Skip to content Skip to navigation Skip to collection information

OpenStax_CNX

You are here: Home » Content » Matrix Analysis » The Laplace Transform

Navigation

Table of Contents

Lenses

What is a lens?

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?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

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

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Rice Digital Scholarship

    This collection is included in aLens by: Digital Scholarship at Rice University

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

Also in these lenses

  • Lens for Engineering

    This module and collection are included inLens: Lens for Engineering
    By: Sidney Burrus

    Click the "Lens for Engineering" link to see all content selected in this lens.

Recently Viewed

This feature requires Javascript to be enabled.
 

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.

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

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?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

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

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

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?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

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

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks