Laplace Transform 

The process of multiplying a function f = f(t)by `^`(e, `+`(`-`(st)))and then integrating the product `*`(`^`(e, `+`(`-`(st))), `*`(f(t))) from 0 to t = infinitymake up a function F = F(s) of the variable s (if the resulting integral exists) named the Laplace transformation. 

 

Laplace Transform 

The Laplace transformation is an operation denoted by the symbol L, which associates with each function   satisfying suitable conditions for a unique function F = F(s) called the Laplace transform of f(t) defined by  

 

provided that the limit exists. 

Image 

Laplace Transform 

 

Sufficient conditions that will guarantee the existence of `*`(L, `*`({f(t)})) are that  f  be piecewise continuous on and that f be of exponential order for A function f is said to be of exponential order if there exist numbers `>`(M, 0) and such that `<=`(abs(f(t)), `^`(Me, ct)) for  

 

We say that a function f 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 f exists from both the left and the right. 

 

 

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

> restart:MathMaple:-ini():
 

We can define the operator L by 

> L:=f->laplace(f,t,s):
 

and L(f(t)) 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(f(t)) = F(s)
 

> 'L'(g(t))=L(g(t));
 

L(g(t)) = G(s)
 

 

Linearity of the Laplace Transform 

Let fand gbe functions whose Laplace transforms exists and a and b are  constants. Then 

`and`(`*`(L, `*`({`+`(`*`(a, `*`(f(t))), `*`(b, `*`(g(t))))})) = `+`(`*`(a, `*`(L, `*`({f(t)}))), `*`(b, `*`(L, `*`({g(t)})))), `+`(`*`(a, `*`(L, `*`({f(t)}))), `*`(b, `*`(L, `*`({g(t)})))) = `+`(`*`(... 

Linearity of the Laplace Transform 

 

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

L(`+`(`*`(a, `*`(f(t))), `*`(b, `*`(g(t))))) = `+`(`*`(a, `*`(F(s))), `*`(b, `*`(G(s))))
 

 

Laplace transforms of some basic functions 

f(t) 

`*`(L, `*`({f(t)})) = F(s) 

f(t) 

`*`(L, `*`({f(t)})) = F(s) 

> 1;
 

1
 

> L(%);
 

`/`(1, `*`(s))
 

> exp(a*t);
 

exp(`*`(a, `*`(t)))
 

> L(%);
 

`/`(1, `*`(`+`(s, `-`(a))))
 

> t;
 

t
 

> L(%);
 

`/`(1, `*`(`^`(s, 2)))
 

> sin(a*t);
 

sin(`*`(a, `*`(t)))
 

> L(%);
 

`/`(`*`(a), `*`(`+`(`*`(`^`(s, 2)), `*`(`^`(a, 2)))))
 

> t^2;
 

`*`(`^`(t, 2))
 

> L(%);
 

`+`(`/`(`*`(2), `*`(`^`(s, 3))))
 

> cos(a*t);
 

cos(`*`(a, `*`(t)))
 

> L(%);
 

`/`(`*`(s), `*`(`+`(`*`(`^`(s, 2)), `*`(`^`(a, 2)))))
 

> t^n;
 

`^`(t, n)
 

> L(%) assuming n>0;
 

`*`(GAMMA(`+`(n, 1)), `*`(`^`(s, `+`(`-`(n), `-`(1)))))
 

> convert(%,factorial);
 

`*`(factorial(n), `*`(`^`(s, `+`(`-`(n), `-`(1)))))
 

> exp(b*t)*sin(a*t);
 

`*`(exp(`*`(b, `*`(t))), `*`(sin(`*`(a, `*`(t)))))
 

> L(%);
 

`/`(`*`(a), `*`(`+`(`*`(`^`(`+`(s, `-`(b)), 2)), `*`(`^`(a, 2)))))
 

> sinh(a*t);
 

sinh(`*`(a, `*`(t)))
 

> L(%);
 

`/`(`*`(a), `*`(`+`(`*`(`^`(s, 2)), `-`(`*`(`^`(a, 2))))))
 

> exp(b*t)*cos(a*t);
 

`*`(exp(`*`(b, `*`(t))), `*`(cos(`*`(a, `*`(t)))))
 

> L(%);
 

`/`(`*`(`+`(s, `-`(b))), `*`(`+`(`*`(`^`(`+`(s, `-`(b)), 2)), `*`(`^`(a, 2)))))
 

> cosh(a*t);
 

cosh(`*`(a, `*`(t)))
 

> L(%);
 

`/`(`*`(s), `*`(`+`(`*`(`^`(s, 2)), `-`(`*`(`^`(a, 2))))))
 

> t^n*exp(a*t);
 

`*`(`^`(t, n), `*`(exp(`*`(a, `*`(t)))))
 

> L(%) assuming n>0;
 

`*`(GAMMA(`+`(n, 1)), `*`(`^`(`+`(s, `-`(a)), `+`(`-`(n), `-`(1)))))
 

> convert(%,factorial);
 

`*`(factorial(n), `*`(`^`(`+`(s, `-`(a)), `+`(`-`(n), `-`(1)))))
 

 

Example 1 

Use the definition of the Laplace transform to find  

a)  f(t) = `+`(`*`(3, `*`(`^`(t, 2))), `*`(8, `*`(t)), `-`(6)) , b)  f(t) 

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));
 

Int(`*`(exp(`+`(`-`(`*`(s, `*`(t))))), `*`(`+`(`*`(3, `*`(`^`(t, 2))), `*`(8, `*`(t)), `-`(6)))), t = 0 .. infinity) = `+`(`/`(`*`(6), `*`(`^`(s, 3))), `/`(`*`(8), `*`(`^`(s, 2))), `-`(`/`(`*`(6), `*`...
 

> 'L'(f(t))=F(s);
 

L(`+`(`*`(3, `*`(`^`(t, 2))), `*`(8, `*`(t)), `-`(6))) = `+`(`/`(`*`(6), `*`(`^`(s, 3))), `/`(`*`(8), `*`(`^`(s, 2))), `-`(`/`(`*`(6), `*`(s))))
 

b) 

> f:=t->sin(t)*cos(2*t):
 

> A:=Int(exp(-s*t)*f(t),t=0..infinity):
A=value(A);
 

Int(`*`(exp(`+`(`-`(`*`(s, `*`(t))))), `*`(sin(t), `*`(cos(`+`(`*`(2, `*`(t))))))), t = 0 .. infinity) = `/`(`*`(`+`(`*`(`^`(s, 2)), `-`(3))), `*`(`+`(`*`(`^`(s, 2)), 9), `*`(`+`(`*`(`^`(s, 2)), 1))))
 

> 'L'(f(t))=F(s);
 

L(`*`(sin(t), `*`(cos(`+`(`*`(2, `*`(t))))))) = `/`(`*`(`+`(`*`(`^`(s, 2)), `-`(3))), `*`(`+`(`*`(`^`(s, 2)), 9), `*`(`+`(`*`(`^`(s, 2)), 1))))
 

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));
 

Int(`*`(exp(`+`(`-`(`*`(s, `*`(t))))), `*`(piecewise(`and`(`<=`(0, t), `<`(t, 1)), 1, `and`(`<=`(1, t), `<=`(t, 3)), -1))), t = 0 .. infinity) = `+`(`-`(`/`(`*`(`+`(`-`(1), `*`(2, `*`(exp(`+`(`-`(s)))...
Int(`*`(exp(`+`(`-`(`*`(s, `*`(t))))), `*`(piecewise(`and`(`<=`(0, t), `<`(t, 1)), 1, `and`(`<=`(1, t), `<=`(t, 3)), -1))), t = 0 .. infinity) = `+`(`-`(`/`(`*`(`+`(`-`(1), `*`(2, `*`(exp(`+`(`-`(s)))...
 

 

> 'L'(f(t))=F(s);
 

L(piecewise(`and`(`<=`(0, t), `<`(t, 1)), 1, `and`(`<=`(1, t), `<=`(t, 3)), -1)) = `/`(`*`(`+`(1, `-`(`*`(2, `*`(exp(`+`(`-`(s)))))), exp(`+`(`-`(`*`(3, `*`(s))))))), `*`(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));
 

L(piecewise(`and`(`<=`(0, t), `<`(t, Pi)), sin(t), `<=`(Pi, t), 2)) = `+`(`/`(1, `*`(`+`(`*`(`^`(s, 2)), 1))), `/`(1, `*`(exp(`*`(Pi, `*`(s))), `*`(`+`(`*`(`^`(s, 2)), 1)))), `/`(`*`(2), `*`(s, `*`(ex...
L(piecewise(`and`(`<=`(0, t), `<`(t, Pi)), sin(t), `<=`(Pi, t), 2)) = `+`(`/`(1, `*`(`+`(`*`(`^`(s, 2)), 1))), `/`(1, `*`(exp(`*`(Pi, `*`(s))), `*`(`+`(`*`(`^`(s, 2)), 1)))), `/`(`*`(2), `*`(s, `*`(ex...