PPT-Laplace Transforms of Discontinuous Forcing Functions

MAT 275 We need a better way to describe functions with discontinuities We use the Heaviside Function which is The graph looks like this Its off 0 when then is on 1 when . 1 11 915 1 1 12 sin kt 13 cos kt 14 at 15 sinh kt 16 cosh kt 17 at bt 18 ae at be bt 19 te at 20 at 1 21 at sin kt 22 at cos kt 23 at sinh kt 24 at cosh kt 25 sin kt ks 26 cos kt 27 sinh kt ks 28 cosh kt 29 sin at arctan 30 960t 31 960t 32 erfc TJ Dodson School of Mathematics Manchester University 1 What are Laplace Transforms and Why This is much easier to state than to motivate We state the de64257nition in two ways 64257rst in words to explain it intuitively then in symbols so that we ca integrable. functions. Section 5.2b. Do Now: Exploration 1 on page 264. It is a fact that. With this information, determine the values of the following. integrals. Explain your answers (use a graph, when necessary).. Motivation. The Bilateral Transform. Region of Convergence (ROC). Properties of the ROC. Rational Transforms. Resources:. MIT 6.003: Lecture 17. Wiki: Laplace Transform. Wiki: Bilateral Transform. Wolfram: Laplace Transform. Familiar . Properties. Initial and Final Value Theorems. Unilateral Laplace Transform. Inverse Laplace Transform. Resources:. MIT 6.003: Lecture 18. MIT 6.003: Lecture 19. Wiki: Inverse Laplace Transform. Let f(x) be defined for 0≤x<∞ and let s denote an arbitrary real variable. . The Laplace transform of f(x) designated by either £{f(x)} or F(s), is. for all values of s for which the improper integral converges.. . Continuity. Coley Hawk. Definition/Laws of Continuity. A function ‘f’ is considered continuous at a number ‘a’ if all three of the following criteria are true.. lim. . x. a. - f(x)= . lim. Mass Systems. Andrew Makepeace. Penn State Erie, The . Behrend. College. Advisor Dr. Joe . Previte. Forced Spring Mass System. m. y’’ + b y’ + k y = F(t). m. – mass. b. – friction constant. MAT 275. Example: . Find the solution of the IVP. Solution: . Rewrite the forcing function using the . notation:. Now apply the Laplace Transform Operator to both sides and simplify:.  . (c) ASU-SoMSS - Scott Surgent. Report errors to surgent@asu.edu. . Given an . integrable. function . we define the . Laplace Transform of .  .  . to be the function . .  .  . Where . , the domain of . , is the . domain . of . for which the integral converges. . Let . be a function. Its . Laplace Transform. , written . , is a function in variable . s. , defined by. Case 1 (Constants). . Let . , where . c. is any constant. Then. The integral . is found using limits:. Derivative Rule, Shift Rule, Gamma . Function . & . f. (. ct. ) Rule. MAT 275. Derivative Rule:. If . , then . .. Proof: . Using the definition of the Laplace Transform, we have . .. Differentiate both sides with respect to . Ming Chuang. 1. , . Linjie. Luo. 2. , Benedict Brown. 3. ,. Szymon. Rusinkiewicz. 2. , and . Misha. Kazhdan. 1. 1. Johns Hopkins University . 2. Princeton University. 3. Katholieke. . Universiteit. L. aplace . Transform. UNIT – IV. UNIT- V. PARTIAL DIFFERENTIAL EQUATIONS OF SECOND ORDER INTRODUCTION: . . An . equation is said to be of order two, if it involves at least one of the differential coefficients .