# About the Use of Generalized Forms of Derivatives in the Study of Electromagnetic Problems
## 2 Lamb-Bateman Equation, Operational Methods, and Fractional Calculus
Regarding the use of this family of operators we note, e.g., that the solution of non-homogeneous differential equation  
\[\hat{L}\ u(x)=f(x),\qquad\qquad\qquad\hat{L}=\int_{0}^{\infty}e^{-y^{2}\partial_ {x}}dy\, \tag{2}\]2  
where \(y(x)\) is the unknown, can be formally written as  
\[\hat{L}\ u(x)=f(x),\qquad\qquad\qquad\hat{L}=\int_{0}^{\infty}e^{-y^{2}\partial_ {x}}dy\, \tag{2}\]3  
The use of the Laplace transform identity applied to the operator  
\[\hat{L}\ u(x)=f(x),\qquad\qquad\qquad\hat{L}=\int_{0}^{\infty}e^{-y^{2}\partial_ {x}}dy\, \tag{2}\]4  
and of the exponential shift operator (3) yields  
\[\hat{L}\ u(x)=f(x),\qquad\qquad\qquad\hat{L}=\int_{0}^{\infty}e^{-y^{2}\partial_ {x}}dy\, \tag{2}\]5  
By exploiting Observation 1 we can provide the solution to the following problem involving partial fractional derivatives. [Heat-like Equation] The solution of evolution-like equations involving partial fractional derivatives as, e.g.,  
\[\left\{ \begin{aligned} &\frac{\partial}{\partial t}F(x,t)=-\frac{ \partial^{\alpha}}{\partial x^{\alpha}}F(x,t)\\ & F(x,t)|_{t=0}=f(x),\qquad\qquad 0<\alpha<1\end{aligned} \right. \tag{17}\]  
can be viewed as a kind of heat equation [34; 35; 36; 29] and solved by the use of an appropriate transform. The use of the formalism of the evolution operator [29; 37] yields  
\[F(x,t):=F_{\alpha}(x,t)=\hat{U}_{\alpha}(t)f(x),\qquad\qquad\qquad\hat{U}_{ \alpha}(t)=e^{-t\frac{\partial^{\alpha}}{\partial x^{\alpha}}}. \tag{18}\]  
The action of the evolution operator \(\hat{U}(t)\) on the initial function \(\gamma(x)\) can be obtained via the so-called Levy transform method [13; 14]. To this aim, we note that the following integral transform,concerning the stretching integral, is known as one-sided Levy stable distribution (Equation (19) is also known as Kohr Hausch-Williams-Watts function [38].) \(g_{\gamma}(\eta)\)[14] and acts in the following way_  
\[e^{-\tau x\gamma}=\int_{0}^{\infty}e^{-(\tau x\gamma)^{\gamma-1}\eta}\;g_{ \gamma}(\eta)\;d\eta=\int_{0}^{\infty}e^{-\tau^{\gamma-1}x\;\eta}\;g_{\gamma}( \eta)\;d\eta,\qquad\forall\gamma,x,\tau\in\mathbb{R}:0<\gamma<1,\;\;\;x,\tau>0. \tag{19}\]  
_We can therefore manage to transform the exponential evolution operator in Equation (18) as_  
\[\hat{U}_{\kappa}(t)=\int_{0}^{\infty}e^{-\mu^{-1}\eta\frac{\partial}{\partial x }g_{\kappa}(\eta)d\eta}, \tag{20}\]  
_which eventually yields, for the solution of our problem,_  
\[F_{\alpha}(x,t)=\int_{0}^{\infty}g_{\alpha}(\eta)f\Big{(}x-t^{\alpha-1}\eta \Big{)}d\eta. \tag{21}\]  
Other forms of fractional derivatives have appeared in the literature. **Remark 2**. : _If, in Equation (21), we consider the case with \(\alpha=\frac{1}{2}\), the Levy distribution writes_  
\[g_{\frac{1}{2}}(\eta)=\frac{1}{2\sqrt{\pi}\;\eta^{\frac{1}{2}}}e^{-\frac{1}{ 4\eta}} \tag{22}\]  
_which, for an initial Gaussian distribution, yields the temporal behaviour reported in Figure 1 which displays the interesting feature that, while time increases, the initial gaussian distribution develops the long tail shape characteristic of a Levy distribution._  
The same procedure can be applied when dealing with slightly more complicated problems, like in the following problem. **Problem 2**. : _Let, \(\forall x\in\mathbb{R}\), \(\forall t\in\mathbb{R}^{+}\)_  
\[\left\{ \begin{array}{l}\partial_{t}F(x,t)=-\sqrt{1+\partial_{x}^{v}}\;F(x,t),\\ F(x,0)=f(x)\end{array} \right. \tag{23}\]  
_whose formal solution reads_  
\[F(x,t):=F_{v}(x,t)=e^{-t\sqrt{1+\partial_{x}^{v}}}f(x). \tag{24}\]  
Figure 1: Solution \(F(x,t)\) of Equation (21) at different times \(t\), for \(\alpha=\frac{1}{2}\) and initial function \(f(. )\) being a Gaussian. An analogous technique has been discussed in Ref. [13], where a generalized form of the heat equation (\(\alpha=2\)) has been considered in terms of a double transformation (Levy & Gauss-Weierstrass). **Remark 3**.