| Citation: |
Azam Ghasemi, Abbas Saadatmandi. A BERNOULLI-REPRODUCING KERNEL METHOD FOR A CLASS OF NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3260-3281. doi: 10.11948/20230508
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A BERNOULLI-REPRODUCING KERNEL METHOD FOR A CLASS OF NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS
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Abstract
We have developed a highly accurate numerical method called Bernoulli-RKM to solve nonlinear singular boundary value problems (SBVPs). This approach uses Bernoulli polynomials and the traditional reproducing kernel method (RKM) and applies the quasi-linearization method to linearize the SBVPs. We have discussed the error and convergence of Bernoulli-RKM and provided numerical examples to demonstrate its potential in solving nonlinear SBVPs. Additionally, we have compared our results with those in existing literature.
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References
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Azam Ghasemi, Abbas Saadatmandi. A BERNOULLI-REPRODUCING KERNEL METHOD FOR A CLASS OF NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3260-3281. doi: 10.11948/20230508
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Figure 1.
Graph of
$ ER_{n, 1}(x) $ $ n=14 $ $ n=18 $ -
Figure 2.
Graph of
$ ER_{n, 3}(x) $ $ {\bf Case}\ 1 $ $ {\bf Case}\ 3 $ -
Figure 3.
Graph of
$ \log_{10}(MER_{n, 3}) $ $ n=4, 8, 12, 16, 20, $ -
Figure 4.
Graph of absolute error function
$ |\nu(x)-\nu_{n, 4}(x)| $ $ n=25 $ -
Figure 5.
Graph of
$ ER_{15, 1}(x) $ $ \varphi=2, $ $ m=0.5 $ -
Figure 6.
Graph of
$ \log_{10}(MER_{n, 1}) $ $ n=7, 11, 15, 19, 23, $ -
Figure 7.
Graph of absolute error function
$ |\nu(x)-\nu_{n, 4}(x)| $ $ n=12 $ -
Figure 8.
Graph of
$ \log_{10}(MAE_{n, 5}) $ $ n=4, 8, 12, 16, 20, $ -
Figure 9.
Graph of
$ \log_{10}(MAE_{n}) $ $ n $