| Citation: |
Mo Faheem, Arshad Khan, Akmal Raza. AN EFFICIENT WAVELET COLLOCATION METHOD BASED ON HERMITE POLYNOMIALS FOR A CLASS OF 2D QUASI-LINEAR ELLIPTIC EQUATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1198-1221. doi: 10.11948/20220530
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AN EFFICIENT WAVELET COLLOCATION METHOD BASED ON HERMITE POLYNOMIALS FOR A CLASS OF 2D QUASI-LINEAR ELLIPTIC EQUATIONS
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Abstract
A wavelet collocation method based on Hermite polynomials is proposed for the study of a class of 2D quasi-linear elliptic partial differential equations (PDEs) that arise frequently in applied mathematics, electromagnetic theory, nonlinear optics, weather forecasting, etc. The application of Hermite wavelets and their integration to 2D quasi-linear elliptic PDEs yielded a system of equations. For the theoretical aspect, the upper bound of the error norm is established to guarantee the convergence of the method. The proposed approach has an exponential rate of convergence, so it converges very rapidly. The proposed method can be uniformly adapted to investigate the solution of 2D singularly perturbed elliptic PDEs without modifying the current scheme. Some numerical simulations have been done to validate the theoretical findings. The maximum absolute errors are calculated for different numbers of collocation grids. The comparison of numerical findings with the existing methods concludes the superiority of the proposed method.
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References
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Mo Faheem, Arshad Khan, Akmal Raza. AN EFFICIENT WAVELET COLLOCATION METHOD BASED ON HERMITE POLYNOMIALS FOR A CLASS OF 2D QUASI-LINEAR ELLIPTIC EQUATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1198-1221. doi: 10.11948/20220530
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Figure 1.
A comparison of HWM and exact solution of Example 6.1 for
$ \mathit{k}_{1}=\mathit{k}_{2}=3,\;\mathit{M}_{1}=\mathit{M}_{2}=5 $ -
Figure 2.
Behavior of AEs of HWM for Example 6.1 for different
$ \mathit{k}_{1},\;\mathit{k}_{2} $ $ \mathit{M}_{1}=\mathit{M}_{2}=5 $ -
Figure 3.
A comparison of HWM and exact solution of Example 6.2 for
$ \mathit{k}_{1}=\mathit{k}_{2}=4,\;\mathit{M}_{1}=\mathit{M}_{2}=5 $ -
Figure 4.
Behavior of AEs of HWM for Example 6.2 for different
$ \mathit{k}_{1},\;\mathit{k}_{2} $ $ \mathit{M}_{1}=\mathit{M}_{2}=5 $ -
Figure 5.
A comparison of HWM and exact solution of Example 6.3 for
$ \mathit{k}_{1}=\mathit{k}_{2}=3,\;\mathit{M}_{1}=\mathit{M}_{2}=4 $ -
Figure 6.
Behavior of AEs of HWM for Example 6.3 for different
$ \mathit{k}_{1},\;\mathit{k}_{2} $ $ \mathit{M}_{1}=\mathit{M}_{2}=4 $ -
Figure 7.
A comparison of HWM and exact solution of Example 6.4 for
$ \mathit{k}_{1}=\mathit{k}_{2}=4,\;\mathit{M}_{1}=\mathit{M}_{2}=4 $ -
Figure 8.
Behavior of AEs of HWM for Example 6.4 for different
$ \mathit{k}_{1},\;\mathit{k}_{2} $ $ \mathit{M}_{1}=\mathit{M}_{2}=4 $ -
Figure 9.
A comparison of HWM and exact solution of Example 6.5 for
$ \mathit{k}_{1}=\mathit{k}_{2}=4,\;\mathit{M}_{1}=\mathit{M}_{2}=4 $ -
Figure 10.
AEs of HWM for Example 6.5 for different
$ \mathit{k}_{1},\;\mathit{k}_{2} $ $ \mathit{M}_{1}=\mathit{M}_{2}=4 $ -
Figure 11.
HWM and exact solutions for
$ \mathit{M}_{1}=\mathit{M}_{2}=2,\; \mathit{k}_{1}=\mathit{k}_{2}=5 $ $ \epsilon $ -
Figure 12.
Effect of
$ \epsilon $ $ \mathit{k}_{1}=\mathit{k}_{2}=5,\;\mathit{M}_{1}=\mathit{M}_{2}=2 $ -
Figure 13.
Effect of
$ \epsilon $ $ \mathit{k}_{1}=\mathit{k}_{2}=6,\;\mathit{M}_{1}=\mathit{M}_{2}=2 $ -
Figure 14.
Comparison of Exact and HWM solutions of Example 6.6 at
$ \mathit{y}=0.5 $ $ \epsilon $