lab7A - Lab 7: Laplace Transforms using Maple The Laplace...

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Page 1 - 7 Use single quotes to remove any assumptions about a variable. Lab 7: Laplace Transforms using Maple The Laplace transform sometimes involves challenging integrals. Maple can usually do these quite easily for you. Recall that the single-sided Laplace transform of the function f ( t ) is defined by the improper integral: L [ f ( t )] = f ( t ) e ! st dt 0 " # For example, the transform of the unit step function u ( t ), assuming Re( s ) > 0, is found to be: L [ u ( t )] = u ( t ) e ! st dt 0 " # = e ! st dt = 0 " # e ! st ! s 0 " = 1 s Let's see how this could be performed in Maple. Example 1: Laplace Transform of Unit Step Function in Maple Using Maple, find the Laplace transform of the unit step function u ( t ). Method 1: You could of course do all the work from scratch as in this first method. > alias(u=Heaviside); interface(imaginaryunit=j); > L := s -> int( u(t)*exp(-s*t), t=0. . infinity); L(s); Maple of course cannot evaluate the limit, because it will not make any assumptions about the complex number s without our permission. To help it perform the limit, we need to assume that the real part of s is greater than 0 so that the exponential will go to zero. > assume( Re(s)>0); L(s); This returns the expected value: Note that the attached tilde simply is Maple's way of denoting that we have made an assumption about s . We will want to unassume before calculating any additional transforms. Method 2: Maple has a built-in Laplace transform function which makes things even easier. The Laplace and inverse Laplace function are in a special library of integral transform routines named inttrans which needs to be loaded first. > with(inttrans); Type in the following help command to see how the laplace command works. Notice that the laplace command can even be used directly with differential equations. > ? inttrans[laplace] OK, let's unassume s and then transform the step function u ( t ) using the built-in laplace command. > s := 's': > laplace( u(t),t,s); returns 1/s Notice that even with the assumption about s removed, the laplace function gives us the answer in the form we want!
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Page 2 - 7 Next note that there is also a built-in inverse laplace transform called invlaplace . > invlaplace( 1/s, s,t); returns 1 Notice however that the inverse laplace returned only 1 instead of u ( t ). This is because u ( t ) = 1 for t > 0 and we are using the one-sided laplace transform. Example 2: Laplace Transform of a Delayed Unit Step Function u(t-t0) a. Using Maple, find the Laplace transform of the unit step function u ( t –1) which has been delayed by one time unit. > laplace( u(t-1),t,s);
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lab7A - Lab 7: Laplace Transforms using Maple The Laplace...

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