FAQ about FourierTransform package (This only applies to V3.01)

<< Calculus`FourierTransform`

Problems

The following examples will return unevaluated.

FourierTransform[Exp[-Abs[t]], t, w]

FourierCosTransform[UnitStep[1 - t], t, w]

Solutions

Use the new options: DefiniteIntegral and Assumptions

FourierTransform[Exp[-Abs[t]], t, w, DefiniteIntegral -> True]

FourierCosTransform[UnitStep[1 - t], t, w, DefiniteIntegral -> True, 
      Assumptions -> {w > 0}]

Details

The usage of FourierTransform is:

?FourierTransform

Attributes[FourierTransform]

New options for FourierTransform, notice that the default "DefiniteIntegral->False"

Options[FourierTransform] // MatrixForm

Convection Problems:

Quit[]

<< Calculus`Master`

FourierTransform[f'[t], t, w]

FourierTransform[f''[t], t, w]

eqn = (U[w]'[t] == -w^2*U[w][t] + w*I*U[w][t])

ic2 = 
      (U[w][0] == 
          FourierTransform[1 - UnitStep[x], x, w, 
            Assumptions -> {Im[x] == 0, Im[w] == 0}])

DSolve[{eqn, ic2}, U[w][t], t]

InverseFourierTransform[
      (E^(t (I w - (w^2)))) 
        (-(I/w) + π DiracDelta[w]), w, x, 
      DefiniteIntegral -> True, Assumptions -> {x > 0, Im[w] == 0, t > 0}]

\(nu[x0_, t0_] := 
      0.5 - Chop[
          I/\((2  Pi)\)*
            \((NIntegrate[
                  E\^\(t\ \((I\ w - w\^2)\) - I\ w\ x\)\/w /. {t -> t0, 
                      x -> x0}, {w, \(-Infinity\), \(-0.000000001\)}] + 

                NIntegrate[
                  E\^\(t\ \((I\ w - w\^2)\) - I\ w\ x\)\/w /. {t -> t0, 
                      x -> x0}, {w, 0.000000001, Infinity}])\)]\)

nt1 = Table[{x, nu[x, 0.1]}, {x, -10, 10}]

nt2 = Table[{x, nu[x, 0.5]}, {x, -10, 10}]

nt3 = Table[{x, nu[x, 1.5]}, {x, -10, 10}]

\(p1 = ListPlot[nt1, PlotJoined -> True]; 

    p2 = ListPlot[nt2, PlotJoined -> True]; 

    p3 = ListPlot[nt3, PlotJoined -> True]; 
Show[p1, p2, p3]\)

eqn = (U[w]'[t] == -w^2*U[w][t] + w*I*U[w][t])

ic3 = (U[w][0] == FourierTransform[Exp[-x^2], x, w])

DSolve[{eqn, ic3}, U[w][t], t]

u[x_, t_] = 
      InverseFourierTransform[
        (E^(-((w^2)/4) + t (I w - (w^2)))) (Sqrt[π]), w, x, 
        DefiniteIntegral -> True, 
        Assumptions -> {t > 0, Im[t] == 0, Im[x] == 0, Im[t] - Im[x] == 0}]

Plot[Evaluate[Table[u[x, t], {t, 0, 1, 0.2}]], {x, -10, 10}, 
      PlotRange -> All]

(D[u[x, t], t] - D[u[x, t], {x, 2}] + D[u[x, t], x]) // 
      Simplify

u[x_, t_, d_, v_] = 
      Exp[-(x - v t)^2/(1 + 4 d t)]/
        Sqrt[1 + 4 d t]

Plot3D[u[x, t, 1, 1], {x, -10, 10}, {t, 0, 1}, 
      PlotPoints -> 40, PlotRange -> All]

Plot3D[u[x, t, 10, 1], {x, -10, 10}, {t, 0, 1}, 
      PlotPoints -> 40, PlotRange -> All]

Plot3D[u[x, t, 100, 1], {x, -10, 10}, {t, 0, 1}, 
      PlotPoints -> 40, PlotRange -> All]

Plot3D[u[x, t, 1, 10], {x, -5, 15}, {t, 0, 1}, 
      PlotPoints -> 40, PlotRange -> All]

Plot3D[u[x, t, 1, 100], {x, -5, 15}, {t, 0, 0.1}, 
      PlotPoints -> 40, PlotRange -> All]