2022_Fractional Differential Equations Via Conformable Fractional Derivatives
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| date | 2023-01-03 |
| format | General Document |
| id | 16194 |
| institution | UniSZA |
| originalfilename | 16194_3b5612629eb3a3d.pdf |
| person | Saber Talat Radwan Al-Syouri |
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| spelling | 16194 https://intelek.unisza.edu.my/intelek/pages/view.php?ref=16194 https://intelek.unisza.edu.my/intelek/pages/search.php?search=!collection3 General Document Malaysia Library Staff (Top Management) Library Staff (Management) Library Staff (Support) Terengganu Faculty of Informatics & Computing English application/pdf 1.5 Server storage Scanned document Universiti Sultan Zainal Abidin UniSZA Private Access UNIVERSITI SULTAN ZAINAL ABIDIN SAMBox 2.4.24; modified using iTextSharp™ 5.5.10 ©2000-2016 iText Group NV (AGPL-version) Copyright©PWB2025 233 2023-01-03 16194_3b5612629eb3a3d.pdf Fractional calculus Saber Talat Radwan Al-Syouri 2022_Fractional Differential Equations Via Conformable Fractional Derivatives Fractional Calculus (FC) is one of the most prominent fields of study in applied science, thus a quantum leap is required in the field of FC to explore the studies through Conformable Fractional Derivatives (CFD). However, CFD has loopholes to verify the linearity properties, arithmetic operations, and prove basic theorems such as Chain Rule and others in FC. Thus, to bridge these gaps, it is necessary to propose a relationship between classical calculus and FC, which builds the Fractional Partial Derivative (FPD) and improves the numerical solution of Newton’s method. The study defines a relationship between classical calculus and FC by improving CFD. Moreover, CFD will be used to verify that all classical calculus properties satisfy the fractional calculus and prove the related theorems such as the Chain Rule, Rolle Theorem, Mean Value Theorem, rule of composition, and Anti-derivative Theorem. In addition, through CFD the general solution ݕఈ = ܽ(ݔ(ݕ + ܾ)ݔ (with the functional factor will be determined. Next, the study will introduce the Fractional Partial Derivatives (FPD) that are improved through CFD. After that, the improved CFD model is applied to the Fractional Newton’s Method (FNM) to generate a Modified FNM (MFNM) to reduce the dimension of fractional operator's equations. Nevertheless, MFNM was evaluated by comparing it to FNM, and the proposed modification method demonstrates good efficiency and a high convergence region. The relationship based on CFD provides logical evidence for all fundamental properties and most of FC theorems, such as the Chain Rule, Rolle Theorem, Mean Value Theorem, a product or division of functions. and combination or composition of functions. CFD also establishes a general solution of ݕ ఈ = ܽ(ݔ (ݕ + ܾ)ݔ(, when ܽ(ݔ(, ܾ)ݔ (are continuous function factors and ݕ = ݂)ݔ (is any integrable function. However, all FPD properties are satisfied by applying CFD to FPD to show the efficiency of Conformable Fractional Partial Derivative (CFPD) in comparison to FPD. Finally, the results of MFNM application indicate that the analytical solution provides an advanced field of focus for all investigated problems to offer better performance with fewer or equal number of iterations than FNM. The results obtained using CFD show that there are relationships between classical calculus and FC. Based on the result, it is found that many properties and theorems have been proven by CFD, which produces MFNM as a proposal to improve the FNM approximate solution process. In fact, CFD proves robustness and ability to solve problems in a larger region, which makes it a good alternative to solving FDEs, and a lower error rate for a more accurate solution by MFNM. Based on the results obtained, it can be concluded that the proposed MFNM is reliable and effective compared to FNM and Newton’s methods. Dissertations, Academic Fractional Differential Equations Fractional Calculus Conformable Fractional Derivatives Thesis |
| spellingShingle | 2022_Fractional Differential Equations Via Conformable Fractional Derivatives |
| state | Terengganu |
| subject | Fractional calculus Dissertations, Academic |
| summary | Fractional Calculus (FC) is one of the most prominent fields of study in applied science, thus a quantum leap is required in the field of FC to explore the studies through Conformable Fractional Derivatives (CFD). However, CFD has loopholes to verify the linearity properties, arithmetic operations, and prove basic theorems such as Chain Rule and others in FC. Thus, to bridge these gaps, it is necessary to propose a relationship between classical calculus and FC, which builds the Fractional Partial Derivative (FPD) and improves the numerical solution of Newton’s method. The study defines a relationship between classical calculus and FC by improving CFD. Moreover, CFD will be used to verify that all classical calculus properties satisfy the fractional calculus and prove the related theorems such as the Chain Rule, Rolle Theorem, Mean Value Theorem, rule of composition, and Anti-derivative Theorem. In addition, through CFD the general solution ݕఈ = ܽ(ݔ(ݕ + ܾ)ݔ (with the functional factor will be determined. Next, the study will introduce the Fractional Partial Derivatives (FPD) that are improved through CFD. After that, the improved CFD model is applied to the Fractional Newton’s Method (FNM) to generate a Modified FNM (MFNM) to reduce the dimension of fractional operator's equations. Nevertheless, MFNM was evaluated by comparing it to FNM, and the proposed modification method demonstrates good efficiency and a high convergence region. The relationship based on CFD provides logical evidence for all fundamental properties and most of FC theorems, such as the Chain Rule, Rolle Theorem, Mean Value Theorem, a product or division of functions. and combination or composition of functions. CFD also establishes a general solution of ݕ ఈ = ܽ(ݔ (ݕ + ܾ)ݔ(, when ܽ(ݔ(, ܾ)ݔ (are continuous function factors and ݕ = ݂)ݔ (is any integrable function. However, all FPD properties are satisfied by applying CFD to FPD to show the efficiency of Conformable Fractional Partial Derivative (CFPD) in comparison to FPD. Finally, the results of MFNM application indicate that the analytical solution provides an advanced field of focus for all investigated problems to offer better performance with fewer or equal number of iterations than FNM. The results obtained using CFD show that there are relationships between classical calculus and FC. Based on the result, it is found that many properties and theorems have been proven by CFD, which produces MFNM as a proposal to improve the FNM approximate solution process. In fact, CFD proves robustness and ability to solve problems in a larger region, which makes it a good alternative to solving FDEs, and a lower error rate for a more accurate solution by MFNM. Based on the results obtained, it can be concluded that the proposed MFNM is reliable and effective compared to FNM and Newton’s methods. |
| title | 2022_Fractional Differential Equations Via Conformable Fractional Derivatives |
| title_full | 2022_Fractional Differential Equations Via Conformable Fractional Derivatives |
| title_fullStr | 2022_Fractional Differential Equations Via Conformable Fractional Derivatives |
| title_full_unstemmed | 2022_Fractional Differential Equations Via Conformable Fractional Derivatives |
| title_short | 2022_Fractional Differential Equations Via Conformable Fractional Derivatives |
| title_sort | 2022_fractional differential equations via conformable fractional derivatives |