1. Introduction
The study of mathematical models and physical systems has
captured the attention of numerous researchers and mathematicians.
They strive to develop theoretical frameworks, numerical methods,
and analytical tools, particularly in the solvability of partial
differential equations (PDEs). These endeavors are still ongoing as
researchers are motivated to uncover the underlying mechanisms,
especially in the field of fractional partial differential equations
[
1,
2,
3,
4,
5,
6,
7].
Fractional operators and corresponding differential and integral
equations are often chosen because of the memory effects provided by
their analytical properties [
8,
9,
10,
11,
12,
13,
14,
15]. However, when it comes to a
mathematical problem, the solution is derived from a real model, the
theoretical results of which are often not directly applicable to
the given problem. So, problems like the integer case are solved
numerically [
16,
17,
18,
19].
The scientific community has shown great interest in studying
dynamic states, such as the explosion of solutions. Currently, the
biggest challenge lies in identifying or estimating the occurrence
of finite-time blow-up phenomena [
20,
21,
22,
23,
24,
25]. This area of study is
considered highly important, with roots tracing back to the 19th
century, when Henri Poincaré and Paul Painlevé highlighted
the possibility of finite-time singularities in ordinary
differential equations (ODEs). Subsequently, after 50 years, Alan
Turing, Grigoriy A. Kolmogorov, Andrei N. Tikhonov, and Vladimir M.
Volpert made significant contributions to the understanding of
finite-time blow-up in nonlinear PDEs.
All of these factors have motivated our own research on
finite-time blow-up phenomena in nonlinear problems. Specifically,
we examine the following equation:
Here, represents a fractional exponent ranging from 0 to
1, and our goal is to explore the combined effects of
time-fractional diffusion, represented by the Caputo fractional
derivative , and a nonlinear reaction term . The fractional derivative accounts for the memory
effect and the non-local nature of time evolution in the equation,
which is crucial for studying finite-time blow-up. To support our
research, we present a numerical study utilizing the finite
difference scheme and compare it with the classical case to validate
our expectations regarding the involvement of the exponent
.
2. Formulation of the Problem
This article concerns finite-time blow-up solutions of the
fractional nonlinear diffusion equation, which are given as follows:
with the initial condition
and the mixed boundary conditions
where
is a given function and
, and
a is positive constant
called the thermal diffusivity.
The topic of existence and uniqueness is not new for scholars.
Many of them have treated it differently. The most well known
approaches are the Faedo–Galarkin method, the energy
inequality method, the linearization method, and other techniques [
26].
For analyzing the existence of the main problem, we can simply
use the same idea as in the article by Oussaeif and Bouziani [
27], where the
source term is generally given by
A positive term f represents a heat source due
to an exothermic reaction. Otherwise, the reaction is endothermic.
If f depends on the gradient, then convection
effects are taken into consideration.
3. Finite-Time Blow-Up Solution
This section is devoted to studying the finite time of explosion
solution for the following fractional problem:
3.1. Analytical Estimation
This method was devised by Kaplan in the case of a bounded
domain
. Suppose that
and that
is convex. The first eigenfunction
of the Laplacian and the corresponding
eigenvalue
are the solution of the following:
the solution of the Sturm–Liouville problem is:
Multiplying
by
, integrating over
, we arrive at the relation
By using the Jensen’s inequality, we obtain
The differential inequality leads immediately to the following
It becomes evident that the estimate of the finite-time blowup
depends on the solution of the precedent inequality, but the
studies do not have a large field, despite considerable recent and
ongoing progress, and are still not sufficient to solve the
Bernoulli equation. Therefore, we concentrate on the general idea
in the article to obtain an estimation for a finite-time blow-up [
28].
So, for solving the equation, we must apply the following
variable change
Therefore, it follows from
and
that
As may be apparent to the reader, we can solve it easily, and
we obtain the following solution:
Next, for
we give the main result:
The solution u must blow up. Notice that
, and the initial data are sufficiently large.
3.2. Numerical Simulation of Explosion Phenomena
Accurately detecting blow-up solutions is a difficult task.
Assuming that we can follow the true behavior of the solution with
reasonable accuracy, how do we find the blow-up time
? For that reason, the research area is aimed at
developing numerical methods that preserve specific properties of
fundamental differential equations. Much progress has been made in
the field over the past decade.
To establish the numerical approximation scheme, let
and , where is the integration time . As usual, , where is the grid size in the spatial direction.
3.2.1. The Explicit Scheme
So, we obtain the following scheme:
3.2.2. The Linear Implicit Scheme
Then, by putting
we get
By using the limit condition
We write the system in the matrix form:
with
3.3. Numerical Experiment
This is a brief expository presentation of the various
approaches to the computational treatment of cases of blow-up
problems using two methods: the explicit method and the implicit
method. To validate our results, we based them on the following
problem:
3.3.1. Example 1
By inputting the previous problem
we obtain the following
Figure 1:
Then, if we input the problem
we obtain the following
Figure 2:
Furthermore, we take
in the problem (P
2), and we obtain
the following
Figure 3:
3.3.2. Example 2
By taking
in the problem (P
2), we obtain the
following
Figure 4:
Furthermore, if we take
in the problem (P
2), we obtain the
following
Figure 5:
Then, by putting
in the problem (P
2), we obtain the
following
Figure 6:
3.3.3. Example 3
By putting
in the problem (P
2), we obtain the
following
Figure 7:
Then, by taking
in the problem (P
2), we obtain the
following
Figure 8:
Furthermore, if we take
in the problem (P
2), we obtain the
following
Figure 9:
4. Comparison Results between Integer and Fractional Problems
This section deals with the comparison of the numerical
simulation of the finite-time blow-up solution between the
fractional problem (P) where
and the following classical problem:
By using the same eigenvalue problem in the second section and
applying the Kaplan method, we obtain that the finite-time blow-up
of the classical problem (
Pc) is given by
We know that has an impact on the finite-time blow-up; this
prompted us to investigate a combination between as a fraction between 0 and 1 . For this, we plan to add the numerical results to
the research as well as the real difference between the fractional
and integer time derivatives where .
We chose
for the same last example (
P2) where we show the comparison of the finite-time
blow-up solution between the fractional problem (
P2) with
and the following classical problem:
For the following different examples:
4.1. Comparison Example 1
By putting
in the problem (
P2)
and
in the problem (
Pce), we obtain the following
Figure 10:
4.2. Comparison Example 2
For
,
and
, we obtain the following
Figure 11:
4.3. Comparison Example 3
For
,
and
, we obtain the following
Figure 12:
5. Conclusions
In this work, we developed some prior findings on the blow-up
phenomena in fractional reaction–diffusion problems that
incorporate a positive power of the unknown variable in the
equation. The finite-time blow-up solution is shown numerically, and
the behavior solution of the fractional problem is also shown. This
article underscores the pivotal role of the exponent in defining the finite-time blow-up phenomenon in
conjunction with the power p and the initial
condition. The interdependence of these factors necessitates careful
consideration and analysis when studying nonlinear problems.
Finally, to confirm and validate the efficiency of the obtained
results, we present some numerical examples and compare them to the
classical problem with integer order. Future research should
continue to explore and refine our understanding of these
relationships, leading to further advancements in the field of
fractional partial differential equations and their applications.
This would certainly help to implement the finite-time blow-up
phenomenon in the financial and economic fields.
Author Contributions
Methodology, Z.C., T.-E.O., A.O. and A.A.; Software, Z.C.,
I.A.F., Y.A.A.-K., A.A.-H., T.-E.O., A.O. and A.A.; Validation,
T.H., Z.C., I.A.F., Y.A.A.-K., A.A.-H., T.-E.O., A.O. and A.A.;
Formal analysis, T.H., Z.C., I.A.F., Y.A.A.-K., A.A.-H., T.-E.O.,
A.O. and A.A.; Investigation, T.H., I.A.F., Y.A.A.-K., A.A.-H.,
T.-E.O., A.O. and A.A.; Writing—original draft, Z.C.,
T.-E.O., A.O. and A.A.; Writing—review & editing, Z.C.,
I.A.F., Y.A.A.-K., A.A.-H., T.-E.O., A.O. and A.A.; Visualization,
Z.C., T.-E.O., A.O. and A.A. All authors have read and agreed to the
published version of the manuscript.
Funding
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
All data that support the findings of this study are included
within the article.
Conflicts of Interest
The authors declare no conflict of interest.
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