Numerical Solutions of Fractional Differential Equations by Using Fractional Explicit Adams Method
Differential equations of fractional order are believed to be more challenging to compute compared to the integer-order differential equations due to its arbitrary properties. This study proposes a multistep method to solve fractional differential equations. The method is derived based on the concep...
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MDPI AG
2020-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/10/1675 |
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| author | Nur Amirah Zabidi Zanariah Abdul Majid Adem Kilicman Faranak Rabiei |
| author_facet | Nur Amirah Zabidi Zanariah Abdul Majid Adem Kilicman Faranak Rabiei |
| author_sort | Nur Amirah Zabidi |
| collection | DOAJ |
| description | Differential equations of fractional order are believed to be more challenging to compute compared to the integer-order differential equations due to its arbitrary properties. This study proposes a multistep method to solve fractional differential equations. The method is derived based on the concept of a third-order Adam–Bashforth numerical scheme by implementing Lagrange interpolation for fractional case, where the fractional derivatives are defined in the Caputo sense. Furthermore, the study includes a discussion on stability and convergence analysis of the method. Several numerical examples are also provided in order to validate the reliability and efficiency of the proposed method. The examples in this study cover solving linear and nonlinear fractional differential equations for the case of both single order as <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> and higher order, <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mfenced separators="" open="[" close=")"><mn>1</mn><mo>,</mo><mn>2</mn></mfenced></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> denotes the order of fractional derivatives of <inline-formula><math display="inline"><semantics><mrow><msup><mi>D</mi><mi>α</mi></msup><mi>y</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The comparison in terms of accuracy between the proposed method and other existing methods demonstrate that the proposed method gives competitive performance as the existing methods. |
| first_indexed | 2024-03-10T15:55:07Z |
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| id | doaj.art-905d616c9e9c43daaf008875bfec4cee |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T15:55:07Z |
| publishDate | 2020-10-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-905d616c9e9c43daaf008875bfec4cee2023-11-20T15:45:19ZengMDPI AGMathematics2227-73902020-10-01810167510.3390/math8101675Numerical Solutions of Fractional Differential Equations by Using Fractional Explicit Adams MethodNur Amirah Zabidi0Zanariah Abdul Majid1Adem Kilicman2Faranak Rabiei3Institute for Mathematical Research, Universiti Putra Malaysia, UPM Serdang 43400, MalaysiaInstitute for Mathematical Research, Universiti Putra Malaysia, UPM Serdang 43400, MalaysiaInstitute for Mathematical Research, Universiti Putra Malaysia, UPM Serdang 43400, MalaysiaSchool of Engineering, Monash University Malaysia, Jalan Lagoon Selatan, Bandar Sunway, Selangor 47500, MalaysiaDifferential equations of fractional order are believed to be more challenging to compute compared to the integer-order differential equations due to its arbitrary properties. This study proposes a multistep method to solve fractional differential equations. The method is derived based on the concept of a third-order Adam–Bashforth numerical scheme by implementing Lagrange interpolation for fractional case, where the fractional derivatives are defined in the Caputo sense. Furthermore, the study includes a discussion on stability and convergence analysis of the method. Several numerical examples are also provided in order to validate the reliability and efficiency of the proposed method. The examples in this study cover solving linear and nonlinear fractional differential equations for the case of both single order as <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> and higher order, <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mfenced separators="" open="[" close=")"><mn>1</mn><mo>,</mo><mn>2</mn></mfenced></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> denotes the order of fractional derivatives of <inline-formula><math display="inline"><semantics><mrow><msup><mi>D</mi><mi>α</mi></msup><mi>y</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The comparison in terms of accuracy between the proposed method and other existing methods demonstrate that the proposed method gives competitive performance as the existing methods.https://www.mdpi.com/2227-7390/8/10/1675multistep methodfractional differential equationlinear FDEnonlinear FDEsingle order FDEhigher order FDE |
| spellingShingle | Nur Amirah Zabidi Zanariah Abdul Majid Adem Kilicman Faranak Rabiei Numerical Solutions of Fractional Differential Equations by Using Fractional Explicit Adams Method Mathematics multistep method fractional differential equation linear FDE nonlinear FDE single order FDE higher order FDE |
| title | Numerical Solutions of Fractional Differential Equations by Using Fractional Explicit Adams Method |
| title_full | Numerical Solutions of Fractional Differential Equations by Using Fractional Explicit Adams Method |
| title_fullStr | Numerical Solutions of Fractional Differential Equations by Using Fractional Explicit Adams Method |
| title_full_unstemmed | Numerical Solutions of Fractional Differential Equations by Using Fractional Explicit Adams Method |
| title_short | Numerical Solutions of Fractional Differential Equations by Using Fractional Explicit Adams Method |
| title_sort | numerical solutions of fractional differential equations by using fractional explicit adams method |
| topic | multistep method fractional differential equation linear FDE nonlinear FDE single order FDE higher order FDE |
| url | https://www.mdpi.com/2227-7390/8/10/1675 |
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