Optimal Auxiliary Function Method for Analyzing Nonlinear System of Belousov–Zhabotinsky Equation with Caputo Operator
Abstract
:1. Introduction
2. Preliminaries
- 1.
- 2.
- 3.
- 4.
3. General Procedure of OAFM
- Step 1: To solve Equation (3), we will use an approximate solution that has two components, such as:
- Step 3: For the purpose of determining the first approximation based on the linear equation.
- Step 4: The nonlinear term seen in expanding form (6) is,
- Step 5: To quickly solve Equation (9) and accelerate the convergence of the first-order approximations , we propose the following alternative equation:
- Step 6: We find a first-order solutions by using the inverse operator after putting the Auxiliary function into Equation (10).
- Step 7: Several methods are used to find the numerical values of the convergence control parameters , including least squares, Galerkin’s, Ritz, and collocation. To eliminate mistakes, we employ the least squares approach.
4. Applications
4.1. Problem 1
4.2. Problem 2
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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OAFM | Exact | Abs.error | |
---|---|---|---|
0.1 | −0.4987 | −0.4987 | 5.1 |
0.2 | −0.495 | −0.4950 | 3.3 |
0.3 | −0.4889 | −0.4889 | 1.5 |
0.4 | −0.4805 | −0.4805 | 2.7 |
0.5 | −0.470 | −0.4700 | 2.0 |
0.6 | −0.4576 | −0.4575 | 3.7 |
0.7 | −0.4434 | −0.4434 | 5.2 |
0.8 | −0.4279 | −0.4278 | 6.6 |
0.9 | −0.411 | −0.411 | 7.8 |
1 | −0.3933 | −0.3932 | 8.9 |
OAFM | Exact | Abs.error | |
---|---|---|---|
0.1 | −0.4487 | −0.4487 | 4.9 |
0.2 | −0.3953 | −0.3953 | 8.2 |
0.3 | −0.34 | −0.3399 | 1.0 |
0.4 | −0.2831 | −0.283 | 1.1 |
0.5 | −0.2251 | −0.225 | 1.2 |
0.6 | −0.1663 | −0.1661 | 1.2 |
0.7 | −0.1071 | −0.1069 | 1.2 |
0.8 | −0.0479 | −0.0478 | 1.2 |
0.9 | 0.0109 | 0.01102 | 1.2 |
1 | 0.06889 | 0.06901 | 1.1 |
OAFM | Exact | Abs.error | |
---|---|---|---|
0.1 | 0.01 | 0.01 | 4.06 |
0.15 | 0.0225 | 0.0225 | 2.72 |
0.2 | 0.04001 | 0.04 | 7.12 |
0.25 | 0.06251 | 0.0625 | 1.30 |
0.3 | 0.09002 | 0.09 | 1.90 |
0.35 | 0.12252 | 0.1225 | 2.10 |
0.4 | 0.16002 | 0.16 | 1.70 |
0.45 | 0.2025 | 0.2025 | 2.74 |
0.5 | 0.24998 | 0.25 | 2.20 |
OAFM | Exact | Abs.error | |
---|---|---|---|
0.10 | 0.005 | 0.005 | 2.000 |
0.15 | 0.01125 | 0.01125 | 2.0001 |
0.20 | 0.0200 | 0.0200 | 2.0025 |
0.25 | 0.03125 | 0.03125 | 2.0242 |
0.30 | 0.0450 | 0.04500 | 2.1515 |
0.35 | 0.06125 | 0.06125 | 2.7125 |
0.40 | 0.0800 | 0.0800 | 4.7134 |
0.45 | 0.10125 | 0.10125 | 1.0770 |
0.50 | 0.1250 | 0.12500 | 2.6870 |
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Alshehry, A.S.; Yasmin, H.; Ahmad, M.W.; Khan, A.; Shah, R. Optimal Auxiliary Function Method for Analyzing Nonlinear System of Belousov–Zhabotinsky Equation with Caputo Operator. Axioms 2023, 12, 825. https://doi.org/10.3390/axioms12090825
Alshehry AS, Yasmin H, Ahmad MW, Khan A, Shah R. Optimal Auxiliary Function Method for Analyzing Nonlinear System of Belousov–Zhabotinsky Equation with Caputo Operator. Axioms. 2023; 12(9):825. https://doi.org/10.3390/axioms12090825
Chicago/Turabian StyleAlshehry, Azzh Saad, Humaira Yasmin, Muhammad Wakeel Ahmad, Asfandyar Khan, and Rasool Shah. 2023. "Optimal Auxiliary Function Method for Analyzing Nonlinear System of Belousov–Zhabotinsky Equation with Caputo Operator" Axioms 12, no. 9: 825. https://doi.org/10.3390/axioms12090825
APA StyleAlshehry, A. S., Yasmin, H., Ahmad, M. W., Khan, A., & Shah, R. (2023). Optimal Auxiliary Function Method for Analyzing Nonlinear System of Belousov–Zhabotinsky Equation with Caputo Operator. Axioms, 12(9), 825. https://doi.org/10.3390/axioms12090825