User talk:OfirMarom

Fractional differential equations are a generalization of differential equations through fractional calculus where the order of the derivative is allowed to take non-integer values. The are almost as old as integer-order differential equations. Fractional diffusion equations have broad applications in fields such as fluid mechanics, biology, hydrodynamics, optic fibres and finance.

The rise in popularity of fractional differential equations is largely due to empirical evidence that suggests that integer order derivatives fail to capture the diffusive characteristics of certain natural processes. For example, in finance, it is often assumed that the log stock price changes follow Brownian motion. This model results in a differential equation that when solved return the price of European derivative written on the stock. However, empirical evidence suggest that Brownian motion does not accurately capture the dynamics of these price movements; however, using models which result in fractional differential equations capture these dynamics far more accurately.

In general it is not possible to find closed form solutions to fractional differential equations and numerical approximations or semi-analytic methods must be used. A popular numerical method to solve fractional differential equations is the finite difference scheme by utilizing the Grünwald-Letnikov derivative to create a finite difference scheme that generalizes the classical finite difference method to the fractional case. A popular semi-analytic solution to solve fractional differential equations is the Adomian decomposition method, which solves these equations using a power series expansion. Adomian decomposition method sometimes yields closed form solutions to fractional differential equations.

Fractional differential equations are characterised by a $$ _{x}D_{b}^{\alpha} = \frac{\partial^{\alpha}}{\partial_{+} r ^{\alpha}} \, $$ or $$ _{a}D_{x}^{\alpha} = \frac{\partial^{\alpha}}{\partial_{-} r ^{\alpha}} \, $$ operator, where $$ a  \, $$ and $$ b \, $$ are constants, that operate on time and spacial variables. These operators represents the left-handed and right-handed fractional Riemann-Liouville derivative of order $$ \alpha \, $$ respectively. The Riemann-Liouville fractional derivatives are derived by differenatiating the Riemann-Liouville integral. The parameter $$ \alpha \, $$ represent the order of the fractional derivative and control the rate of diffusion of the process. The left-handed Riemann-Liouville derivative of a function $$ f(r) \, $$ at a point $$ r \, $$ depends on the function's values to the left of  $$ r \, $$. Similarly, The right-handed Riemann-Liouville derivative of a function $$ f(r) \, $$ at a point $$ r \, $$ depends on the function's values to the right of  $$ r \, $$.

Examples of Fractional Differential Equations
In general, the one-dimentional fractional differential equation for a function $$ U  \, $$ in  $$ x  \, $$ and  $$ t  \, $$ is:

$$

\frac{\partial^{\beta} U(x,t) }{\partial t ^{\beta}} = A(x,t) = \frac{\partial^{\alpha} U(x,t)}{\partial_{+} x ^{\alpha}} + B(x,t)   \frac{\partial^{\alpha} U(x,t)}{\partial_{-} x ^{\alpha}} + C(x,t), \, $$

(where $$ \alpha \, $$ and $$ \beta \, $$ are real numbers. ???)

Solving Fractional Differential Equations
Fractional Differential Equations can be solved by using a finite difference scheme that discretises Grünwald-Letnikov derivativeon a finite interval $$ L < r < R \, $$. According to this derivative,

$$ \frac{\partial^{\alpha} U(r,t) }{\partial_{-} r ^{\alpha}} = \, $$ The case $$1 < \alpha < 2 \, $$ represent superdiffusion where particals diffuse faster than in the classical case, whereas $$ 0 < \alpha < 1 \, $$ represents subdiffusion where particles diffuse slower that in the classical case.