace Transform
Lecturer: 黄志琦
http://zhiqihuang.top/mmp
MMP §14 Laplace Transform Zhiqi Huang
![Page 2: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/2.jpg)
Introduction Laplace Transform Homework Appendix
Outline
Introduction
Laplace Transform
Homework
Appendix
MMP §14 Laplace Transform Zhiqi Huang
![Page 3: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/3.jpg)
Introduction Laplace Transform Homework Appendix
思考题
h(t) =
0, if t < 0;12 , if t = 01, if t > 0.
如图的函数称为单位跃阶函数(Heaviside step function),它的导函数是什么?
MMP §14 Laplace Transform Zhiqi Huang
![Page 4: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/4.jpg)
Introduction Laplace Transform Homework Appendix
一个经典的电路问题
如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直流电源串联,在t = 0时刻合上开关K,求在t > 0时刻电路中的电流I (t)。
MMP §14 Laplace Transform Zhiqi Huang
![Page 5: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/5.jpg)
Introduction Laplace Transform Homework Appendix
一种思路
设电容上积累的电荷为Q,则由并联部分电压关系得到
I1 =Q
CR, I2 =
dQ
dt
总电压方程为:
Ld
dt
(Q
CR+
dQ
dt
)+
Q
C= E
即
Q ′′ +1
CRQ ′ +
1
CLQ =
EL
我们要求这个二阶常微分方程在Q(0) = 0,Q ′(0) = 0的初始条件下的解。
MMP §14 Laplace Transform Zhiqi Huang
![Page 6: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/6.jpg)
Introduction Laplace Transform Homework Appendix
拉普拉斯变换的使用流程
t = 0的初始条件 + 线性常微分方程组↓
拉普拉斯变换 (查表,幼儿园操作)↓
拉普拉斯空间的代数方程↓
求解代数方程 (小学操作)↓
拉普拉斯逆变换 (查表,幼儿园操作)↓
原问题的解
MMP §14 Laplace Transform Zhiqi Huang
![Page 7: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/7.jpg)
Introduction Laplace Transform Homework Appendix
拉拉拉普普普拉拉拉斯斯斯变变变换换换
f ′(t)LT=⇒ pF (p)− f (0)
MMP §14 Laplace Transform Zhiqi Huang
![Page 8: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/8.jpg)
Introduction Laplace Transform Homework Appendix
拉普拉斯变换
函数f (t)的拉普拉斯变换F (p)定义为
F (p) =
∫ ∞0−
f (t)e−ptdt .
(这里的0−理解为比零稍稍小一点点的数.)或简单写成
fLT=⇒ F .
和傅立叶变换一样,拉普拉斯变换是一个线性变换
MMP §14 Laplace Transform Zhiqi Huang
![Page 9: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/9.jpg)
Introduction Laplace Transform Homework Appendix
积分要从0−开始
普拉斯变换的积分下限0−在物理上很容易理解:只有积分范围覆盖t = 0时刻,我们才能用到t = 0时刻的初始条件。
在不致引起混淆但情况下,可以把0−写成0.
F (p) =
∫ ∞0
f (t)e−ptdt .
MMP §14 Laplace Transform Zhiqi Huang
![Page 10: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/10.jpg)
Introduction Laplace Transform Homework Appendix
导函数的拉普拉斯变换
设fLT=⇒ F,则导函数f ′(t)的拉普拉斯变换为∫ ∞
0−f ′(t)e−ptdt = f (t)e−pt
∣∣∞0−
+p
∫ ∞0−
f (t)e−ptdt = pF (p)−f (0−).
对f ′(t)应用上述结论,得到二阶导函数的拉普拉斯变换为
p(pF (p)− f (0))− f ′(0) = p2F (p)− pf (0−)− f ′(0−).
反复使用这个办法,可以推出任意阶导函数的拉普拉斯变换:
f (n)(t)LT=⇒ pnF (p)−
n−1∑k=0
pn−1−k f (k)(0−).
MMP §14 Laplace Transform Zhiqi Huang
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Introduction Laplace Transform Homework Appendix
查表就ok我们还需要知道怎么把给定的具体函数进行拉普拉斯变换和逆变换。除了一些特别简单的情况外,都可以通过查表来解决。(推荐wikipedia的Laplace Transform词条下的表,比较全。)附录里给出的几个常用公式:
tαeβtLT=⇒ Γ(α + 1)
(p − β)α+1; (α > −1)
cosωtLT=⇒ p
p2 + ω2
sinωtLT=⇒ ω
p2 + ω2
在第一个公式里取 α = β = 0,或者在第二个公式里取ω = 0,就能得到1的拉普拉斯变换为 1
p ;
MMP §14 Laplace Transform Zhiqi Huang
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Introduction Laplace Transform Homework Appendix
回到电路题
Q ′′ +1
CRQ ′ +
1
CLQ =
EL
对上式进行拉普拉斯变换,设Q(t)LT=⇒ Q(p):[
p2Q − pQ(0)− Q ′(0)]
+1
CR
[pQ − Q(0)
]+
1
CLQ =
ELp
利用Q(0) = Q ′(0) = 0,得到
Q =E
Lp(p2 + p
CR + 1CL
)
MMP §14 Laplace Transform Zhiqi Huang
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Introduction Laplace Transform Homework Appendix
回到电路题
记I2 = Q ′(t)的拉普拉斯变换为I2,注意到Q(0) = 0,
I2 = pQ − Q(0) =E
L(p2 + p
CR + 1CL
)做因式分解 p2 + 1
CR p + 1CL = (p − α)(p − β) (即求出一元二次
多项式的两个根α, β)。 如果L = 4CR2,则α = β = − 12CR .
I2 =E
L(p − α)2
利用前面所列的第一条公式进行反演
I2 =ELteαt .
MMP §14 Laplace Transform Zhiqi Huang
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Introduction Laplace Transform Homework Appendix
回到电路题
如果L 6= 4CR2,则
I2 =E
L(α− β)
(1
p − α− 1
p − β
)即
I2 =E
L(α− β)
(eαt − eβt
).
最后,利用
I1 =Q
CR=
∫ t0 I2(t ′)dt ′
CR
以及I = I1 + I2
即可算出电路中的总电流I。
MMP §14 Laplace Transform Zhiqi Huang
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Introduction Laplace Transform Homework Appendix
Homework
I 求f (t) =∫ t0
sin xx dx的拉普拉斯变换。
I 求F (p) = 1p2−2p+2
的拉普拉斯逆变换。
I 如图电动势为E的直流稳压电源和电感L以及电阻R串联,在t = 0时刻合上开关K,求之后电路中的电流I (t)。
MMP §14 Laplace Transform Zhiqi Huang
![Page 16: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/16.jpg)
Introduction Laplace Transform Homework Appendix
附附附录录录
常见性质和变换公式(别有压力,能学多少就学多少)
MMP §14 Laplace Transform Zhiqi Huang
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Introduction Laplace Transform Homework Appendix
利用定义
利用拉普拉斯变换的定义,容易算出:
δ(t)LT=⇒ 1
1LT=⇒ 1
p
MMP §14 Laplace Transform Zhiqi Huang
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Introduction Laplace Transform Homework Appendix
幂函数因子
设有拉普拉斯变换
F (p) =
∫ ∞0
f (t)e−ptdt,
两边对p求导得到
F ′(p) =
∫ ∞0
(−tf (t))e−ptdt,
反复利用上式即得
tnf (t)LT=⇒
(− d
dp
)n
F (p).
MMP §14 Laplace Transform Zhiqi Huang
![Page 19: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/19.jpg)
Introduction Laplace Transform Homework Appendix
幂函数的拉普拉斯变换
利用 1LT=⇒ 1
p,左边乘上tn,右边作用(− d
dp
)n,得到
tnLT=⇒ n!
pn+1, n = 0, 1, 2, . . .
实际上,直接用拉普拉斯变换以及Γ函数的定义,可以得到更一般的结论:
tα−1LT=⇒ Γ(α)
pα, Re(α) > 0.
MMP §14 Laplace Transform Zhiqi Huang
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Introduction Laplace Transform Homework Appendix
缩放
设f (t)的拉普拉斯变换为F (p),则导函数f (at)的拉普拉斯变换为∫ ∞0
f (at)e−ptdt =
∫ ∞0
f (u)e−paud(ua
)=
1
aF(pa
),
即
f (at)LT=⇒ 1
aF(pa
).
MMP §14 Laplace Transform Zhiqi Huang
![Page 21: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/21.jpg)
Introduction Laplace Transform Homework Appendix
指数函数因子
设f (t)的拉普拉斯变换为F (p),则eat f (t)的拉普拉斯变换为∫ ∞0
eat f (t)e−ptdt =
∫ ∞0
f (t)e−(p−a)tdt = F (p − a) ,
即
eat f (t)LT=⇒ F (p − a) .
MMP §14 Laplace Transform Zhiqi Huang
![Page 22: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/22.jpg)
Introduction Laplace Transform Homework Appendix
指数函数的拉普拉斯变换
利用 1LT=⇒ 1
p以及乘指数函数因子的规则,得到
eatLT=⇒ 1
p − a, Re(p − a) > 0.
MMP §14 Laplace Transform Zhiqi Huang
![Page 23: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/23.jpg)
Introduction Laplace Transform Homework Appendix
三角函数的拉普拉斯变换
有了指数函数,三角函数就简单了:
cos (ωt) =e ıωt + e−ıωt
2LT=⇒ 1
2
(1
p − iω+
1
p + iω
)=
p
p2 + ω2
sin (ωt) =e ıωt − e−ıωt
2ıLT=⇒ 1
2ı
(1
p − ıω− 1
p + ıω
)=
ω
p2 + ω2
MMP §14 Laplace Transform Zhiqi Huang
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Introduction Laplace Transform Homework Appendix
源的积分
源的积分比较简单:
设fLT=⇒ F,令g(t) =
∫ t0 f (u)du,并设g
LT=⇒ G。
因g ′(t) = f (t),且g(0) = 0,根据导函数的拉普拉斯变换规则:
F (p) = pG (p)− 0 = pG (p)
即 ∫ t
0f (u)du
LT=⇒ F (p)
p
MMP §14 Laplace Transform Zhiqi Huang
![Page 25: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/25.jpg)
Introduction Laplace Transform Homework Appendix
像的积分
像的积分则稍显复杂。
设fLT=⇒ F,令g(t) = f (t)
t ,并设gLT=⇒ G。
根据幂函数因子的拉普拉斯变换规则:
F (p) = −G ′(p).
一一一般般般情情情况况况下下下,G (∞) = 0。积分得到
f (t)
tLT=⇒
∫ ∞p
F (υ)dυ
思考:你能设计一个不一般的情况让G (∞) = 0不成立吗?
MMP §14 Laplace Transform Zhiqi Huang
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Introduction Laplace Transform Homework Appendix
平移
设f (t)的拉普拉斯变换为F (p),则f (t − a)h(t − a)的拉普拉斯变换为∫ ∞
af (t − a)e−ptdt = e−ap
∫ ∞0
f (u)e−pudu = e−apF (p)
即
f (t − a)h(t − a)LT=⇒ e−apF (p)
MMP §14 Laplace Transform Zhiqi Huang
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Introduction Laplace Transform Homework Appendix
卷积
对拉普拉斯变换, f和g的卷积f ∗ g定义为
(f ? g)(t) =
∫ t
0f (t − τ)g(τ)dτ.
这跟我们学习傅立叶变换时的卷积定义不太一样。但不必担心符号混乱:一般拉普拉斯变换和傅立叶变换不会同时进行。
MMP §14 Laplace Transform Zhiqi Huang
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Introduction Laplace Transform Homework Appendix
拉普拉斯变换的卷积定理
拉普拉斯变换的卷积定理和傅立叶变换的卷积定理在形式上完全相同。
如果fLT=⇒ F , g
LT=⇒ G,则f ? g
LT=⇒ FG .
MMP §14 Laplace Transform Zhiqi Huang
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Introduction Laplace Transform Homework Appendix
卷积定理的证明
(f ? g)LT=⇒
∫ ∞0
e−ptdt
∫ t
0f (t − τ)g(τ)dτ.
做变量替换u = t − τ, υ = τ,注意0 < τ < t的条件转化为u > 0, υ > 0。转换矩阵的行列式为1,所以dtdτ直接改为dudυ:
(f ? g)LT=⇒
∫ ∞0
∫ ∞0
e−p(u+υ)f (u)g(υ)dudυ = F (p)G (p)
MMP §14 Laplace Transform Zhiqi Huang
![Page 30: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/30.jpg)
Introduction Laplace Transform Homework Appendix
思考题
试利用卷积定理以及h(t)LT=⇒ 1
p,证明积分的拉普拉斯变换性
质: ∫ t
0f (u)du
LT=⇒ F (p)
p.
MMP §14 Laplace Transform Zhiqi Huang
![Page 31: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/31.jpg)
Introduction Laplace Transform Homework Appendix
拉普拉斯变换的反演
拉普拉斯变换是否像傅立叶变换一样存在反演公式呢?
MMP §14 Laplace Transform Zhiqi Huang
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Introduction Laplace Transform Homework Appendix
梅林反演公式(了解这样的公式存在即可)
若f (t)LT=⇒ F (p),则f (t) = 1
2πı
∫ β+ı∞β−ı∞ F (p)eptdp。
其中β是足够大的正数。
我们计算拉普拉斯变换的反演一般不不不用用用这个公式。
我就喜欢这样看起来很厉害又不会用到的公式
MMP §14 Laplace Transform Zhiqi Huang
![Page 33: Mathematical Physics x14 Laplace Transformzhiqihuang.top/mmp/lectures/MMP14_LaplaceTransform.pdfIntroduction Laplace Transform Homework Appendix 一个经典的电路问题 如图,电阻R和电容C并联,再依次和电感L以及电动势为E的直](https://reader033.fdocuments.net/reader033/viewer/2022060816/6095af04557f4d0ff565ecdd/html5/thumbnails/33.jpg)
Introduction Laplace Transform Homework Appendix
总结
学了这么多公式,真是太开心了
除了都记不住,没别的毛病
MMP §14 Laplace Transform Zhiqi Huang
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Introduction Laplace Transform Homework Appendix
精简版
tα−1LT=⇒ Γ(α)
pα, Re(α) > 0;
eαtLT=⇒ 1
p − α;
cosωtLT=⇒ p
p2 + ω2
sinωtLT=⇒ ω
p2 + ω2
f ′(t)LT=⇒ pF (p)− f (0);
f ′′(t)LT=⇒ p2F (p)− pf (0)− f ′(0);∫ t
0f (τ)dτ
LT=⇒ F (p)
p.
MMP §14 Laplace Transform Zhiqi Huang