A Novel Approach for the Approximate Solution of Wave Problems in Multi-Dimensional Orders with Computational Applications
Abstract
:1. Introduction
2. Preliminary Definitions of IT
3. Formulation of HITM
4. Convergence Analysis
- (1)
- ;
- (2)
- is forever in the neighborhood of meaning
- (3)
- .
- (1)
- Consider condition (1) by recognition of n such that , and the Banach fixed point theorem states that X has a fixed point (i.e., ). Therefore, we have
- (2)
- Our initial challenge is to demonstrate , which is attained by replacing m. Thus, for with as an initial condition. Consider that for is an induction theory. Thus, we haveNow, ∀, using (1), we obtain
- (3)
- Using condition (2) and , it follows that , and henceThus, converges.
5. Computational Applications
5.1. Example 1
5.2. Example 2
5.3. Example 3
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Approximate | Exact | Absolute Error | ||
---|---|---|---|---|
0.5 | 0.25 | 0.420735 | 0.420735 | 1 × 10−8 |
0.50 | 0.73846 | 0.73846 | 1.7 × 10−7 | |
0.75 | 0.875386 | 0.875384 | 2 × 10−6 | |
1.0 | 0.798027 | 0.797984 | 4.3 × 10−5 | |
1.0 | 0.25 | 0.259035 | 0.259035 | 1 × 10−9 |
0.5 | 0.454649 | 0.454649 | 1.5 × 10−8 | |
0.75 | 0.53895 | 0.538949 | 2.3 × 10−7 | |
1.0 | 0.491323 | 0.491295 | 2.8 × 10−6 |
Approximate | Exact | Absolute Error | ||
---|---|---|---|---|
0.5 | 0.25 | 0.365286 | 0.365286 | 1 × 10−7 |
0.50 | 1.08947 | 1.08947 | 1 × 10−7 | |
0.75 | 2.23054 | 2.23054 | 1.5 × 10−6 | |
1.0 | 3.86685 | 3.86683 | 2 × 10−5 | |
1.0 | 0.25 | 0.542593 | 0.542593 | 1 × 10−8 |
0.5 | 1.47082 | 1.47082 | 1.2 × 10−7 | |
0.75 | 2.81568 | 2.81567 | 2.3 × 10−6 | |
1.0 | 4.64388 | 4.64385 | 3 × 10−5 |
Approximate | Exact | Absolute Error | ||
---|---|---|---|---|
0.5 | 0.25 | 0.0157883 | 0.0157883 | 1 × 10−9 |
0.50 | 0.0325685 | 0.0325685 | 1.2 × 10−9 | |
0.75 | 0.0513948 | 0.0513948 | 1.4 × 10−8 | |
1.0 | 0.0734501 | 0.0734501 | 2 × 10−7 | |
1.0 | 0.25 | 0.252612 | 0.252612 | 1 × 10−9 |
0.5 | 0.521095 | 0.521095 | 1.8 × 10−8 | |
0.75 | 0.822317 | 0.822317 | 2.5 × 10−7 | |
1.0 | 1.1752 | 1.1752 | 2.9 × 10−6 |
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Nadeem, M.; Akgül, A.; Guran, L.; Bota, M.-F. A Novel Approach for the Approximate Solution of Wave Problems in Multi-Dimensional Orders with Computational Applications. Axioms 2022, 11, 665. https://doi.org/10.3390/axioms11120665
Nadeem M, Akgül A, Guran L, Bota M-F. A Novel Approach for the Approximate Solution of Wave Problems in Multi-Dimensional Orders with Computational Applications. Axioms. 2022; 11(12):665. https://doi.org/10.3390/axioms11120665
Chicago/Turabian StyleNadeem, Muhammad, Ali Akgül, Liliana Guran, and Monica-Felicia Bota. 2022. "A Novel Approach for the Approximate Solution of Wave Problems in Multi-Dimensional Orders with Computational Applications" Axioms 11, no. 12: 665. https://doi.org/10.3390/axioms11120665