Mathematical Modeling for Brain Tumors Including Fractional Operator 161

6.3.2

Finite Difference Method

The finite difference approach is one of the “grid-point” approaches. The grid-

point techniques use a space-time grid to cover a computational domain, with

each function’s values represented at grid points. Although the grid points’

space-time distribution is essentially random, it has a big impact on how accu-

rate the approximation is. Generally, the values located within the grid points

are not assumed. The function values at a predetermined set of grid points are

used in the so-called finite-difference formula, which approximates a derivative

of a function. Even if the finite-difference approach is not used to solve the

differential equation, it is still useful to understand its fundamentals. This is

due to the fact that finite-difference formula and other numerical approaches

often estimate the temporal dependence of the functions [9]. The following

actions play a role in applying the approach to a specific differential problem:

a.

Building the problem’s discrete finite-difference model:

•Coverage of the computational domain by a space-time grid,

•Estimates for functions, derivatives, and/or beginning and/or

boundary conditions

•Creation of a finite-difference system, all at the grid points

(i.e., algebraic) equations

b.

Analyzing the model with finite differences:

•The order and consistency of the approximation

•Stability

•Convergence

c.

Numerical computations

6.3.3

Finite Volume Method

Similar to FEM, the finite volume technique (FVM) also relies on an unstruc-

tured mesh, such as a triangle. It is hence appropriate for complicated and

irregular geometry. For issues involving fluid mechanics, FVM is superior to

FEM in a different manner. The numerical techniques we have so far shown

are based on PDEs. As opposed to this, FVM is predicated on the conserva-

tion laws’ integral form as compared with their differential form. Generally,

FVM is more applicable to fluid mechanic problems when it is compared with

other methods [10].