Mathematical Modeling for Brain Tumors Including Fractional Operator 163

6.4.2.1

Fractional Diffusion Equations

In several branches of science and engineering, reaction-diffusion equations

are important tools. Applications of population biology are well studied; the

diffusion term takes into account migration, while the response term simu-

lates growth. The physics model serves as the source of the classical diffusion

term [23–33]. In accordance with recent studies, the typical diffusion equa-

tion is unable to accurately describe a wide range of real-world situations in

which a particle plume spreads more quickly than the classical model predicts

and may show notable asymmetry. These circumstances are known as anoma-

lous diffusion. The fractional diffusion equation, which substitutes a fractional

derivative of order 0 < α < 2 for the typical second derivative in space, is a

well-liked model for anomalous diffusion. The fractional diffusion equation’s

solutions may show asymmetries and spread more quickly than those of the

conventional diffusion equation. Still, these equations’ fundamental solutions

have useful scaling characteristics that draw applications to them. The classi-

cal diffusion equation ^∂u

∂t ^{= D ∂2u}

∂x^{2 is strongly related to statistics’ central limit}

theorem, which asserts that when the number of summands goes to infinity,

the probability distribution of a normalized sum of independently distributed

random variables converges to a normal distribution. The fractional diffusion

equation ^∂u

∂t ^{= D ∂αu}

∂x^{α relates to another central limit theorem. According to}

the usual discovering, individual random leap has a limited standard deviation

[22].

6.4.2.2

Fractional Reaction Equations

The common reaction-diffusion equation in one dimension

∂u(x, t)

∂t

= D^{∂2u(x, t)}

∂x²

+ ^˜f(u(x, t)), u(x, 0) = u0(x)

(6.1)

acts as a model for the spread of invasive species in population biology. Here

u(x, t) is the population density at location x ∈R and time t > 0 [22]. The

diffusion component, which is the first term on the right, models migration.

The reaction term, which simulates growth in population, is the second term;

a typical choice is the Kolmogorov-Fisher equation ^˜f(u(x, t)) = ru(x, t)(1 −

u(x, t)/K) where r is a species’ intrinsic growth rate and K is the carrying

capacity of the ecosystem, or the highest sustainable population density [22].

A fractional reaction-diffusion equation with more generality is:

∂u(x, t)

∂t

= D^{∂αu(x, t)}

∂x^α

+ ^˜f(u(x, t)), u(x, 0) = u0(x)

(6.2)

with 0 < α ≤2.