| Citation: |
Ouidad Boulakour, Ahcene Merad, Fares Bekhouche, Nourhane Attia, Ali Akgül, Evren Hincal. SOLVING NONLINEAR TIME-FRACTIONAL KAWAHARA AND MODIFIED KAWAHARA EQUATIONS USING THE LAPLACE RESIDUAL POWER SERIES METHOD[J]. Journal of Applied Analysis & Computation, 2026, 16(4): 1695-1718. doi: 10.11948/20240590
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SOLVING NONLINEAR TIME-FRACTIONAL KAWAHARA AND MODIFIED KAWAHARA EQUATIONS USING THE LAPLACE RESIDUAL POWER SERIES METHOD
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Abstract
In this study, we efficiently obtain numerical solutions for the nonlinear time-fractional Kawahara and modified Kawahara equations using the Laplace residual power series method (L-RPSM), which combines the Laplace transform with the residual power series approach. We demonstrate the procedure, validity, and applicability of the proposed method by solving both equations with arbitrary initial conditions. To validate the theoretical results, we apply three distinct test problems. The results are presented in tabular form and visually represented in both two- and three-dimensional graphs, showing that the solutions converge to the exact results. Additionally, the results highlight the impact of the time-Caputo fractional operator on the obtained solutions and demonstrate that the proposed method offers more accurate approximations compared to other methods, such as the Adomian decomposition method (ADM), the variational iteration method (VIM), the residual power series method (RPSM), and the homotopy perturbation method (HPM).
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References
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Ouidad Boulakour, Ahcene Merad, Fares Bekhouche, Nourhane Attia, Ali Akgül, Evren Hincal. SOLVING NONLINEAR TIME-FRACTIONAL KAWAHARA AND MODIFIED KAWAHARA EQUATIONS USING THE LAPLACE RESIDUAL POWER SERIES METHOD[J]. Journal of Applied Analysis & Computation, 2026, 16(4): 1695-1718. doi: 10.11948/20240590
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Figure 1.
3D plots of exact and 4th order L-RPS solutions of Example 4.1 for
$ -20 \leq \zeta \leq20 $ $ 0 \leq \varrho \leq10 $ $ \mathfrak{u}(\zeta, \varrho) $ $ \alpha = 1 $ $ \mathfrak{u}_{4}(\zeta,\varrho) $ $ (\alpha = 1) $ $ \mathfrak{u}_{4}(\zeta, \varrho) $ $ (\alpha = 0.75) $ $ \mathfrak{u}_{4}(\zeta, \varrho) $ $ (\alpha = 0.5), $ $ \mathfrak{u}_{4}(\zeta, \varrho) $ $ (\alpha = 0.25) $ -
Figure 2.
Graphs of the exact and 4th order L-RPS solutions for Example 4.1 at different values of
$ \alpha $ $ \alpha = 1 $ $ \alpha = 0.75 $ $ \alpha = 0.5, $ $ \alpha = 0.25 $ $ \zeta = 10 $ $ 0 \leq \varrho \leq1.5 $ $ \varrho =5 $ $ -10 \leq \zeta \leq10 $ -
Figure 3.
3D-plots of exact and 3rd-order L-RPS solutions for Example 4.2, with
$ -20 \leq \zeta \leq20 $ $ 0 \leq \varrho \leq2: $ $ \mathfrak{u}(\zeta, \varrho) $ $ \alpha = 1 $ $ \mathfrak{u}_{3}(\zeta,\varrho) $ $ (\alpha = 1) $ $ \mathfrak{u}_{3}(\zeta, \varrho) $ $ (\alpha = 0.9) $ $ \mathfrak{u}_{3}(\zeta, \varrho) $ $ (\alpha = 0.75) $ $ \mathfrak{u}_{3}(\zeta, \varrho) $ $ (\alpha = 0.5) $ -
Figure 4.
Graphs of the exact and 3rd-order L-RPS solutions for Example 4.2 at different values of
$ \alpha $ $ \alpha = 1 $ $ \alpha = 0.9 $ $ \alpha = 0.75, $ $ \alpha = 0.5 $ $ \zeta = 10 $ $ 0 \leq \varrho \leq2 $ $ \varrho = 2.5, $ $ -20 \leq \zeta \leq20 $ -
Figure 5.
3D plots of exact and 3rd-order L-RPS solutions for Example 4.3, with
$ -10 \leq \zeta \leq10 $ $ 0 \leq \varrho \leq1: $ $ \mathfrak{u}(\zeta, \varrho) $ $ \alpha = 1 $ $ \mathfrak{u}_{3}(\zeta,\varrho) $ $ (\alpha = 1) $ $ \mathfrak{u}_{3}(\zeta, \varrho) $ $ (\alpha = 0.75) $ $ \mathfrak{u}_{3}(\zeta, \varrho) $ $ (\alpha = 0.5) $ $ \mathfrak{u}_{3}(\zeta, \varrho) $ $ (\alpha = 0.25) $ -
Figure 6.
Graphs of the exact and 3rd-order L-RPS solutions for Example 4.3 at different values of
$ \alpha $ $ \alpha = 1 $ $ \alpha = 0.75 $ $ \alpha = 0.5, $ $ \alpha = 0.25 $ $ \zeta = 10 $ $ 0 \leq \varrho \leq1.5 $ $ \varrho =0.3 $ $ -10 \leq \zeta \leq10 $