Laplace Transform

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 h is ℒ{h(s)}≡∫₀^∞ⅇ^−(st)h(t)dt.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

ℒ{(

ⅇ^t

tⅇ^−t

)}=(

s−1

(s+1)²

)

Now, in preparing to apply the Laplace transform to our equation from the dynamic strang quartet module: <diff></diff>x^′=Bx+g, we write it as

ℒ{

(x)}=ℒ{Bx+g}(1)

and so must determine how ℒ acts on derivatives and sums. With respect to the latter it follows directly from the definition that

ℒ{Bx+g}	=	ℒ{Bx}+ℒ{g}
	=	Bℒ{x}+ℒ{g}

.(2)

Regarding its effect on the derivative we find, on integrating by parts, that ℒ{

(x)}=∫₀^∞ⅇ^−(st)

(x(t))dt=x(t)ⅇ^−(st)|₀^∞+s∫₀^∞ⅇ^−(st)x(t)dt. Supposing that x ands are such that x(t)ⅇ^−(st)→0 as t→∞ we arrive at

ℒ{

(x)}=sℒ{x}−x(0).(3)

Now, upon substituting Equation 2 and Equation 3 into Equation 1 we find sℒ{x}−x(0)=Bℒ{x}+ℒ{g}, which is easily recognized to be a linear system for ℒ{x}, namely

(sI−B)ℒ{x}=ℒ{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 s. 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 ℒ{x}=(sI−B)^-1(ℒ{g}+x(0)). The matrix(sI−B)^-1 is typically called the transfer function or resolvent, associated with B, at s. 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 (sI−B)^-1 is well defined except at the roots of the quadratic, s²−4s+3. This quadratic is the determinant of sI−B and is often referred to as the characteristic polynomial of B. Its roots are called the eigenvalues of B.