Analytical Approximations of Well Function by Solving the Governing Differential Equation Representing Unsteady Groundwater Flow in a Confined Aquifer
Abstract
:1. Introduction
2. Brief Overview of Homotopy-Based Methods
2.1. Homotopy Analysis Method
2.2. Homotopy Perturbation Method
2.3. Optimal Homotopy Asymptotic Method
3. Governing Equation and Solution Methodologies
3.1. Theis Solution
3.2. Homotopy-Based Solutions
3.2.1. HAM-Based Solution
3.2.2. HPM-Based Solution
3.2.3. OHAM-Based Solution
4. Results and Discussion
4.1. Validation of the Well Function’s Approximations
4.2. Numerical Convergence and Validation of the HAM-Based Solution
4.3. Validation of HPM-Based Solution
4.4. Validation of OHAM-Based Solution
4.5. Comparison between Different Approximations
5. Concluding Remarks
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Convergence Theorems
Appendix A.1. Convergence Theorem of HAM-Based Solution
Appendix A.2. Convergence Theorem of OHAM-Based Solution
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Computational Time (s) | ||
---|---|---|
2 | 7.15 10−4 | 0.212 |
4 | 5.23 10−5 | 1.135 |
6 | 1.98 10−5 | 2.481 |
8 | 1.01 10−5 | 5.074 |
10 | 8.86 10−6 | 6.843 |
12 | 6.23 10−6 | 10.149 |
Numerical Solution | HAM-Based Approximation | |||
---|---|---|---|---|
4th Order | 7th Order | 10th Order | ||
0.1 | 2.523 10−1 | 3.009 10−1 | 2.821 10−1 | 2.726 10−1 |
1 | 3.036 10−2 | 3.995 10−2 | 3.548 10−2 | 3.305 10−2 |
2 | 6.768 10−3 | 9.302 10−3 | 8.011 10−3 | 7.228 10−3 |
3 | 1.806 10−3 | 2.560 10−3 | 2.146 10−3 | 1.886 10−3 |
4 | 5.230 10−4 | 7.603 10−4 | 6.213 10−4 | 5.377 10−4 |
5 | 1.589 10−4 | 2.359 10−4 | 1.881 10−4 | 1.627 10−4 |
6 | 4.983 10−5 | 7.530 10−5 | 5.870 10−5 | 5.150 10−5 |
7 | 1.598 10−5 | 2.453 10−5 | 1.871 10−5 | 1.689 10−5 |
8 | 5.213 10−6 | 8.111 10−6 | 6.066 10−6 | 5.688 10−6 |
9 | 1.723 10−6 | 2.713 10−6 | 1.993 10−6 | 1.955 10−6 |
10 | 5.753 10−7 | 9.160 10−7 | 6.617 10−7 | 6.814 10−7 |
Numerical Solution | Four Terms of the HPM-Based Approximation | |
---|---|---|
0.1 | 2.523 10−1 | 3.135 10−1 |
0.3 | 1.253 10−1 | 1.533 10−1 |
0.5 | 7.747 10−2 | 9.362 10−2 |
0.7 | 5.173 10−2 | 6.246 10−2 |
0.9 | 3.601 10−2 | 4.346 10−2 |
1.1 | 2.574 10−2 | 3.097 10−2 |
1.3 | 1.875 10−2 | 2.237 10−2 |
1.5 | 1.384 10−2 | 1.563 10−2 |
1.7 | 1.033 10−2 | 8.139 10−3 |
2.0 | 6.768 10−3 | 1.298 10−2 |
Numerical Solution | Three Terms of OHAM-Based Approximation | |
---|---|---|
0.1 | 2.523 10−1 | 2.466 10−1 |
1 | 3.036 10−2 | 3.143 10−2 |
2 | 6.768 10−3 | 6.539 10−3 |
3 | 1.806 10−3 | 2.112 10−3 |
4 | 5.230 10−4 | 8.502 10−4 |
5 | 1.589 10−4 | 3.066 10−4 |
6 | 4.983 10−5 | 1.000 10−4 |
7 | 1.598 10−5 | 3.013 10−5 |
8 | 5.213 10−6 | 8.461 10−6 |
9 | 1.723 10−6 | 2.283 10−6 |
10 | 5.753 10−7 | 5.978 10−7 |
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Kumbhakar, M.; Singh, V.P. Analytical Approximations of Well Function by Solving the Governing Differential Equation Representing Unsteady Groundwater Flow in a Confined Aquifer. Mathematics 2023, 11, 1652. https://doi.org/10.3390/math11071652
Kumbhakar M, Singh VP. Analytical Approximations of Well Function by Solving the Governing Differential Equation Representing Unsteady Groundwater Flow in a Confined Aquifer. Mathematics. 2023; 11(7):1652. https://doi.org/10.3390/math11071652
Chicago/Turabian StyleKumbhakar, Manotosh, and Vijay P. Singh. 2023. "Analytical Approximations of Well Function by Solving the Governing Differential Equation Representing Unsteady Groundwater Flow in a Confined Aquifer" Mathematics 11, no. 7: 1652. https://doi.org/10.3390/math11071652