Uebungen

D'Gl. 2. Ordnung

Remove["Global`*"]

links = 
      LaplaceTransform[y''[t] + y'[t] + y[t], t, 
          s] /. {LaplaceTransform[y[t], t, s] -> Y[s], 
          y[0] -> 1, y'[0] -> 1}

rechts = LaplaceTransform[Sin[t], t, s]

solv = Solve[links == rechts, {Y[s]}]  //  Flatten

U[s] := Y[s] /.  solv;   U[s]

U[s] // Apart

u0[t_] := InverseLaplaceTransform[U[s], s, t] // Simplify;  
    u0[t]

Plot[Evaluate[{u0[t]}], {t, 0, 6  Pi}];

System

Remove["Global`*"]

links1 = 
      LaplaceTransform[y''[t] + z[t], t, 
          s] /. {LaplaceTransform[y[t], t, s] -> Y[s], 
          y[0] -> 1, y'[0] -> 0, 
          LaplaceTransform[z[t], t, s] -> Z[s], 
          z[0] -> 1, z'[0] -> 0}

links2 = 
      LaplaceTransform[y[t] - z'[t], t, 
          s] /. {LaplaceTransform[y[t], t, s] -> Y[s], 
          y[0] -> 1, y'[0] -> 0, 
          LaplaceTransform[z[t], t, s] -> Z[s], 
          z[0] -> 1, z'[0] -> 0}

rechts1 = LaplaceTransform[Sin[t], t, s]

rechts2 = LaplaceTransform[Cos[t], t, s]

solv = 
      Solve[{links1 == rechts1,  links2 == rechts2}, {Y[s], 
            Z[s]}]  //  Flatten

U[s] := Y[s] /.  solv[[1]];   U[s]

U[s] // Apart

V[s] := Z[s] /.  solv[[2]];   V[s]

u[t_] := InverseLaplaceTransform[U[s], s, t] // Simplify;  
    u[t]

v[t_] := InverseLaplaceTransform[V[s], s, t] // Simplify;  
    v[t]

Plot[Evaluate[{u[t], v[t]}], {t, 0, 6  Pi}];

Faltung

Remove["Global`*"]

Rechnung gewöhnlich

links = 
      LaplaceTransform[y''[t] + 3  y'[t] + 4  y[t], t, 
          s] /. {LaplaceTransform[y[t], t, s] -> Y[s], 
          y[0] -> 0, y'[0] -> 0}

rechts = LaplaceTransform[Sin[t], t, s]

solv = Solve[links == rechts, {Y[s]}]  //  Flatten

U[s] := Y[s] /.  solv;   U[s]

U[s] // Apart

InverseLaplaceTransform[U[s], s, t] // Simplify

g[t_] := 
      InverseLaplaceTransform[1/(s^2 + 3  s + 4), s, t] // Simplify;  
    g[t]

Plot[Evaluate[{g[t]}], {t, 0, 2  Pi}];

Rechnung mit Faltung

faltung[u_, v_, t_] := 
        Integrate[
          u[λ] v[t - λ], {λ, 0, t}];

u[t_] := g[t];  v[t_] := Sin[t];
faltung[u, v, t]

Faltung allgemein

Remove["Global`*"]

LaplaceTransform[
        a y''[t] + b y'[t] + c y[t], t, 
        s] /. {LaplaceTransform[y[t], t, s] -> Y[s], 
        y[0] -> y0, y'[0] -> y1}

Beispiel 1

a = 1;  b = 1;  c = 1; 
y0  =  1;  y1  =  1;
f[t_] := Sin[t];

mo[k_] := 
        Module[{ }, 
          r[t_] := InverseLaplaceTransform[(a s y0  + a y1  + 
                      b y0)/(a s^2 + b s + c), s, t] // 
              Simplify; 
Print["r[t] = ", 
            r[t]]; 
g[t_] := 
            InverseLaplaceTransform[1/(a s^2 + b s + c), s, t] // 
              Simplify;  
          Print["g[t] = ", g[t]]; 
faltung[f_, g_, t_] := 
            Integrate[f[λ] g[t - λ], {λ, 0, t}, 
                GenerateConditions -> False]  //  
              Simplify; 
Print["y[t] = ", 
            faltung[f, g, 
              t]];
];
mo[k]

Beispiel 2

a = 1;  b = 1;  c = 1; 
y0  =  1;  y1  =  1;
f[t_] := Cos[t];

    
mo[k_] := 
        Module[{ }, 
          r[t_] := InverseLaplaceTransform[(a s y0  + a y1  + 
                      b y0)/(a s^2 + b s + c), s, t] // 
              Simplify; 
Print["r[t] = ", 
            r[t]]; 
g[t_] := 
            InverseLaplaceTransform[1/(a s^2 + b s + c), s, t] // 
              Simplify;  
          Print["g[t] = ", g[t]]; 
faltung[f_, g_, t_] := 
            Integrate[f[λ] g[t - λ], {λ, 0, t}, 
                GenerateConditions -> False]  //  
              Simplify; 
Print["y[t] = ", 
            faltung[f, g, 
              t]];
];
mo[k]

Beispiel 3

a = 1;  b = 1;  c = 1; 
y0  =  1;  y1  =  1;
f[t_] := 
        E^(-t);

mo[k_] := 
        Module[{ }, 
          r[t_] := InverseLaplaceTransform[(a s y0  + a y1  + 
                      b y0)/(a s^2 + b s + c), s, t] // 
              Simplify; 
Print["r[t] = ", 
            r[t]]; 
g[t_] := 
            InverseLaplaceTransform[1/(a s^2 + b s + c), s, t] // 
              Simplify;  
          Print["g[t] = ", g[t]]; 
faltung[f_, g_, t_] := 
            Integrate[f[λ] g[t - λ], {λ, 0, t}, 
                GenerateConditions -> False]  //  
              Simplify; 
Print["y[t] = ", 
            faltung[f, g, 
              t]];
];
mo[k]

Beispiel 4

a = 1;  b = 1;  c = 1; 
y0  =  1;  y1  =  1;
f[t_] := t;

mo[k_] := 
        Module[{ }, 
          r[t_] := InverseLaplaceTransform[(a s y0  + a y1  + 
                      b y0)/(a s^2 + b s + c), s, t] // 
              Simplify; 
Print["r[t] = ", 
            r[t]]; 
g[t_] := 
            InverseLaplaceTransform[1/(a s^2 + b s + c), s, t] // 
              Simplify;  
          Print["g[t] = ", g[t]]; 
faltung[f_, g_, t_] := 
            Integrate[f[λ] g[t - λ], {λ, 0, t}, 
                GenerateConditions -> False]  //  
              Simplify; 
Print["y[t] = ", 
            faltung[f, g, t] // 
              Simplify];
];
mo[k]