Class Notes 12: Laplace Transform (1/3) Intro & Differential Equations 82 – Engineering Mathematics

Laplace Transform – Introduction

Laplace Transform – Introduction

Pierre Simon Laplace (Marquis) French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste (Celestial Mechanics) (1799-1825). This seminal work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. He formulated Laplace's equation, and invented the Laplace transform. In mathematics, the Laplace transform is one of the best known and most widely used integral transforms. It is commonly used to produce an easily solvable algebraic equation from an ordinary differential equation. It has many important applications in mathematics, physics, optics, electrical engineering, control engineering, signal processing, and probability theory. He restated and developed the nebular hypothesis of the origin of the solar system and was one of the first scientists to postulate the existence of black holes and the notion of gravitational collapse. Pierre-Simon, marquis de Laplace 23 April 1749 - 5 March 1827 Video Clip

Laplace Transform – Motivation Transform between the time domain (t) to the frequency domain (s) Lossless Transform – No information is lost when the transform and the inverse transform is applied Frequency Domain (S) Time Domain (t) Differential Equation Algebraic Equation Time Domain Solution Frequency Domain Solution

Improper Integral – Definition An improper integral over an unbounded interval is defined as the limit of an integral over a finite interval where A is a positive real number. If the integral from a to A exists for each A > a the limit as A → ∞ exists then the improper integral is said to converge to that limiting value. Otherwise, the integral is said to diverge or fail to exist.

Improper Integral – Example

Integral Transform - Laplace Transform – Definition Tool for solving linear diff. eq. – Integral transform k(s,t) - The kernel of the transformation Transform Laplace transform whenever this improper integral converges

Suppose that f is a function for which the following hold: Laplace Transform – Theorem 6.1.2 – Sufficient Conditions for Existence Suppose that f is a function for which the following hold: (1) f is piecewise continuous on [0, b] for all b > 0. (2) | f(t) |  Meat when t  T, with T, M > 0. Then the Laplace Transform of f exists for s > a. Note: A function f that satisfies the conditions specified above is said to have exponential order as t  .

a = t0 < t1 < … < tn = b such that Laplace Transform – Theorem 6.1.2 – Condition No. 1 - Piecewise Continuous A function f is piecewise continuous on an interval [a, b] if this interval can be partitioned by a finite number of points a = t0 < t1 < … < tn = b such that (1) f is continuous on each (tk, tk+1) In other words, f is piecewise continuous on [a, b] if it is continuous there except for a finite number of jump discontinuities.

Laplace Transform – Theorem 6. 1. 2 – Condition No Laplace Transform – Theorem 6.1.2 – Condition No. 1 - Piecewise Continuous

Laplace Transform – Theorem 6. 1. 2 – Condition No Laplace Transform – Theorem 6.1.2 – Condition No. 2 - Exponential Order If f is an increasing function Then | f(t) |  Meat when t  T, with T, M > 0.

Laplace Transform – Theorem 6. 1. 2 – Condition No Laplace Transform – Theorem 6.1.2 – Condition No. 2 - Exponential Order A positive integral power of t is always of exponential order since for c>0

Laplace Transform (Function) – Example

Laplace Transform (Function) – Example

Laplace Transform (Function) – Example

Laplace Transform (Function) – Example second integration by parts

Laplace Transform – Discontinues Function - Example Transformation of piecewise continuous function

Operation Properties – Translation on the t-Axis (Time) Second Translation Theorem

Operation Properties – Translation on the S-Axis (Freq Operation Properties – Translation on the S-Axis (Freq.) First Translation Theorem Example:

Laplace Transform – Linearity Suppose f and g are functions whose Laplace transforms exist for s > a1 and s > a2, respectively. Then, for s greater than the maximum of a1 and a2, the Laplace transform of c1 f (t) + c2g(t) exists. That is, with

Laplace Transform – Linearity – Example

Inverse Laplace Transform – Example

Inverse Laplace Transform – Example

Inverse Laplace Transform – Example (linearity)

Inverse Laplace Transform – Partial Fraction Option 1: 3 equations with 3 unknown variables A, B, C Option 2: Plug (s=1) 16=A(-1)(5) A=-16/5 Plug (s=2) 25=B(1)(6) B=25/6 Plug (s=-4) 1=C(-5)(-6) C=1/30

Inverse Laplace Transform – Partial Fraction

Laplace Transform of a Derivative (First)

Laplace Transform of a Derivative (Second)

Laplace Transform of a Derivative first derivative Initial Conditions second derivative third derivative n-th derivative

Laplace Transform – Solving Linear ODEs Initial Conditions: - constants (I.C.s) - constants Apply Laplace Transform

Laplace Transform – Solving Linear ODEs The Laplace Transform of a linear differential equation with constant coefficients become an algebraic equation in Y(s) Time Domain Differential Equation Frequency Domain Algebraic Equation Laplace Transform

Laplace Transform – Example First Order Linear ODEs

Laplace Transform – Example First Order Linear ODEs

Laplace Transform – Example Second Order Linear ODEs (See previous example)