A Second-Order Adaptive Grid Method for a Singularly Perturbed Volterra Integrodifferential Equation
Abstract
1. Introduction
2. Preliminary Results
3. Truncation Error Analysis
4. Adaptive Grid and Convergence Analysis
5. Numerical Results and Discussion
5.1. Mesh Generation Algorithm
Algorithm 1: Adaptive grid algorithm |
Step 1. Provide an initial uniform mesh with N mesh intervals. Choose a constant that controls the algorithm terminates. Step 2. For a given grid , and the corresponding computed solution , set for each i and and for . Step 3. Define . If holds true, then go to Step 5. Otherwise go to Step 4. Step 4. For , let and be a linear interpolation function through knots . Then, generate a new mesh by for . Let and return to Step 2. Step 5. Take as the final calculation mesh and as the corresponding numerical solution. Then, stop iteration process. |
5.2. Numerical Experiments and Discussion
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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1.95 | 1.98 | 1.99 | 1.99 | 2.00 | - | ||
1 | 1 | 1 | 1 | 1 | 1 | ||
1.60 | 1.78 | 1.96 | 1.93 | 1.97 | - | ||
2 | 2 | 1 | 1 | 1 | 1 | ||
1.79 | 1.78 | 1.69 | 1.64 | 1.75 | - | ||
3 | 2 | 2 | 2 | 2 | 1 | ||
1.80 | 1.89 | 1.99 | 1.87 | 1.85 | - | ||
3 | 3 | 2 | 2 | 3 | 2 | ||
1.75 | 1.99 | 1.94 | 1.99 | 1.99 | - | ||
4 | 4 | 3 | 3 | 2 | 2 | ||
1.97 | 1.85 | 2.01 | 1.96 | 1.99 | - | ||
5 | 4 | 4 | 3 | 3 | 3 |
N | ||||||
---|---|---|---|---|---|---|
S-Mesh [1] | Method [20] | Our Method | S-Mesh [1] | Method [20] | Our Method | |
64 | ||||||
1.81 | 1.92 | 1.74 | 1.82 | 2.03 | 1.95 | |
128 | ||||||
1.84 | 1.87 | 2.01 | 1.86 | 1.97 | 1.95 | |
256 | ||||||
1.93 | 1.79 | 1.84 | 1.93 | 2.04 | 1.88 | |
512 | ||||||
1.98 | 1.65 | 1.81 | 1.99 | 1.99 | 2.07 | |
1024 |
1.51 | 1.75 | 1.93 | 1.92 | 1.96 | - | ||
2 | 2 | 2 | 1 | 1 | 1 | ||
2.01 | 2.05 | 2.00 | 1.69 | 1.07 | - | ||
3 | 3 | 3 | 2 | 2 | 3 | ||
1.74 | 2.21 | 1.99 | 2.02 | 1.99 | - | ||
5 | 8 | 4 | 3 | 3 | 3 | ||
2.01 | 2.20 | 2.01 | 1.99 | 2.02 | - | ||
7 | 9 | 5 | 5 | 4 | 3 |
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Liu, L.; Liang, Y.; Zhang, Y. A Second-Order Adaptive Grid Method for a Singularly Perturbed Volterra Integrodifferential Equation. Fractal Fract. 2022, 6, 636. https://doi.org/10.3390/fractalfract6110636
Liu L, Liang Y, Zhang Y. A Second-Order Adaptive Grid Method for a Singularly Perturbed Volterra Integrodifferential Equation. Fractal and Fractional. 2022; 6(11):636. https://doi.org/10.3390/fractalfract6110636
Chicago/Turabian StyleLiu, Libin, Ying Liang, and Yong Zhang. 2022. "A Second-Order Adaptive Grid Method for a Singularly Perturbed Volterra Integrodifferential Equation" Fractal and Fractional 6, no. 11: 636. https://doi.org/10.3390/fractalfract6110636
APA StyleLiu, L., Liang, Y., & Zhang, Y. (2022). A Second-Order Adaptive Grid Method for a Singularly Perturbed Volterra Integrodifferential Equation. Fractal and Fractional, 6(11), 636. https://doi.org/10.3390/fractalfract6110636