Table 1 The fundamental operations of the DTM
Original function | Transformed function |
|---|---|
\(w(x)=\alpha u(x)\pm \beta v(x)\) | \(W(k)=\alpha U(k)\pm \beta V(k)\) |
\(w(x)=u(x)v(x)\) | \(W(k)=\sum _{m=0}^k U(m)V(k-m)\) |
\(w(x)=\hbox {d}^m u(x)/\hbox {d}x^m\) | \(W(k)= \frac{(k+m)!}{k!}U(k+m)\) |
\(w(x)=x^m\) | \(W(k)= \delta (k-m)=\left\{ \begin{array}{ll} 1, \quad \hbox {if}\; k=m,\\ 0, \quad \hbox {if} \; k\ne m. \end{array} \right.\) |
\(w(x)=\exp (x)\) | \(W(k)= 1/k!\) |
\(w(x)=\sin (\alpha x+\beta )\) | \(W(k)= \alpha ^k/k! \sin (k \pi /2+\beta )\) |
\(w(x)=\cos (\alpha x+\beta )\) | \(W(k)= \alpha ^k/k! \cos (k \pi /2+\beta )\) |