Application of new homotopy analysis method and optimal homotopy asymptotic method for solving fuzzy fractional ordinary differential equations
Physical phenomena that are complex and have hereditary features as well as uncertainty are recognized to be well-described using fuzzy fractional ordinary differential equations (FFODEs). The analytical approach for solving FFODEs aims to give closed-form solutions that are considered exact solutio...
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my-uum-etd.101992023-01-09T01:28:44Z Application of new homotopy analysis method and optimal homotopy asymptotic method for solving fuzzy fractional ordinary differential equations 2022 Dulfikar jawad, Hashim Ali, Fareed Jameel Teh, Yuan Ying Awang Had Salleh Graduate School of Arts & Sciences Awang Had Salleh Graduate School of Arts & Sciences QA299.6-433 Analysis QA76.76 Fuzzy System. Physical phenomena that are complex and have hereditary features as well as uncertainty are recognized to be well-described using fuzzy fractional ordinary differential equations (FFODEs). The analytical approach for solving FFODEs aims to give closed-form solutions that are considered exact solutions. However, for most FFODEs, the analytical solutions are not easily derived. Moreover, most complex physical phenomena tend to lack analytical solutions. The approximation approach can handle this drawback by providing open-form solutions where several FFODEs are solvable using the approximate-numerical class of methods. However, those methods are mostly employed for linear or linearized problems, and they cannot directly solve FFODES of high order. Meanwhile, the approximate-analytic class of methods under the approximation approach are not only applicable to nonlinear FFODEs without the need for linearization or discretization, but also can determine solution accuracy without requiring the exact solution for comparison. However, existing approximateanalytical methods cannot ensure convergence of the solution. Nevertheless, to solve non-fuzzy fractional ordinary differential equations, there exist perturbation-based methods: the fractional homotopy analysis method (F-HAM) and the optimal homotopy asymptotic method (F-OHAM), that possess convergence-control ability. Therefore, this research aims to develop new convergence-controlled approximateanalytical methods, fuzzy F-HAM (FF-HAM) and fuzzy F-OHAM (FF-OHAM), for solving first-order and second-order fuzzy fractional ordinary initial value problems and fuzzy fractional ordinary boundary value problems. In the theoretical development, the establishment of the convergence of the solutions is done based on the convergence-control parameters. In the experimental work, the convergence of solutions is determined using properties of fuzzy numbers. FF-HAM and FF-OHAM are not only able to solve difficult nonlinear problems but are also able to solve highorder problems directly without reducing them into first-order systems. The developed methods demonstrate the excellent performance of the developed methods in comparison to other methods, where FF-HAM and FF-OHAM are individually superior in terms of accuracy. 2022 Thesis https://etd.uum.edu.my/10199/ https://etd.uum.edu.my/10199/1/s903922_01.pdf text eng 2025-03-24 staffonly https://etd.uum.edu.my/10199/2/s903922_02.pdf text eng public other doctoral Universiti Utara Malaysia |
| institution |
Universiti Utara Malaysia |
| collection |
UUM ETD |
| language |
eng eng |
| advisor |
Ali, Fareed Jameel Teh, Yuan Ying |
| topic |
QA299.6-433 Analysis QA76.76 Fuzzy System. |
| spellingShingle |
QA299.6-433 Analysis QA76.76 Fuzzy System. Dulfikar jawad, Hashim Application of new homotopy analysis method and optimal homotopy asymptotic method for solving fuzzy fractional ordinary differential equations |
| description |
Physical phenomena that are complex and have hereditary features as well as uncertainty are recognized to be well-described using fuzzy fractional ordinary differential equations (FFODEs). The analytical approach for solving FFODEs aims to give closed-form solutions that are considered exact solutions. However, for most FFODEs, the analytical solutions are not easily derived. Moreover, most complex physical phenomena tend to lack analytical solutions. The approximation approach can handle this drawback by providing open-form solutions where several FFODEs are solvable using the approximate-numerical class of methods. However, those methods are mostly employed for linear or linearized problems, and they cannot directly solve FFODES of high order. Meanwhile, the approximate-analytic class of methods under the approximation approach are not only applicable to nonlinear FFODEs without the need for linearization or discretization, but also can determine solution accuracy without requiring the exact solution for comparison. However, existing approximateanalytical methods cannot ensure convergence of the solution. Nevertheless, to solve non-fuzzy fractional ordinary differential equations, there exist perturbation-based methods: the fractional homotopy analysis method (F-HAM) and the optimal homotopy asymptotic method (F-OHAM), that possess convergence-control ability. Therefore, this research aims to develop new convergence-controlled approximateanalytical methods, fuzzy F-HAM (FF-HAM) and fuzzy F-OHAM (FF-OHAM), for solving first-order and second-order fuzzy fractional ordinary initial value problems and fuzzy fractional ordinary boundary value problems. In the theoretical development, the establishment of the convergence of the solutions is done based on the convergence-control parameters. In the experimental work, the convergence of solutions is determined using properties of fuzzy numbers. FF-HAM and FF-OHAM are not only able to solve difficult nonlinear problems but are also able to solve highorder problems directly without reducing them into first-order systems. The developed methods demonstrate the excellent performance of the developed methods in comparison to other methods, where FF-HAM and FF-OHAM are individually superior in terms of accuracy. |
| format |
Thesis |
| qualification_name |
other |
| qualification_level |
Doctorate |
| author |
Dulfikar jawad, Hashim |
| author_facet |
Dulfikar jawad, Hashim |
| author_sort |
Dulfikar jawad, Hashim |
| title |
Application of new homotopy analysis method and optimal homotopy asymptotic method for solving fuzzy fractional ordinary differential equations |
| title_short |
Application of new homotopy analysis method and optimal homotopy asymptotic method for solving fuzzy fractional ordinary differential equations |
| title_full |
Application of new homotopy analysis method and optimal homotopy asymptotic method for solving fuzzy fractional ordinary differential equations |
| title_fullStr |
Application of new homotopy analysis method and optimal homotopy asymptotic method for solving fuzzy fractional ordinary differential equations |
| title_full_unstemmed |
Application of new homotopy analysis method and optimal homotopy asymptotic method for solving fuzzy fractional ordinary differential equations |
| title_sort |
application of new homotopy analysis method and optimal homotopy asymptotic method for solving fuzzy fractional ordinary differential equations |
| granting_institution |
Universiti Utara Malaysia |
| granting_department |
Awang Had Salleh Graduate School of Arts & Sciences |
| publishDate |
2022 |
| url |
https://etd.uum.edu.my/10199/1/s903922_01.pdf https://etd.uum.edu.my/10199/2/s903922_02.pdf |
| _version_ |
1776103764279689216 |