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Table 1 The fundamental operations of RDTM

From: Modified differential transform method for solving linear and nonlinear pantograph type of differential and Volterra integro-differential equations with proportional delays

Original form Transformed form
u(t) \(\mathcal{U}(k)=\frac{1}{k!}[\frac{\partial ^{k}}{\partial {t}^{k}}u(t)]_{t=0}\)
w(t)=λu(tγv(t) \(\mathcal{W}(k)=\lambda \mathcal{U}(k)\pm \gamma \mathcal{V}(k)\)
\(w(t)=t^{r}\) \(\mathcal{W}(t)=\delta (k-r)\), \(\delta (k)=\bigl\{ \scriptsize{ \begin{array}{l@{\quad}l} 1, & k=0,\\ 0, & k\neq 0, \end{array} }\)
\(w(t)= t^{r} u(t)\) \(\mathcal{W}(k)=\mathcal{U}(k-r)\)
w(t)=u(t)v(t) \(\mathcal{W}(k)=\sum _{r=0}^{k}\mathcal{V}(r)\mathcal{U}(k-r)= \sum _{r=0} ^{k}\mathcal{U}(r)\mathcal{V}(k-r)\)
\(w(t)=\frac{\partial ^{r}}{\partial {t}^{r}}u(t)\) \(\mathcal{W}(k)=(k+1)\cdots (k+r)\mathcal{U}(k+r)=\frac{(k+r)!}{k!}\mathcal{U}(k+r)\)
w(t)=sin(λt) \(\mathcal{W}(k)=\frac{\lambda ^{k}}{k!}\sin (\frac{\pi k}{2!})\)
w(t)=cos(λt) \(\mathcal{W}(k)=\frac{\lambda ^{k}}{k!}\cos (\frac{\pi k}{2!})\)
\(w(t)=e^{\lambda t}\) \(\mathcal{W}(k)=\frac{\lambda ^{k}}{k!}\)