1 Fibonacci Example

\[ f(x+2) = f(x+1) + f(x) \]

Translation Rules:

\[\displaylines{{\color{red}f(x) \overset{T}{\Rightarrow} T [f(x+1)] } \\ {\color{red}f(x) \overset{T^2}{\Rightarrow} T^2 [f(x+2)] }\\ {\color{red}f(x) \overset{T^0}{\Rightarrow} T^0 [f(x)]} } \]

Solve Inverse Translation Rule:

\[ let\ T = a*_\]

\[ f(x+1) = a * f(x)\] \[ f(x) = a^{x} k\]

Translate Equation into Operations:

• \(T\) is NOT a function, it is more like a functor

\[ f(x+2) = f(x+1) + f(x) \Rightarrow T^2[f(x)] = T^1 [f(x)] + T^0 [f(x)] \] \[ T^2 [f(x)] = T [f(x)] + 1 [f(x)] \]

Solve Operational Eqn:

\[ ( T^2 - T - 1 ) [f(x)] = 0\] \[ (T - \frac{1 + \sqrt{5}}{2})(T - \frac{1 - \sqrt{5}}{2}) [f(x)] = 0\] $$ (T - )(T - (-)^-1)[f(x)] = 0

Use Inverse Translation Rule:

\[ f(x) = \phi^x k_1 + (-\phi)^{-x}k_2\]

2 Operational Calculus in ODE

\[ f''(x) + 5 f'(x) + 6 f(x) = 0 \]

Translation Rules:

\[ f''(x) \Rightarrow T^2 f(x) \] \[ f'(x) \Rightarrow T^1 f(x) \] \[ f(x) \Rightarrow T^0 f(x) \]

Translate Equation into Operations:

\[ T^2 [f(x)] + 5T^1 [f(x)] + 6T^0 [f(x)] = 0 \] \[ (T^2 + 5T + 6) [f(x)] = 0\]

Solve Operational Eqn:

\[(T + 2)(T + 3)f(x) = 0\]

\[ T f(x) = -2 f(x) \] \[ T f(x) = -3 f(x) \]

Solve:

\[ f'(x) = -2 f(x) \] \[ f'(x) = -3 f(x) \]

3 Operational Calculus in Differential Calculus

Lets focus on numerator of limit definition \(\lim_{\Delta \to 0}\frac{f(x+\Delta) - f(x)}{\Delta}\)

\[f(x+\Delta) - f(x)\]

Translation Rules:

\[ f(x+\Delta) \Rightarrow T f(x)\] \[ f(x) \Rightarrow 1 f(x)\]

Translate Equation into Operations:

\[f(x+\Delta) - f(x) \Rightarrow T f(x) - 1 f(x)\] \[T - 1\]