19.4.3 Defining symbolic transform pairs
Some algebraic transformations of a function behave predictably under
the Fourier or Laplace transform. For example, if g(x)=f(x−a), then
F(g)(s)=e−2π i a sF(f)(s). The
addtable
command lets you assign a function name to the
Fourier (see Section 19.4.2) or Laplace
(see Section 12.4.2) transform of another function name, without specifying the either
function. This allows you to alter the original function and see the
effect on the Fourier (or Laplace) transform.
-
addtable takes five arguments:
-
transform, which can be fourier or
laplace and indicates the type of transform.
- f(x), where f is a symbol representing an
unspecified function of the variable x.
- F(s), where F is a symbol representing the
transform of f and s is the new variable.
- x, the variable used by f.
- s, the variable used by F.
- addtable(transform,f(x),F(s),x,s)
returns 1 if F is assigned as the transform of f, and 0
otherwise. In the case that F is assigned as the transform
of f, then the transform (fourier or laplace) of
manipulations of f will be returned in terms of F and conversely.
Examples
addtable(fourier,y(x),Y(s),x,s) |
addtable(fourier,g(x,t),G(s,t),x,s) |
ODE solving with Fourier transforms.
Fourier transforms can be used for solving linear differential
equations with constant coefficients. For example, to obtain a
particular solution to the equation
where δ is the Dirac delta function, you can first
transform both sides of the above equation.
L:=fourier(y(x)+4*diff(y(x),x,4),x,s);
R:=fourier(Dirac(x),x,s) |
Then you can solve the equation L=R for Y(s). Generally, you should
apply csolve instead of solve.
Finally, you can apply ifourier to obtain y(x).
|
| | e | | | ⎛
⎜
⎜
⎝ | cos | ⎛
⎜
⎜
⎝ | | ⎞
⎟
⎟
⎠ | +sin | ⎛
⎜
⎜
⎝ | | ⎞
⎟
⎟
⎠ | ⎞
⎟
⎟
⎠ |
| | | | | | | | | | |
|
The above solution can be combined with solutions of the corresponding
homogeneous equation to obtain the general solution.