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Course by: Steven J. Cox. E-mail the author

The Laplace Transform

Module by: Doug Daniels, Steven J. Cox. E-mail the authors

Summary: This module examines the Laplace Transform, an analytical tool that produces exact solutions for small, closed-form, tractable systems. We use the Laplace transform to move toward a solution for the nerve fiber potentials modeled by the dynamic Strang Quartet in the earlier module of the same name.

The Laplace Transform is typically credited with taking dynamical problems into static problems. Recall that the Laplace Transform of the function hh is hs0e(st)htdt . h s t 0 s t h t . MATLAB is very adept at such things. For example:

Example 1: The Laplace Transform in MATLAB


>> syms t

>> laplace(exp(t))

ans = 1/(s-1)

>> laplace(t*(exp(-t))

ans = 1/(s+1)^2


The Laplace Transform of a matrix of functions is simply the matrix of Laplace transforms of the individual elements.

Example 2: Laplace Transform of a matrix of functions

ettet=1s11s+12 t t t 1 s 1 1 s 1 2

Now, in preparing to apply the Laplace transform to our equation from the dynamic strang quartet module: x=Bx+g , x B x g , we write it as

dxdt=Bx+g t x B x g
(1)
and so must determine how acts on derivatives and sums. With respect to the latter it follows directly from the definition that
Bx+g=Bx+g=Bx+g . B x g B x g B x g .
(2)
Regarding its effect on the derivative we find, on integrating by parts, that dxdt=0e(st)dxtdtdt=xte(st)|0+s0e(st)xtdt . t x t 0 s t t x t 0 x t s t s t 0 s t x t . Supposing that xx and ss are such that xte(st)0 x t s t 0 as t t we arrive at
dxdt=sxx0 . t x s x x 0 .
(3)
Now, upon substituting Equation 2 and Equation 3 into Equation 1 we find sxx0=Bx+g , s x x 0 B x g , which is easily recognized to be a linear system for x x , namely
(sIB)x=g+x0 . s I B x g x 0 .
(4)
The only thing that distinguishes this system from those encountered since our first brush with these systems is the presence of the complex variable ss. This complicates the mechanical steps of Gaussian Elimination or the Gauss-Jordan Method but the methods indeed apply without change. Taking up the latter method, we write x=sIB-1(g+x0) . x s I B g x 0 . The matrix sIB-1 s I B is typically called the transfer function or resolvent, associated with BB, at ss. We turn to MATLAB for its symbolic calculation. (for more information, see the tutorial on MATLAB's symbolic toolbox). For example,

Example 3


>> B = [2 -1; -1 2]

>> R = inv(s*eye(2)-B)

R =

[ (s-2)/(s*s-4*s+3), -1/(s*s-4*s+3)]

[ -1/(s*s-4*s+3), (s-2)/(s*s-4*s+3)]


We note that sIB-1 s I B is well defined except at the roots of the quadratic, s24s+3 s 2 4 s 3 . This quadratic is the determinant of sIB s I B and is often referred to as the characteristic polynomial of BB. Its roots are called the eigenvalues of BB.

Example 4

As a second example let us take the BB matrix of the dynamic Strang quartet module with the parameter choices specified in fib3.m, namely

B=( -0.1350.1250 0.5-1.010.5 00.5-0.51 ) B -0.1350.1250 0.5-1.010.5 00.5-0.51
(5)
The associated sIB-1 s I B is a bit bulky (please run fib3.m) so we display here only the denominator of each term, i.e.,
s3+1.655s2+0.4078s+0.0039 . s 3 1.655 s 2 0.4078 s 0.0039 .
(6)
Assuming a current stimulus of the form i 0 t=t3et610000 i 0 t t 3 t 6 10000 and E m =0 E m 0 brings gs=0.191s+16400 g s 0.191 s 1 6 4 0 0 and so Equation 6 persists in x=sIB-1g=0.191s+164(s3+1.655s2+0.4078s+0.0039)s2+1.5s+0.270.5s+0.260.2497 x s I B g 0.191 s 1 6 4 s 3 1.655 s 2 0.4078 s 0.0039 s 2 1.5 s 0.27 0.5 s 0.26 0.2497

Now comes the rub. A simple linear solve (or inversion) has left us with the Laplace transform of xx. The accursed

Theorem 1: No Free Lunch Theorem

We shall have to do some work in order to recover xx from x x .

confronts us. We shall face it down in the Inverse Laplace module.

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

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Any individual member, a community, or a respected organization.

What are tags?

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