| Citation: |
Ali Khalouta. THE STUDY OF NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS VIA THE KHALOUTA-ATANGANA-BALEANU OPERATOR[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3175-3196. doi: 10.11948/20230483
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THE STUDY OF NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS VIA THE KHALOUTA-ATANGANA-BALEANU OPERATOR
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Abstract
This paper studies nonlinear fractional partial differential equations via the Khalouta-Atangana-Baleanu operator. Using Banach's fixed point theorem we obtain new results on the existence and uniqueness of solutions to the proposed problem. Furthermore, two new semi-analytical methods called Khalouta homotopy perturbation method (KHHPM) and Khalouta variational iteration method (KHVIM) are presented to find new approximate analytical solutions to our nonlinear fractional problem. The first of the two new proposed methods, KHHPM, is a hybrid method that combines homotopy perturbation method and Khalouta transform in the sense of Atangana-Baleanu-Caputo derivative. The other method, KHVIM is also a hybrid method that combines variational iteration method and Khalouta transform in the sense of Atangana-Baleanu-Caputo derivative. Convergence and absolute error analysis of KHHPM and KHVIM were also performed. A numerical example is provided to support our results. The results obtained showed that the proposed methods are very impressive, effective, reliable, and easy methods for dealing with complex problems in various fields of applied sciences and engineering.
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References
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Ali Khalouta. THE STUDY OF NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS VIA THE KHALOUTA-ATANGANA-BALEANU OPERATOR[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3175-3196. doi: 10.11948/20230483
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Figure 1.
3D plots of the approximate KHHPM-solutions for different values of
$ \alpha $ $ \alpha=0.8 $ $ \alpha=0.9 $ $ \alpha=1 $ -
Figure 2.
3D plots of the approximate KHVIM-solutions for different values of
$ \alpha $ $ \alpha=0.8 $ $ \alpha=0.9 $ $ \alpha=1 $ -
Figure 3.
2D plots of the approximate KHHPM-solutions, KHVIM-solutions and exact solution for different values of
$ \alpha $ $ \varkappa=0.5 $