Ch_02 - Laplace Transform - Summary

Ch_02 - Laplace Transform - Summary - Chapter 2 The Laplace...

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Chapter 2 The Laplace Transformation
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Table of Contents 1. Definition of Transform 1.1 The Laplace Transform 1.2 The Laplace Transform of Simple Functions 1.3 Sectionally Continuous Functions. Exponential Order 1.4 The General Theorem of the Laplace Transform 1.5 Properties of the Laplace Transform 1.6 The Transforms of Periodic Functions 2. The Inverse Laplace Transform 2.1 Partial Fraction Expansion Technique 2.2 Inversion Theorem for Pole Singularities 2.3 Heaviside’s Expansion Theorem 2.4 Convolution Theorem Method 2.5 Qualitative Nature of Solutions 3. Other Related Functions and Transforms 3.1 Some Special Functions 3.2 Other Transforms 4. Applications of the Laplace Transform 4.1 Some Examples of the Laplace Transform 4.2 Applications to Ordinary Differential Equations Appendix A.1 Partial Fraction Expansion Technique A.2 Inversion Theorem for Pole Singularities A.3 Graphs of Typical Integral Functions A.4 Forced Oscillations
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The Laplace Transformation The Laplace transformation is the name given to a particular “operational method”' of solving differential equations. Essentially, an operational method is a technique whereby an ordinary differential equation is converted into an equivalent algebraic form which can be solved by the laws of elementary algebra. The method will also convert a partial differential equation into an equivalent easily solvable ordinary differential equation. There are many such operational methods but the Laplace transformation is a particularly useful mathematical tool for solving engineering problems because the boundary conditions are introduced into the equation prior to its solution. The application of operational methods for the solution of engineering problems was first made by Heaviside (1850-1925), and although his methods were largely intuitive they were successful. This stimulated the interest of other mathematicians who developed the theoretical basis of Heaviside 's methods and extended them into what is known as the Laplace Transformation . 1. Definition of Transform Differential Operator D   ' () f tf t D Integral Operator I  0 x f t d t I Linear Operator L     11 2 2 () c ft cft c ft cft  LL L for all functions of some class and for every constant c . If L satisfies the following conditions, then, the transform of f t is “linear”. i.e.,     f tg t f t g t cf t c f t L
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The Laplace Transformation Definition of Transform General linear integral transformation of function () f t with respect to the kernel (,) Kst is  () () (,) b a Tft f tKs td t if 0 a , b  and (,) st e , then {() } is the Laplace Transform of f t . 1.1 The Laplace Transform If f t is a continuous function of an independent variable t for all values of t greater than zero, then the integral with respect to t of the product of ft with (,) s t between the limits a and b is defined as transformation, i.e., ˆ ()(,) b a f sf t t s d t Especially, if 0 a , b and (,) st s te , then it is defined as the Laplace Transform
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Ch_02 - Laplace Transform - Summary - Chapter 2 The Laplace...

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