Theory and Modern Applications
Table 1 Basic operations of the FRDTM [ 39 ]
Functional form | Transformed form |
|---|---|
u(x,t) | \(\frac{1}{\Gamma(k\alpha+1)} [\frac{\partial^{\alpha k } }{\partial t^{\alpha k} } u(x,t) ]_{ t=0} \) |
γu(x,t)±βv(x,t) | \(\gamma U_{k} (x)\pm \beta V_{k} (x)\), where γ and β are constants |
u(x,t).v(x,t) | \(\sum_{i=0}^{k}U_{i} (x)V_{k-i} (x) \) |
u(x,t).v(x,t).w(x,t) | \(\sum_{i=0}^{k}\sum_{j=0}^{i}U_{j} (x) V_{i-j} (x) W_{k-i} (x)\) |
\(\frac{\partial^{n\alpha} }{\partial t^{n\alpha } } u(x,t)\) | \(\frac{\Gamma (k\alpha+n\alpha +1 )}{\Gamma (k\alpha+1 )} U_{k+n} (x)\) |
\(\frac{\partial^{n} }{\partial x^{n} } u(x,t)\) | \(\frac{\partial^{n} }{\partial x^{n} } U_{k}(x)\) |
\(x^{m} t^{n} u(x,t)\) | \(x^{m} U_{k-n} (x )\) |
\(x^{m} t^{n} \) | \(x^{m} \delta (k\alpha-n )\), where \(\delta(k\alpha -n)=\left \{ \begin{array}{l@{\quad}l} 1, & \alpha k=n \\ 0, & \alpha k\ne n \end{array} \right \}\) |