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

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

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

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

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

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

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

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

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

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

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

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 + 2 - 4 £(y) = £(y) = Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Chapter 5 Laplace Transforms Fourier Transforms Definition : 1. 2. for any constant  3. If and are sectionally continuous on [-,], then Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

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

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

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

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