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- 1. Department Of Computer Engineering Laplace Transform 1 G.H Patel College of Engineering and Technology Made By : Neel Shah
- 2. Contents 1. Introduction 2. Properties and Theorem 3. Applications of Laplace Transform in Mathematics and Sciences 4. Applications of Laplace Transform in Engineering G.H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY 2
- 3. Introduction Let f(t) be a given function defined for all t ≥ 0 then, ∞ 푒−푠푡 푓 푡 푑푡 0 If the above Integral exists and suppose it is F(s). Then, F(s) is known as Laplace transform of f(t) Denoted By: F(s) = L[f(t)] And, the original function is known as Inverse Transform of F(s) Denoted By: f(t) = L-1[F(s)] G.H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY 3
- 4. Properties and Theorems 1. Linearity 2. First Shifting Theorem 3. Unit Step Function and Second Shifting Theorem 4. Differentiation of Laplace transform 5. Integration of Laplace transform 6. Laplace Transform of Periodic Function 7. Laplace Transform of an Integral 8. Laplace transform of a Differential 9. Convolution G.H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY 4
- 5. Linearity • Let f(t) be a given function defined for all t ≥ 0 such that, f(t) = ag(t) + bh(t) Then, L[f(t)] = aL[g(t)] + bL[h(t)] This can be proved by the formal Integral Definition. G.H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY 5
- 6. First Shifting Theorem • If f(t) has the transform F(s) (where s > k), then 푒푎푡 f(t) has the transform F(s – a ) {Where s – a > k } G.H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY 6
- 7. Unit Step Function • Simply u(t - a) is function whose function value is ZERO for t < a. And has a jump size of 1 at t = a. This function is called Unit Step Function or Heaviside Function. 푢 푡 − 푎 = 0, 푡 < 푎 1, 푡 > 푎 G.H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY 7
- 8. Second Shifting Theorem • If f(t) has the transform F(s) then, G.H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY 8 • AND
- 9. Differentiation of Laplace transform • If f(t) is a function and • F(s) is its Laplace transform then, • L 푡푛푓(푡) = −푑푛[퐹 푠 ] 푑푥푛 G.H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY 9
- 10. Integration of Laplace transform • If f(t) is a function and • F(s) is its Laplace transform then, • L 1 푡 ∞ 퐹 푠 푑푠 푓(푡) = 푠 G.H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY 10
- 11. Laplace Transform of Periodic Function • If f(t) is a periodic function with period a then, • L 푓(푡) = 1 푎 푒−푠푡 푓 푡 푑푡 1− 푒−푎푠 0 G.H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY 11
- 12. Laplace Transform of an Integral G.H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY 12 • Let F(s) be the Laplace transform of the f(t) . • If f is piecewise and continuous function then,
- 13. Laplace transform of a Differential • Let f(t) is continuous and n times differentiable function then, G.H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY 13
- 14. Convolution G.H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY 14
- 15. Applications of Laplace Transform in Mathematics and Sciences G.H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY 15 • A very simple application of Laplace transform in the area of physics could be to find out the harmonic vibration of a beam which is supported at its two ends. • The Laplace transform can be applied to solve the switching transient phenomenon in the series or parallel RL,RC or RLC circuits. • Concept of pulse in Mechanics and Electricity. • To Solve Differential Equations in Mathematics
- 16. Applications of Laplace Transform in Engineering G.H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY 16 • Control Engineering • Communication • Signal Analysis and Design • System Analysis { Application of Computer Engineering } • Solving Differential Equations • Electrical Engineering
- 17. References Advanced Engineering Mathematics By : Erwin Kreyszig [8th Edition] Lecture Notes of Prof. Sarina Adhikari( Department of Electrical Engineering and Computer Science) [University of Tennessee]. Prof . M. C. Anumaka (Department of Electrical Electronics Engineering) [Imo State University, Owerri, Imo State, Nigeria] A Text Book of Engineering Mathematics By : Dr. K . N Srivastava Dr. G . K Dhawan [ 2nd Edition] (Year : 1987) G.H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY 17