FOURIEROVA METODA

vlastní funkce a vlastní čísla

λ[k_]=k^2-3;
v[k_,x_]=Sin[k x];

Show[
GraphicsRow[
Table[
Plot[v[k,x],{x,0,π},PlotRange->All,Ticks->None,PlotLabel->"λ="<>ToString[TraditionalForm[λ[k]]]],
{k,1,5}
]
]
]

pravá strana

f[x_]=(π-x)/2;

Plot[f[x],{x,0,π},AxesLabel->{"x","f(x)"}]

Fourierovy koeficienty

norma2[k_]=Integrate[v[k,x]^2,{x,0,π}]
d[k_]:=Integrate[f[x]v[k,x],{x,0,π}]/norma2[k]
c[k_]:=d[k]/λ[k]

ListPlot[Table[norma2[k],{k,1,20}],AxesLabel->{"k","||(Subscript[v, k])((||)^2)"}]
ListPlot[Table[d[k],{k,1,20}],AxesLabel->{"k","(Subscript[d, k])"},PlotRange->{-2,2}]
ListPlot[Table[λ[k],{k,1,20}],AxesLabel->{"k","(Subscript[λ, k])"}]
ListPlot[Table[c[k],{k,1,20}],AxesLabel->{"k","(Subscript[c, k])"},PlotRange->{-2,2}]

řešení okrajové úlohy

y[n_,x_]:=Sum[c[k]*v[k,x],{k,1,n}]
Show[
GraphicsRow[
Table[
Plot[Evaluate[y[k,x]],{x,0,π},PlotRange->All,Ticks->None,PlotLabel->"n="<>ToString[k]],
{k,1,5}
]
]
]

Plot[Evaluate[y[10,x]],{x,0,π},PlotRange->All,PlotLabel->"n=10"]

Manipulate[Plot[Evaluate[y[n,x]],{x,0,π}],{n,1,10}]