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Table 1 One-dimensional DTM fundamental operations

From: A quasi-analytical solution of homogeneous extended surfaces heat diffusion equation

Functions

Transform functions

f(x)=

F(k)

af 1(x) + bf 2(x)

F(k) = aF 1(k) + bF 2(k)

\( \frac{df}{dx} \)

F(k) = (k + 1)F(k + 1)

\( \frac{d^2f}{dx^2} \)

F(k) = (k + 1)(k + 2)F(k + 2)

x n

\( F(k)=\delta \left(k-n\right)=\left\{\begin{array}{c}1\kern0.75em si\kern0.75em k=n\\ {}0\kern0.75em si\kern0.5em k\ne n\end{array}\right. \)

exp(λx)

\( F(k)=\frac{\lambda^k}{k!} \)

f 1(x)f 2(x)

\( F(k)=\sum_{l=0}^k{F}_1(l){F}_2\left(k-l\right) \)

f 1(x)f 2(x)f 3(x)

\( F(k)=\sum_{h=0}^k\sum_{l=0}^h{F}_1(l){F}_2\left(h-l\right){F}_3\left(k-h\right) \)