TY - JOUR
T1 - Numerical Analysis of Augmented FVM for Nonlinear Time Fractional Degenerate Parabolic Equation 
AU - Ali , Muhammad Aamir
AU - Zhang , Zhiyue
AU - Xie , Jianqiang
JO - International Journal of Numerical Analysis and Modeling
VL - 3
SP - 340
EP - 360
PY - 2025
DA - 2025/03
SN - 22
DO - http://doi.org/10.4208/ijnam2025-1015
UR - https://global-sci.org/intro/article_detail/ijnam/23882.html
KW - Time fractional degenerate parabolic equations, finite volume method, Puiseux series.
AB - <p style="text-align: justify;">Utilizing the nonlinear Time Fractional Degenerate Parabolic Equation (TFDPE) in
modeling provides a comprehensive approach to studying phenomena exhibiting both fractional
order dynamics and degenerate parabolic behavior, facilitating accurate predictions and insights
across diverse scientific domains. However, the numerical solution of TFDPE is a challenging task
and traditional numerical methods cannot solve this equation because of the spatial singularity
influence. In this paper, we find the numerical solution of nonlinear TFDPE with both strongly
and weakly degenerate cases using the higher order augmented finite volume method on uniform
grids. To handle the singularity of TFDPE, we choose an intermediate point near the singular
point and split the whole domain into singular and regular subdomains. Then, we find the solution
on singular subdomain using the Puiseux series while on the regular subdomain we find the solution
by finite volume schemes. The main idea is to recover the Puiseux series on singular subdomain
using the Picard iteration methods which is also a challenging because of the time fractional
derivative in the original equation. The solution on the singular subdomain is in the form of
Puiseux series, which has multiple undetermined augmented variables and these variables play
a role in organically combining the singular and regular subdomains. To approximate the time
fractional derivative, we use the second order weighted and shifted Grünwald difference (WSGD)
scheme and give the comparison of our results with $L_1$-scheme. We use the discrete energy method
to prove that the schemes has temporal second order while spatial second and fourth-order on
the whole domain and for the augmented variables in discrete $L_2$-norm. Finally, we give some
numerical examples to confirm the accuracy and order of convergence of the proposed schemes
for the whole domain and the augmented variables. We also give an interesting example with
coefficient blow-up at the degenerate point and show the schemes are working the same as the
other cases.</p>