Cal3-Dinh_Hai LaplaceTransform SLIDES-GV (2017).pdf

Cal3-Dinh_Hai LaplaceTransform SLIDES-GV (2017).pdf -...

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CALCULUS 3 Chapter 2 LAPLACE TRANSFORMS Assoc. Profs. Dr. N. Dinh & N. N. Hai INTERNATIONAL UNIVERSITY October 11, 2017 Assoc. Profs. Dr. N. Dinh & N. N. Hai CALCULUS 3 Chapter 2 LAPLACE TRANSFORMS
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2.1 INTRODUCTION, DEFINITIONS AND EXAMPLES 2.1.1 Introduction Laplace transform plays a key role in the modern approach to the analysis and design of engineering systems. The stimulus: the work of Oliver Heaviside (English electrical engineer, 1850-1925). Intuitive method but work on practice and was accepted by engineers. Assoc. Profs. Dr. N. Dinh & N. N. Hai CALCULUS 3 Chapter 2 LAPLACE TRANSFORMS
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2.1 INTRODUCTION, DEFINITIONS AND EXAMPLES 2.1.1 Introduction (cont’s) It was then recognized that an integral transformation invented almost a century before by P.S. Laplace (French, 1749-1825) provided a theoretical foundation for Heaviside’s work. It was also recognized that the use of this integral method (Laplace transform) provides more systematic alternative for investigating differential equations than the method provided by Heaviside. Assoc. Profs. Dr. N. Dinh & N. N. Hai CALCULUS 3 Chapter 2 LAPLACE TRANSFORMS
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2.1 INTRODUCTION, DEFINITIONS AND EXAMPLES 2.1.1 Introduction (cont’s) The Laplace transform is an ideal tools for the investigation of electrical circuits and mechanical vibration . It also finds particular applications in signals and linear system analysis . Assoc. Profs. Dr. N. Dinh & N. N. Hai CALCULUS 3 Chapter 2 LAPLACE TRANSFORMS
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2.1 INTRODUCTION, DEFINITIONS AND EXAMPLES 2.1.1 Introduction (cont’s) Basic idea (for instance, in solving ODEs) Step 1. Given “hard” equation is transformed into a “simple” equation (subsidiary equation). Step 2. The subsidiary equation is solved by purely algebraic manipulations. Step 3. The solution of the subsidiary equation is transformed back to obtain the solution of the given equation. Assoc. Profs. Dr. N. Dinh & N. N. Hai CALCULUS 3 Chapter 2 LAPLACE TRANSFORMS
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2.1.2 DEFINITION OF LAPLACE TRANSFORM Definition 1.1 (Laplace transform) Let f ( t ) be a function on [0 , ). The Laplace transform of f is the function F defined by the integral ( ) = Z 0 - ( ) (1) The domain of F ( s ) is all the values of s (in general, s is a complex number) for which the integral in (1) exists. Assoc. Profs. Dr. N. Dinh & N. N. Hai CALCULUS 3 Chapter 2 LAPLACE TRANSFORMS
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2.1.2 DEFINITION OF LAPLACE TRANSFORM The Laplace transform of f is denoted by F or L ( f ), or L { f } . Thus, L ( f ) = F ( s ) = Z 0 e - st f ( t ) dt . Note The symbol L denotes the Laplace transform operator . It transforms f ( t ) into a function F ( s ) of the complex variable s . The domain of f ( t ) is called time domain . The domain of F ( s ) is called frequency domain . Assoc. Profs. Dr. N. Dinh & N. N. Hai CALCULUS 3 Chapter 2 LAPLACE TRANSFORMS
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2.1.2 DEFINITION OF LAPLACE TRANSFORM Figure 2.1 The Laplace Transform operator. Assoc. Profs. Dr. N. Dinh & N. N. Hai CALCULUS 3 Chapter 2 LAPLACE TRANSFORMS
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2.1.2 DEFINITION OF LAPLACE TRANSFORM In almost of the cases of applications, we use Laplace transformations where s takes real values. For the sake of simplicity, in this chapter we assume that s R . The students who are
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  • Spring '17
  • Janet Harris
  • Calculus, Dr. N. Dinh

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