PPT-Using Laplace Transforms to Solve IVPs with Discontinuous Forcing Functions

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 ASUSoMSS Scott Surgent Report errors to surgentasuedu. A:. Evaluate . the following . limit . and tell . if it . is continuous, removable discontinuity, or . nonremovable. discontinuity . at the given value. . . (. Check for infinite limits if the limit of the value does not exist.). and Feedback Processes. Earth. ’. s Climate System. What have we learned?. Earth is a planet. Planetary temperature is determined by . Brightness of our star. Earth-sun distance. Albedo of the planet. 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. Parker MacCready, January 2011. Reference: . Admiralty Manual of Tides. , . Doodson. & Warburg, 1941, His Majesty’s Stationery Office. Orbital Force Balance. Tides are forced by the sun and moon, with similar physics. The Moon’s forcing is about twice that of the sun.. 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.. Assigned work: . pg 51 #4adef, bcdf,7,8,10-13. A continuous curve is a curve without breaks, holes or jumps. Usually if we talk about a curve being discontinuous it is at a specific point. These are discontinuous….. Feedbacks. . “Our understanding of the climate system is complicated by . feedbacks. that either . amplify. or . damp. perturbations…”. Feedbacks. . Describe the inter-related growth or decay of the components of a system.. :. Potential . Vorticity. :. = 0 for isentropic motions. Equivalent Potential . Vorticity. :. For a horizontally homogeneous atmosphere with vertical wind shear, vortex lines and Theta-E surfaces are initially horizontal (EPV=0). Since EPV is conserved, vortex lines remain on the original theta-E surfaces if they are subsequently tilted up or down…. . . 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. . Douglas Wilhelm Harder, . M.Math. . LEL. Department of Electrical and Computer Engineering. University of Waterloo. Waterloo, Ontario, Canada. ece.uwaterloo.ca. dwharder@alumni.uwaterloo.ca. © 2012 by Douglas Wilhelm Harder. Some rights reserved.. 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.