1. Chapter 5 Laplace Transforms
拉普拉斯轉換乃算子演算法(operational calculus),它將微積分演算變成代數演算.(為特殊的傅立葉轉換)
拉普拉斯轉換在工程上用於機械以及電力的驅動力問題,特別是當驅動力為不連續,脈衝或是正弦,餘弦及更複雜的周期性函數.
拉普拉斯轉換可直接解問題,求解初值問題時無需先求通解,且解非齊次微分方程時亦無需先求對應之齊次方程式之解.
偏微分方程式也能以拉普拉斯轉換處理.
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

2. Chapter 5 Laplace Transforms
The Laplace transform £[f(t)] of a function f(t) is defined to be
£[f(t)]
£[f(t)] = F(s)
£-1[F(s)] = f(t)
The Laplace transform of f(t) = t is
If s > 0
£[f(t)]
£(tn)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

3. Chapter 5 Laplace Transforms
The Laplace transform of f(t) = cos(wt) is
£[f(t)]
利用分部積分
If s > 0
£[cos(wt)]
£[sin(wt)]
同理可證
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

4. Chapter 5 Laplace Transforms
Theorem : 1.
£[f(t)+g(t)] = £[f(t)]+ £[g(t)]
Whenever all three Laplace transform exist
£(eiwt) = £(coswt + i sinwt) = £(coswt)+ i£(sinwt)
2. For any real number a,
£[af(t)] = a£[f(t)]
Whenever both sides exist 3.
£-1[F(s)+G(s)] = £-1[F(s)]+ £-1[G(s)] 4.
£-1[aF(s)] = a£-1[F(s)]
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

5. Chapter 5 Laplace Transforms
f(t) =
£[f(t)]
£[f(t)]
{£[f(t)]}2
Let x = r cosθ , y = r sin θ
£[f(t)]
Gamma function :
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

6. Chapter 5 Laplace Transforms
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

7. Chapter 5 Laplace Transforms
Rule 1: if then
£[f(t)] = F(s)
£[eatf(t)] = F(s-a)
For s > a
£[cos(wt)]
£[eatcos(wt)]
Rule 2: Let a be a positive constant. Let f(t) be given, with f(t) = 0 if t < 0. Define g(t) by
g(t) = f(t-a), then
£[g(t)] = e-as £[f(t)]
f(t)
g(t) t a t
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

8. Chapter 5 Laplace Transforms
Unit step function
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

9. Chapter 5 Laplace Transforms
Unit step function
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

10. Chapter 5 Laplace Transforms
Dirac’s Delta function
以兩個單位階梯函數來表示
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

11. Chapter 5 Laplace Transforms
函數 f (t) 其存在拉普拉斯轉換的充分條件為存在正實數 M 與 α , 使得在任何 t  0之下有
Rule 3: if for t  t0, f(t) is continuous for t  0, and f ‘(t) is piecewise
continuous on [0, k] for every k > 0, then
£[f ’(t)] = s £[f (t)]-f (0)
For s > α
£[cos(wt)]
£[sin(wt)] = £ = £
{s £[cos(wt)]-cos (0)}
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

12. Chapter 5 Laplace Transforms
Rule 4:
£[f (n)(t)] = sn £[f (t)]-sn-1f (0)-sn-2f ’(0)- ……..-f (n-1)(0)
Solve the initial value problem :
y’’- 4y = t ; y(0) = 1, y’(0) = -2
£(y’’) - 4£(y) = £(t)
£(t) =
By rule 4 : £(y’’) = s2£(y) – sy(0) – y’(0) = s2 £(y) – s + 2
s2 £(y) – s £(y) =
£(y) =
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

13. Chapter 5 Laplace Transforms
The Laplace transform of f(t) = cos(wt) 則
£[cos(wt)]
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

14. Chapter 5 Laplace Transforms
The Laplace transform of f(t) = sin2t 則
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

15. Chapter 5 Laplace Transforms
Rule 5: if f (t + ω) = f (t), so that f (t) has period ω, then
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

16. Chapter 5 Laplace Transforms
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

17. Chapter 5 Laplace Transforms
Rule 6:
求 f (t)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

18. Chapter 5 Laplace Transforms
解初值微分方程式問題
a, b 為常數, r(t)為輸入(驅動力), y(t)為輸出(系統的響應)
Step 1 : 取Laplace, 令
Subsidiary equation
(輔助方程式)
Step 2 :
Transfer function (轉移函數)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

19. Chapter 5 Laplace Transforms
解初值微分方程式問題 若
Step 3 :
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

20. Chapter 5 Laplace Transforms
Inverse Laplace Transforms
Consider the problem of finding , where P(s) and Q(s) are polynomials
Having no common factor and Q(s) has higher degree than P(s).
Heaviside’s formulas
Case 1 : If Q(s) contains an unrepeated linear factor (s-a), then f(t) contains
the term
where
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

21. Chapter 5 Laplace Transforms
Inverse Laplace Transforms
Case 2 : If k  2 and Q(s) contains the linear factor (s-a)k, but not (s-a)k+1 ,
then the corresponding term in f(t) is
where
Case 3 : If Q(s) contains the unrepeated quadratic factor (s-a)2+b2, then f(t)
contains the terms
In which r is the real part of H(a+ib), i is the imaginary part of H(a+ib)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

22. Chapter 5 Laplace Transforms
Inverse Laplace Transforms
Case 4 : If Q(s) contains the quadratic factor [(s-a)2+b2]2, but not [(s-a)2+b2]3
then f(t) contains the terms
In which r is the real part of H(a+ib), i is the imaginary part of H(a+ib)
r is the real part of H’(a+ib),  i is the imaginary part of H’(a+ib)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

23. Chapter 5 Laplace Transforms
Inverse Laplace Transforms
Solve
Q(s) has unrepeated linear factor (s-5) and (s+1), and an unrepeated quadratic factor
(s2-2s+5), which is (s-1)2+4.
For (s-5) f(t) has a form
For (s+1) f(t) has a form
For (s-1)2+4 
f(t) has a form
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

24. Chapter 5 Laplace Transforms
Convolution theorem
If and , then
Commutative property :
Find
and
and
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

25. Chapter 5 Laplace Transforms
Fourier Sine Transforms
Suppose that f(x) is defined on [0, ]. We define the Fourier sine transform to be If
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

26. Chapter 5 Laplace Transforms
Fourier Sine Transforms
Theorem 1. 2.
for any constant 
Let f(x) and f ’(x) be piecewise continuous on [0,]. Assume also
that limx  f(x) = limx  f ‘(x) = 0, then
Proof
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

27. Chapter 5 Laplace Transforms
Fourier Cosine Transforms
Suppose that f(x) is defined on [0, ]. We define the Fourier cosine transform to be If
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

28. Chapter 5 Laplace Transforms
Fourier Cosine Transforms
Theorem 1. 2.
for any constant 
Let f(x) and f ’(x) be piecewise continuous on [0,]. Assume also
that limx  f(x) = limx  f ‘(x) = 0, then
Proof --- Homework
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

29. Chapter 5 Laplace Transforms
Fourier Transforms
Definition : 1. 2.
for any constant  If
and are sectionally continuous on [-,], then
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

30. Chapter 5 Laplace Transforms
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

31. Chapter 5 Laplace Transforms
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

32. Chapter 5 Laplace Transforms
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

33. Chapter 5 Laplace Transforms
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung