Laplace Transform 

The process of multiplying a function by and then integrating the product from 0 to make up a function of the variable (if the resulting integral exists) named the Laplace transformation. 

Laplace Transform 

The Laplace transformation is an operation denoted by the symbol , which associates with each function   satisfying suitable conditions for a unique function called the Laplace transform of defined by  

provided that the limit exists. 

Laplace Transform 

Sufficient conditions that will guarantee the existence of are that    be piecewise continuous on and that be of exponential order for A function is said to be of exponential order if there exist numbers and such that for  

We say that a function is piecewise continuous if it has  only finitely many points of discontinuity over any finite interval and at every point of discontinuity the limit of exists from both the left and the right. 

• laplace applies the Laplace transform of with respect to

The command is in the inttrans package, a collection of commands designed to compute integral transforms.  

>	restart:MathMaple:-ini():

 

We can define the operator by  > L:=f->laplace(f,t,s):

and by  > alias(F(s)=L(f(t))):

To facilitate the use of the Laplace transform, these two and more definitions are included in the initialization file.  

> 'L'(f(t))=L(f(t));

> 'L'(g(t))=L(g(t));

Linearity of the Laplace Transform 

Let and be functions whose Laplace transforms exists and and are  constants. Then 

Linearity of the Laplace Transform 

>	'L'(a*f(t)+b*g(t))=L(a*f(t)+b*g(t));

 

Laplace transforms of some basic functions
> 1;	> L(%);	> exp(a*t);	> L(%);
> t;	> L(%);	> sin(a*t);	> L(%);
> t^2;	> L(%);	> cos(a*t);	> L(%);
> t^n;	> L(%) assuming n>0;     > convert(%,factorial);	> exp(b*t)*sin(a*t);	> L(%);
> sinh(a*t);	> L(%);	> exp(b*t)*cos(a*t);	> L(%);
> cosh(a*t);	> L(%);	> t^n*exp(a*t);	> L(%) assuming n>0;     > convert(%,factorial);

Example 1 

Use the definition of the Laplace transform to find  

a)   , b)   

c)   ,  d)    

Solution 

>	assume(s>0):

 

Using definition	Direct solution
a)  > f:=t->3*t^2+8*t-6:   > A:=Int(exp(-s*t)*f(t),t=0..infinity): A=expand(value(A));	> 'L'(f(t))=F(s);
b)  > f:=t->sin(t)*cos(2*t):   > A:=Int(exp(-s*t)*f(t),t=0..infinity): A=value(A);	> 'L'(f(t))=F(s);
c)  > f:=t->piecewise(t>=0 and t<1,1,t>=1 and t<=3,-1):   > A:=Int(exp(-s*t)*f(t),t=0..infinity): A=simplify(value(A));	> 'L'(f(t))=F(s);
d)  > f:=t->piecewise(t>=0 and t<Pi,sin(t),t>=Pi,2):   > A:=Int(exp(-s*t)*f(t),t=0..infinity): A=expand(value(A));	> 'L'(f(t))=expand(F(s));