Hermite Cubic Spline Collocation Method for Nonlinear Fractional Differential Equations with Variable-Order
In this paper, we develop a Hermite cubic spline collocation method (HCSCM) for solving variable-order nonlinear fractional differential equations, which apply <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup&...
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MDPI AG
2021-05-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/5/872 |
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| author | Tinggang Zhao Yujiang Wu |
| author_facet | Tinggang Zhao Yujiang Wu |
| author_sort | Tinggang Zhao |
| collection | DOAJ |
| description | In this paper, we develop a Hermite cubic spline collocation method (HCSCM) for solving variable-order nonlinear fractional differential equations, which apply <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>1</mn></msup></semantics></math></inline-formula>-continuous nodal basis functions to an approximate problem. We also verify that the order of convergence of the HCSCM is about <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>h</mi><mrow><mo movablelimits="true" form="prefix">min</mo><mo>{</mo><mn>4</mn><mo>−</mo><mi>α</mi><mo>,</mo><mi>p</mi><mo>}</mo></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula> while the interpolating function belongs to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>C</mi><mi>p</mi></msup><mrow><mo>(</mo><mi>p</mi><mo>≥</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <i>h</i> is the mesh size and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> the order of the fractional derivative. Many numerical tests are performed to confirm the effectiveness of the HCSCM for fractional differential equations, which include Helmholtz equations and the fractional Burgers equation of constant-order and variable-order with Riemann-Liouville, Caputo and Patie-Simon sense as well as two-sided cases. |
| first_indexed | 2024-03-10T11:27:07Z |
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| id | doaj.art-bc30d6bd8ec047d68d09b1295b940901 |
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| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T11:27:07Z |
| publishDate | 2021-05-01 |
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| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-bc30d6bd8ec047d68d09b1295b9409012023-11-21T19:35:14ZengMDPI AGSymmetry2073-89942021-05-0113587210.3390/sym13050872Hermite Cubic Spline Collocation Method for Nonlinear Fractional Differential Equations with Variable-OrderTinggang Zhao0Yujiang Wu1School of Mathematics, Lanzhou City University, Lanzhou 730070, ChinaSchool of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, ChinaIn this paper, we develop a Hermite cubic spline collocation method (HCSCM) for solving variable-order nonlinear fractional differential equations, which apply <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>1</mn></msup></semantics></math></inline-formula>-continuous nodal basis functions to an approximate problem. We also verify that the order of convergence of the HCSCM is about <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>h</mi><mrow><mo movablelimits="true" form="prefix">min</mo><mo>{</mo><mn>4</mn><mo>−</mo><mi>α</mi><mo>,</mo><mi>p</mi><mo>}</mo></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula> while the interpolating function belongs to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>C</mi><mi>p</mi></msup><mrow><mo>(</mo><mi>p</mi><mo>≥</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <i>h</i> is the mesh size and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> the order of the fractional derivative. Many numerical tests are performed to confirm the effectiveness of the HCSCM for fractional differential equations, which include Helmholtz equations and the fractional Burgers equation of constant-order and variable-order with Riemann-Liouville, Caputo and Patie-Simon sense as well as two-sided cases.https://www.mdpi.com/2073-8994/13/5/872collocation methodfractional calculushermite cubic splinefractional burgers equation |
| spellingShingle | Tinggang Zhao Yujiang Wu Hermite Cubic Spline Collocation Method for Nonlinear Fractional Differential Equations with Variable-Order Symmetry collocation method fractional calculus hermite cubic spline fractional burgers equation |
| title | Hermite Cubic Spline Collocation Method for Nonlinear Fractional Differential Equations with Variable-Order |
| title_full | Hermite Cubic Spline Collocation Method for Nonlinear Fractional Differential Equations with Variable-Order |
| title_fullStr | Hermite Cubic Spline Collocation Method for Nonlinear Fractional Differential Equations with Variable-Order |
| title_full_unstemmed | Hermite Cubic Spline Collocation Method for Nonlinear Fractional Differential Equations with Variable-Order |
| title_short | Hermite Cubic Spline Collocation Method for Nonlinear Fractional Differential Equations with Variable-Order |
| title_sort | hermite cubic spline collocation method for nonlinear fractional differential equations with variable order |
| topic | collocation method fractional calculus hermite cubic spline fractional burgers equation |
| url | https://www.mdpi.com/2073-8994/13/5/872 |
| work_keys_str_mv | AT tinggangzhao hermitecubicsplinecollocationmethodfornonlinearfractionaldifferentialequationswithvariableorder AT yujiangwu hermitecubicsplinecollocationmethodfornonlinearfractionaldifferentialequationswithvariableorder |