TopicsSections 7.1 to 7.6 The textbook topics, definitions, examples and theorems

Edwards-Penney Ch 7, 7.1 to 7.5 (21.0 K, txt, 22 Feb 2015)

Laplace TheoryForward table Backward tableBasic Theorems of Laplace TheoryFunctions of exponential order Vector space theory Piecewise continuous and piecewise smooth functions Existence theorem for Laplace integrals Euler solution atoms have a Laplace integral Forward table Backward table Lerch's theorem Linearity. Shift theorem L(exp(at)f(t)) = L(f(t))|s->(s-a) Parts theorem L(y')=sL(y)-y(0) Parts formula derivation. Unit step, pulse and ramp General piecewise continuous function as a sum of modulated pulses Convolution on (-inf,inf) and Laplace theory convolution definition Convolution theorem [no proof ever ...] Maple Lab 2 session LCB115, last 20 min. Lab due next Monday. The s-differentiation theorem (d/ds)L(f(t))=L((-t)f(t)). Solving differential equations by Laplace's method. Slide: Solving y' = -1, y(0)=2 with Laplace's method Laplace's method and quadrature for higher order equations and systems Solving x'' + 4x = t exp(-t), x(0)=1, x'(0)=0 by the Laplace methodLaplace theory references: Laplace and Newton calculus. Photos. (188.3 K, pdf, 14 Mar 2016)Slides: Intro to Laplace theory. Calculus assumed. (144.9 K, pdf, 13 Mar 2016)Slides: Laplace rules (144.9 K, pdf, 14 Mar 2016)Slides: Laplace table proofs (151.9 K, pdf, 14 Mar 2016)Slides: Laplace examples (133.7 K, pdf, 14 Mar 2016)Slides: Piecewise functions and Laplace theory (98.5 K, pdf, 14 Mar 2016)Slides: DE systems, examples, theory (730.9 K, pdf, 09 Apr 2014)Manuscript: Laplace resolvent method (81.0 K, pdf, 14 Mar 2016)Slides: Laplace second order systems (273.7 K, pdf, 14 Mar 2016)Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 14 Mar 2016)Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008)Slides: Sliding plates example (105.8 K, pdf, 20 Aug 2008)Slides: Heaviside's method 2008 (352.3 K, pdf, 07 Jan 2014)Manuscript: Laplace theory 2008-2017 (546.8 K, pdf, 24 Feb 2017)Manuscript: Ch7 Laplace solutions 7.1 to 7.4 (from EP 2250 book) (1068.7 K, pdf, 22 Feb 2015)Transparencies

DEF:Piecewise Continuous FunctionExistence of the Laplace integral. DEF. Unit step u(t-a)=1 for t>=a, else zero DEF. Ramp t->(t-a)u(t-a) Backward table problems: examples Forward table problems: examples Computing Laplace integrals L(f(t)) with rules Solving an equation L(y(t))=expression in s for y(t) Complex roots and quadratic factors Partial fraction methods Trig identities and their use in Laplace calculations Hyperbolic functions and Laplace calculations Why the forward and backward tables don't have cosh, sinh entries

Piecewise FunctionsUnit Step: u(t)=1 for t>=0, u(t)=0 for t<0. Pulse: pulse(t,a,b)=u(t-a)-u(t-b) Ramp: ramp(t-a)=(t-a)u(t-a) L(u(t-a)) = (1/s) exp(-as) [for a >= 0 only]Integral TheoremL(int(g(x),x=0..t)) = s L(g(t)) Applications to computing ramp(t-a) L(ramp(t-a)) = (1/s^2) exp(-as) [for a >= 0 only]Piecewise defined periodic wavesSquare wave: f(t)=1 on [0,1), f(t)=-1 on [1,2), 2-periodic Triangular wave: f(t)=|t| on [-1,1], 2-periodic Sawtooth wave: f(t)=t on [0,1], 1-periodic Rectified sine: f(t)=|sin(kt)| Half-wave rectified sine: f(t)=sin(kt) when positive, else zero. Parabolic wavePeriodic function theoremProof details (Monday) Laplace of the square wave. Problem 7.5-25. Answer: (1/s)tanh(as/2) Applications of Laplace's method from 7.3, 7.4, 7.5Convolution theoremDEF. Convolution of f and g = f*g(t) = integral of f(x)g(t-x) from x=0 to x=t THEOREM. L(f(t))L(g(t))=L(convolution of f and g) Application: L(cos t)L(sin t) = L(0.5 t sin(t))