Haar wavelet Arctic Puffin optimization method (HWAPOM): Application to logistic models with fractal-fractional Caputo-Fabrizio operator
This study introduces a novel hybrid numerical methodology for approximating differential equations involving the fractal-fractional Caputo-Fabrizio (FFCF) operator, which is an essential tool for modelling complex dynamical systems involving memory effects. The proposed method integrates the Haar w...
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| Format: | Article |
| Language: | English |
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Elsevier
2025-03-01
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| Series: | Partial Differential Equations in Applied Mathematics |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2666818125000427 |
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| author | Najeeb Alam Khan Sahar Altaf Nadeem Alam Khan Muhammad Ayaz |
| author_facet | Najeeb Alam Khan Sahar Altaf Nadeem Alam Khan Muhammad Ayaz |
| author_sort | Najeeb Alam Khan |
| collection | DOAJ |
| description | This study introduces a novel hybrid numerical methodology for approximating differential equations involving the fractal-fractional Caputo-Fabrizio (FFCF) operator, which is an essential tool for modelling complex dynamical systems involving memory effects. The proposed method integrates the Haar wavelet with the Arctic Puffin optimization (APO) algorithm, a meta-heuristic optimization inspired by the foraging behavior of Arctic Puffins. The Haar wavelet, well-known for its compact support and piecewise constant characteristics, is based on the Haar basis functions used to construct an operational matrix for the FFCF operator. These matrices transform the differential equations into a system of algebraic equations involving unknown coefficients, and then optimize them using the APO algorithm, ensuring efficient and accurate solutions. Two nonlinear quadratic and cubic logistic models were examined to demonstrate the effectiveness of this method. The accuracy of the designed method was validated by comparing its results with those obtained using the modified Homotopy Perturbation method (MHPM). Error metrics, such as mean absolute error, maximum absolute error, and the experimental convergence rate, are calculated at various collocation points and presented in a tabular format. The findings revealed the method's high accuracy, rapid convergence, and computational efficiency. Overall, the proposed method offers a powerful tool for solving complex differential equations, as evidenced by its strong agreement with MHPM results. The study results were further reinforced through statistical performance metrics and their visual representations, confirming the reliability of the method, low computational cost, and its potential for broad application in numerical computations. |
| format | Article |
| id | doaj-art-9290bfd2ee1c4e25bed49a55a706b0a4 |
| institution | DOAJ |
| issn | 2666-8181 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Partial Differential Equations in Applied Mathematics |
| spelling | doaj-art-9290bfd2ee1c4e25bed49a55a706b0a42025-08-20T02:43:43ZengElsevierPartial Differential Equations in Applied Mathematics2666-81812025-03-011310111410.1016/j.padiff.2025.101114Haar wavelet Arctic Puffin optimization method (HWAPOM): Application to logistic models with fractal-fractional Caputo-Fabrizio operatorNajeeb Alam Khan0Sahar Altaf1Nadeem Alam Khan2Muhammad Ayaz3Department of Mathematics, University of Karachi, Karachi 75270, PakistanDepartment of Mathematics, University of Karachi, Karachi 75270, Pakistan; Department of Mathematics, Karachi Institute of Economics and Technology, Karachi 75160, PakistanDepartment of Computer Science, Iqra University, Karachi, PakistanDepartment of Mathematics, University of Karachi, Karachi 75270, PakistanThis study introduces a novel hybrid numerical methodology for approximating differential equations involving the fractal-fractional Caputo-Fabrizio (FFCF) operator, which is an essential tool for modelling complex dynamical systems involving memory effects. The proposed method integrates the Haar wavelet with the Arctic Puffin optimization (APO) algorithm, a meta-heuristic optimization inspired by the foraging behavior of Arctic Puffins. The Haar wavelet, well-known for its compact support and piecewise constant characteristics, is based on the Haar basis functions used to construct an operational matrix for the FFCF operator. These matrices transform the differential equations into a system of algebraic equations involving unknown coefficients, and then optimize them using the APO algorithm, ensuring efficient and accurate solutions. Two nonlinear quadratic and cubic logistic models were examined to demonstrate the effectiveness of this method. The accuracy of the designed method was validated by comparing its results with those obtained using the modified Homotopy Perturbation method (MHPM). Error metrics, such as mean absolute error, maximum absolute error, and the experimental convergence rate, are calculated at various collocation points and presented in a tabular format. The findings revealed the method's high accuracy, rapid convergence, and computational efficiency. Overall, the proposed method offers a powerful tool for solving complex differential equations, as evidenced by its strong agreement with MHPM results. The study results were further reinforced through statistical performance metrics and their visual representations, confirming the reliability of the method, low computational cost, and its potential for broad application in numerical computations.http://www.sciencedirect.com/science/article/pii/S2666818125000427Fractal-fractionalCaputo-Fabrizio derivativeHaar waveletArctic Puffin optimizationModified homotopy perturbation method (MHPM) |
| spellingShingle | Najeeb Alam Khan Sahar Altaf Nadeem Alam Khan Muhammad Ayaz Haar wavelet Arctic Puffin optimization method (HWAPOM): Application to logistic models with fractal-fractional Caputo-Fabrizio operator Partial Differential Equations in Applied Mathematics Fractal-fractional Caputo-Fabrizio derivative Haar wavelet Arctic Puffin optimization Modified homotopy perturbation method (MHPM) |
| title | Haar wavelet Arctic Puffin optimization method (HWAPOM): Application to logistic models with fractal-fractional Caputo-Fabrizio operator |
| title_full | Haar wavelet Arctic Puffin optimization method (HWAPOM): Application to logistic models with fractal-fractional Caputo-Fabrizio operator |
| title_fullStr | Haar wavelet Arctic Puffin optimization method (HWAPOM): Application to logistic models with fractal-fractional Caputo-Fabrizio operator |
| title_full_unstemmed | Haar wavelet Arctic Puffin optimization method (HWAPOM): Application to logistic models with fractal-fractional Caputo-Fabrizio operator |
| title_short | Haar wavelet Arctic Puffin optimization method (HWAPOM): Application to logistic models with fractal-fractional Caputo-Fabrizio operator |
| title_sort | haar wavelet arctic puffin optimization method hwapom application to logistic models with fractal fractional caputo fabrizio operator |
| topic | Fractal-fractional Caputo-Fabrizio derivative Haar wavelet Arctic Puffin optimization Modified homotopy perturbation method (MHPM) |
| url | http://www.sciencedirect.com/science/article/pii/S2666818125000427 |
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