| Citation: |
Mo Faheem, Arshad Khan, Fathalla Ali Rihan. A WAVELET COLLOCATION METHOD FOR NEUTRAL DELAY DIFFERENTIAL EQUATIONS ON METRIC STAR GRAPH[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 2124-2151. doi: 10.11948/20240402
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A WAVELET COLLOCATION METHOD FOR NEUTRAL DELAY DIFFERENTIAL EQUATIONS ON METRIC STAR GRAPH
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Abstract
This paper proposes a Haar wavelet collocation approach to solve neutral delay differential equations on a metric star graph (NDDE-MSG) with $\kappa$ edges. The application of Haar wavelet, together with its integration on NDDE-MSG, yields a system of equations, which on solving gives unknown wavelet coefficients and subsequently the solution. The upper bound of the global error norm is established to demonstrate that the proposed method converges exponentially. We conduct some numerical experiments to test the computational convergence of our approach. In this study, the authors explore the numerical solution for NDDE on metric star graphs for the first time.
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References
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Mo Faheem, Arshad Khan, Fathalla Ali Rihan. A WAVELET COLLOCATION METHOD FOR NEUTRAL DELAY DIFFERENTIAL EQUATIONS ON METRIC STAR GRAPH[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 2124-2151. doi: 10.11948/20240402
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Figure 1.
Simplified schematic representation of a metric star graph consisting
$ \kappa $ -
Figure 2.
(2(a)) Graph of
$ P_{V_{2}^{J_{\mathit{x}}, J_{\mathit{t}}}}u_{r}(\mathit{x, t}), \;r=1, 2, 3 $ $ \mathit{t}=\mathit{t}_{N_{\mathit{t}}} $ $ J_{\mathit{x}}=J_{\mathit{t}}=5 $ $ u_{1}(\mathit{x, t}) $ $ \mathit{t}=\mathit{t}_{N_{\mathit{t}}} $ $ J_{\mathit{x}}=J_{\mathit{t}}=5 $ -
Figure 3.
Impact of resolution parameters
$ J_{\mathit{x}}, \;J_{\mathit{t}} $ $ u(\mathit{x, t}) $ $ \mathit{t}=\mathit{t}_{N_{\mathit{t}}} $ $ J_{\mathit{x}}=J_{\mathit{t}}=4 $ $ J_{\mathit{x}}=J_{\mathit{t}}=5 $ -
Figure 4.
(4(a)) Graph of
$ P_{V_{2}^{J_{\mathit{x}}, J_{\mathit{t}}}}u_{r}(\mathit{x, t}), \;r=1, 2, 3 $ $ \mathit{t}=\mathit{t}_{N_{\mathit{t}}} $ $ J_{\mathit{x}}=J_{\mathit{t}}=5 $ $ u_2(\mathit{x, t}) $ $ \mathit{t}=\mathit{t}_{N_{\mathit{t}}} $ $ J_{\mathit{x}}=J_{\mathit{t}}=5 $ -
Figure 5.
Impact of resolution parameters
$ J_{\mathit{x}}, \;J_{\mathit{t}} $ $ u(\mathit{x, t}) $ $ \mathit{t}=\mathit{t}_{N_{\mathit{t}}} $ $ J_{\mathit{x}}=J_{\mathit{t}}=3 $ $ J_{\mathit{x}}=J_{\mathit{t}}=4 $ -
Figure 6.
(6(a)) Graph of
$ P_{V_{2}^{J_{\mathit{x}}, J_{\mathit{t}}}}u_{r}(\mathit{x, t}), \;r=1, 2, 3 $ $ \mathit{t}=\mathit{t}_{N_{\mathit{t}}} $ $ J_{\mathit{x}}=J_{\mathit{t}}=5 $ $ u_2(\mathit{x, t}) $ $ \mathit{t}=\mathit{t}_{N_{\mathit{t}}} $ $ J_{\mathit{x}}=J_{\mathit{t}}=5 $ -
Figure 7.
Impact of resolution parameters
$ J_{\mathit{x}}, \;J_{\mathit{t}} $ $ u(\mathit{x, t}) $ $ \mathit{t}=\mathit{t}_{N_{\mathit{t}}} $ $ J_{\mathit{x}}=J_{\mathit{t}}=4 $ $ J_{\mathit{x}}=J_{\mathit{t}}=5 $ -
Figure 8.
(8(a)) Graph of
$ P_{V_{2}^{J_{\mathit{x}}, J_{\mathit{t}}}}u_{r}(\mathit{x, t}), \;r=1, 2, 3 $ $ \mathit{t}=\mathit{t}_{N_{\mathit{t}}} $ $ J_{\mathit{x}}=J_{\mathit{t}}=5 $ $ u_{1}(\mathit{x, t}) $ $ \mathit{t}=\mathit{t}_{N_{\mathit{t}}} $ $ J_{\mathit{x}}=J_{\mathit{t}}=5 $