Abstract
This work is an experiment to solve the fractional diffusion equation in two dimensions with a novel hybrid method. The method involves an amalgamation of the well-known differential transform method and the differential quadrature method. This work is not about the superiority of one method over the other, instead this is an idea that can be worked upon for possible greatness. Numerical examples are discussed with tables and figures.
1 Introduction
The process of diffusion happens on its own, without stirring, shaking, or wafting. The phenomenon of osmosis and respiration happening in the living entities are the most prominent examples of diffusion. Mathematically this process had been modeled as a partial differential equation, and with time, it has evolved into a two-dimensional fractional differential equation, which is given as follows:
on a finite domain
where
This article is arranged as follows: Section 2 presents the brief introduction of some prerequisites, followed by the proposed methodology and the numerical experiments. The results have been presented using tables and graphs.
2 Prerequisites
In this section, some much required topics are discussed in brief. For a fractional derivative, there exist many definitions [1,2], and the most accepted Caputo’s definition is considered in this article. The semi-analytical and numerical methods, viz. differential transform method [3] and differential quadrature method [4], respectively, have been discussed for the two-dimensional variable
2.1 Caputo fractional derivative
Let
where
2.2 Differential transform method
In this method, a polynomial series solution [3,10] is obtained by an iterative procedure. For an analytic function of two space and one time variable
The inverse differential transform of
2.3 Differential quadrature method
For any function
and the qth derivative at a point
where
3 Proposed methodology
A hybrid method has an advantage of having the properties of all the methods it is composed of. Here, we propose an amalgamation of the DTM and DQM as follows.
Applying DTM to the Eq. (1), the following expression is obtained, where the
The space differentials can be approximated by using DQM (8) and (9) and Eq. (10) can be written as follows:
Then, by using the inverse DTM, the approximate solution of the Eq. (1) can be written as follows,
Relation in
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Relation in
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4 Numerical experiment
To check the methodology, two problems solved in [16] are considered. Throughout this section, to show the precision of the proposed technique, the maximum absolute error between exact and approximate solutions is considered.
4.1 Example 1
Consider Eq. (1) on the domain
and with the initial condition
On taking the differential transform on both sides of Eq. (1),
The differential transforms of
Thus, the recurrence relation in (11) can be written as follows:
or
Thus,
It can be observed from Figures 1 and 2 that for
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Maximum absolute error |
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0.1 | 0.0100 |
0.25 | 0.0077 |
0.4 | 0.0148 |
0.5 | 0.0212 |
0.75 | 0.0100 |
1 | 0.0900 |
We also solved this problem by DTM only. Its approximate series solution is obtained as follows:
where
where
where
As we increase the number of terms of the series solution and decrease the time variable, the solutions with DTM (Table 3) as well as with D(TQ)M (Table 4) are getting better. A comparison of the exact solution and approximate solution using the hybrid method and using only DTM, at all the mesh points for a
Number of terms |
|
Maximum absolute error with DTM |
---|---|---|
3 | 1 | 1 |
0.5 | 0.25 | |
0.25 | 0.0625 | |
0.1 | 0.0100 | |
0.01 |
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4 | 1 | 1 |
0.5 | 0.25 | |
0.25 | 0.0625 | |
0.1 | 0.0100 | |
0.01 |
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5 | 1 |
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0.5 | 0.0798 | |
0.25 | 0.0463 | |
0.1 | 0.0151 | |
0.01 |
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6 | 1 |
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0.5 | 0.0798 | |
0.25 | 0.0463 | |
0.1 | 0.0151 | |
0.01 |
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Number of terms |
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Max. absolute error with hybrid method |
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3 | 1 | 1 |
0.5 | 0.25 | |
0.25 | 0.0625 | |
0.1 | 0.0100 | |
0.01 |
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4 | 1 | 1 |
0.5 | 0.25 | |
0.25 | 0.0625 | |
0.1 | 0.0100 | |
0.01 |
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5 | 1 | 0.0081 |
0.5 | 0.0798 | |
0.25 | 0.0463 | |
0.1 | 0.0151 | |
0.01 |
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6 | 1 | 0.0081 |
0.5 | 0.0798 | |
0.25 | 0.0463 | |
0.1 | 0.0151 | |
0.01 |
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4.2 Example 2
Consider Eq. (1) on the domain
and with the initial condition
The recurrence relation in (11) can be written as follows:
where
Thus,
We also solved this problem by DTM only. The series solution for some values of
When
When
When
When
The comparison of errors due to DTM and D(TQ)M can be observed in Tables 5 and 6, whereas Tables 6 and 7 compare the errors due to D(TQ)M at
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1 | 0.2500 | 0.2500 | 0.2500 |
2/3 | 0.0707 | 0.0998 | 0.0880 |
1/2 | 0.0129 | 0.1307 | 0.0543 |
2/5 | 0.0587 | 0.2015 | 0.1963 |
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1 | 0.75000 | 0.68762 | 0.67520 |
2/3 | 0.03872 | 0.35016 | 0.55380 |
1/2 | 0.07775 | 0.85338 | 1.5244 |
2/5 | 0.18509 | 1.31846 | 2.5550 |
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1 | 0.99990 | 0.99989 | 0.99989 |
2/3 | 0.93084 | 0.93093 | 0.93092 |
1/2 | 0.83427 | 0.83476 | 0.83478 |
2/5 | 0.73638 | 0.73260 | 0.73353 |
It is observed that the series obtained due to DTM is absolutely convergent for various values of
5 Results
From the numerical examples, it can be observed that DTM is an efficient method for solving two-dimensional fractional diffusion equation. The hybrid method, approximates the solution for small values of time as presented in Tables 3 and 4. In the first example, the solution obtained from the hybrid method is same as exact solution for
In second example, as the value of
Thus, this work is an experiment of combining a semi analytical method with a numerical method. The experiment can be called successful as the results are near to the exact solution for small values of time. However, the method can be explored for its efficiency for large value of time, along with its theoretical analysis.
Acknowledgments
I am grateful to my colleague, Dr. Prince Singh, for his unconditional guidance and help. I also acknowledge the constructive comments and suggestions by the reviewers.
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Funding information: The authors state no funding involved.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Conflict of interest: Authors state no conflict of interest.
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© 2022 Pratiksha Devshali and Geeta Arora, published by De Gruyter
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