Numerical Solutions of Variable Coefficient Higher-Order Partial Differential Equations Arising in Beam Models
Abstract
:1. Introduction
2. Motivation
3. Haar Wavelets and Their Integrals
Function Approximation
4. Description of the Method
Note
5. Stability
6. Illustrative Examples
6.1. Problem 5.1
6.2. Problem 5.2
6.3. Problem 5.3
7. Initial Disturbance and Noisy Data
8. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Methods | Points | t | |||||
---|---|---|---|---|---|---|---|
Present | 64 | 0.02 | 6.54 × 10 | 1.24 × 10 | 1.71 × 10 | 2.01 × 10 | 2.11 × 10 |
64 | 0.05 | 5.22 × 10 | 9.94 × 10 | 1.36 × 10 | 1.60 × 10 | 1.69 × 10 | |
64 | 1 | 7.14 × 10 | 1.35 × 10 | 1.87 × 10 | 2.20 × 10 | 2.31 × 10 | |
128 | 0.02 | 2.60 × 10 | 4.94 × 10 | 6.81 × 10 | 8.00 × 10 | 8.42 × 10 | |
128 | 0.05 | 2.59 × 10 | 4.94 × 10 | 6.80 × 10 | 7.99 × 10 | 8.40 × 10 | |
128 | 1 | 6.83 × 10 | 1.29 × 10 | 1.78 × 10 | 2.10 × 10 | 2.21 × 10 | |
Mittal [39] | 91 | 0.02 | 3.20 × 10 | 6.08 × 10 | 8.37 × 10 | 9.84 × 10 | 1.04 × 10 |
91 | 0.05 | 3.59 × 10 | 6.83 × 10 | 9.39 × 10 | 1.10 × 10 | 1.16 × 10 | |
91 | 1 | 6.32 × 10 | 1.20 × 10 | 1.65 × 10 | 1.94 × 10 | 2.04 × 10 | |
181 | 0.02 | 3.55 × 10 | 6.76 × 10 | 9.30 × 10 | 1.09 × 10 | 1.15 × 10 | |
181 | 0.05 | 3.99 × 10 | 7.58 × 10 | 1.04 × 10 | 1.23 × 10 | 1.29 × 10 | |
181 | 1 | 7.00 × 10 | 1.33 × 10 | 1.83 × 10 | 2.16 × 10 | 2.27 × 10 | |
Caglar [40] | 121 | 0.02 | 4.80 × 10 | 9.70 × 10 | 1.40 × 10 | 1.90 × 10 | 2.40 × 10 |
191 | 0.02 | 5.20 × 10 | 2.10 × 10 | 3.10 × 10 | 4.20 × 10 | 5.20 × 10 | |
521 | 0.02 | 4.90 × 10 | 9.90 × 10 | 1.40 × 10 | 1.90 × 10 | 2.40 × 10 | |
Aziz et al. [12] | 20 | 0.05 | 9.30 × 10 | 8.00 × 10 | 2.80 × 10 | 1.00 × 10 | 2.70 × 10 |
Rashidinia [13] | 20 | 0.05 | 2.91 × 10 | 1.73 × 10 | 1.60 × 10 | 2.23 × 10 | 2.60 × 10 |
Mohammadi [41] | 40 | 0.05 | 2.96 × 10 | 1.77 × 10 | 1.64 × 10 | 2.28 × 10 | 2.65 × 10 |
Problem 4.1 | |||
---|---|---|---|
Rate | |||
2 | 1/100 | 90,568 × 10 | |
3 | 1/200 | 2.4428 × 10 | 1.8904 |
4 | 1/400 | 6.8223 × 10 | 1.8402 |
5 | 1/800 | 2.0398 × 10 | 1.7418 |
Methods | Points | |||||||
---|---|---|---|---|---|---|---|---|
Present | 32 | 0.001 | 1.76 × 10 | 5.72 × 10 | 1.80 × 10 | 2.14 × 10 | 2.87 × 10 | 2.39 × 10 |
64 | 0.001 | 1.72 × 10 | 5.75 × 10 | 1.48 × 10 | 1.90 × 10 | 2.95 × 10 | 6.13 × 10 | |
[41] | 100 | 0.01 | 1.78 × 10 | 5.85 × 10 | 1.57 × 10 | 2.00 × 10 | 2.95 × 10 | 1.60 × 10 |
200 | 0.005 | 2.38 × 10 | 7.80 × 10 | 2.09 × 10 | 2.67 × 10 | 3.94 × 10 | 2.57 × 10 |
4 | 6.10 × 10 | 3.38 × 10 | 3.86 × 10 | 2.11 × 10 | 1.02 × 10 | 5.58 × 10 | 8.64 × 10 | 4.72 × 10 |
5 | 1.11 × 10 | 7.36 × 10 | 5.85 × 10 | 3.29 × 10 | 2.24 × 10 | 1.23 × 10 | 2.29 × 10 | 1.25 × 10 |
Rate | |||
---|---|---|---|
2 | 1/100 | 1.5254 × 10 | |
3 | 1/200 | 3.9824 × 10 | 1.9374 |
4 | 1/400 | 9.5864 × 10 | 2.0545 |
5 | 1/800 | 2.1397 × 10 | 2.1635 |
4 | 2.73 × 10 | 1.93 × 10 | 1.29 × 10 | 9.08 × 10 | 3.14 × 10 | 2.20 × 10 | 2.50 × 10 | 1.75 × 10 |
5 | 2.72 × 10 | 1.93 × 10 | 1.27 × 10 | 9.07 × 10 | 3.08 × 10 | 2.20 × 10 | 2.46 × 10 | 1.75 × 10 |
Methods | Points | |||||||
---|---|---|---|---|---|---|---|---|
Present | 32 | 0.01 | 1.40 × 10 | 1.51 × 10 | 1.46 × 10 | 6.95 × 10 | 3.62 × 10 | 2.68 × 10 |
[41] | 100 | 0.01 | 7.82 × 10 | 2.59 × 10 | 7.27 × 10 | 9.43 × 10 | 1.44 × 10 | 7.65 × 10 |
Problem 5.1 | Problem 5.2 | Problem 5.3 | |
---|---|---|---|
1 | 0.99984 | 0.99937 | 0.99937 |
2 | 0.99984 | 0.99929 | 0.99929 |
3 | 0.99984 | 0.99927 | 0.99927 |
4 | 0.99984 | 0.99927 | 0.99927 |
Problem 5.1 | Problem 5.2 | Problem 5.3 | ||||
1 | 4.53 × 10 | 2.02 × 10 | 4.82 × 10 | 3.16 × 10 | 1.30 × 10 | 2.27 × 10 |
2 | 2.10 × 10 | 9.41 × 10 | 1.67 × 10 | 8.62 × 10 | 4.90 × 10 | 3.09 × 10 |
3 | 1.46 × 10 | 6.55 × 10 | 4.14 × 10 | 2.18 × 10 | 3.40 × 10 | 2.26 × 10 |
4 | 1.30 × 10 | 5.82 × 10 | 1.34 × 10 | 5.94 × 10 | 3.15 × 10 | 2.20 × 10 |
Problem 5.1 | Problem 5.2 | Problem 5.3 | ||||
1 | 3.64 × 10 | 1.63 × 10 | 4.78 × 10 | 3.16 × 10 | 1.36 × 10 | 7.63 × 10 |
2 | 1.18 × 10 | 5.30 × 10 | 1.55 × 10 | 8.56 × 10 | 5.02 × 10 | 3.16 × 10 |
3 | 5.38 × 10 | 2.40 × 10 | 3.99 × 10 | 2.18 × 10 | 3.39 × 10 | 2.26 × 10 |
4 | 3.75 × 10 | 1.67 × 10 | 9.02 × 10 | 4.70 × 10 | 3.13 × 10 | 2.20 × 10 |
Noise = 1% | Problem 5.1 | Problem 5.2 | Problem 5.3 | |||
---|---|---|---|---|---|---|
1 | 3.55 × 10 | 1.58 × 10 | 4.77 × 10 | 3.16 × 10 | 1.37 × 10 | 7.67 × 10 |
2 | 1.08 × 10 | 4.84 × 10 | 1.54 × 10 | 8.56 × 10 | 5.03 × 10 | 3.16 × 10 |
3 | 4.35 × 10 | 1.94 × 10 | 3.98 × 10 | 2.18 × 10 | 3.39 × 10 | 2.26 × 10 |
4 | 2.72 × 10 | 1.21 × 10 | 8.47 × 10 | 4.69 × 10 | 3.13 × 10 | 2.20 × 10 |
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Ghafoor, A.; Haq, S.; Hussain, M.; Abdeljawad, T.; Alqudah, M.A. Numerical Solutions of Variable Coefficient Higher-Order Partial Differential Equations Arising in Beam Models. Entropy 2022, 24, 567. https://doi.org/10.3390/e24040567
Ghafoor A, Haq S, Hussain M, Abdeljawad T, Alqudah MA. Numerical Solutions of Variable Coefficient Higher-Order Partial Differential Equations Arising in Beam Models. Entropy. 2022; 24(4):567. https://doi.org/10.3390/e24040567
Chicago/Turabian StyleGhafoor, Abdul, Sirajul Haq, Manzoor Hussain, Thabet Abdeljawad, and Manar A. Alqudah. 2022. "Numerical Solutions of Variable Coefficient Higher-Order Partial Differential Equations Arising in Beam Models" Entropy 24, no. 4: 567. https://doi.org/10.3390/e24040567