A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik Equation
In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></in...
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2024-10-01
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| author | Maria Carmela De Bonis Donatella Occorsio |
| author_facet | Maria Carmela De Bonis Donatella Occorsio |
| author_sort | Maria Carmela De Bonis |
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| description | In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><msup><mi>D</mi><mi>α</mi></msup><mi>f</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><mstyle><mfrac><mn>1</mn><mrow><mo>Γ</mo><mo>(</mo><mi>m</mi><mo>−</mo><mi>α</mi><mo>)</mo></mrow></mfrac></mstyle><msubsup><mo>∫</mo><mn>0</mn><mi>y</mi></msubsup><msup><mrow><mo>(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>m</mi><mo>−</mo><mi>α</mi><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>f</mi><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>d</mi><mi>x</mi><mo>,</mo><mspace width="1.em"></mspace><mi>y</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>−</mo><mn>1</mn><mo><</mo><mi>α</mi><mo>≤</mo><mi>m</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>m</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi><mo>.</mo></mrow></semantics></math></inline-formula> The numerical procedure is based on approximating <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>f</mi><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></msup></semantics></math></inline-formula> by the <i>m</i>-th derivative of a Lagrange polynomial, interpolating <i>f</i> at Jacobi zeros and some additional nodes suitably chosen to have corresponding logarithmically diverging Lebsegue constants. Error estimates in a uniform norm are provided, showing that the rate of convergence is related to the smoothness of the function <i>f</i> according to the best polynomial approximation error and depending on order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. As an application, we approximate the solution of a Volterra integral equation, which is equivalent in some sense to the Bagley–Torvik initial value problem, using a Nyström-type method. Finally, some numerical tests are presented to assess the performance of the proposed procedure. |
| format | Article |
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| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-10-01 |
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| spelling | doaj-art-0058c5e10738474ebb94e00970f1cb942024-11-26T17:50:47ZengMDPI AGAxioms2075-16802024-10-01131175010.3390/axioms13110750A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik EquationMaria Carmela De Bonis0Donatella Occorsio1Department of Basic and Applied Sciences, University of Basilicata, Viale dell’Ateneo Lucano 10, 85100 Potenza, ItalyDepartment of Basic and Applied Sciences, University of Basilicata, Viale dell’Ateneo Lucano 10, 85100 Potenza, ItalyIn this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><msup><mi>D</mi><mi>α</mi></msup><mi>f</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><mstyle><mfrac><mn>1</mn><mrow><mo>Γ</mo><mo>(</mo><mi>m</mi><mo>−</mo><mi>α</mi><mo>)</mo></mrow></mfrac></mstyle><msubsup><mo>∫</mo><mn>0</mn><mi>y</mi></msubsup><msup><mrow><mo>(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>m</mi><mo>−</mo><mi>α</mi><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>f</mi><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>d</mi><mi>x</mi><mo>,</mo><mspace width="1.em"></mspace><mi>y</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>−</mo><mn>1</mn><mo><</mo><mi>α</mi><mo>≤</mo><mi>m</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>m</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi><mo>.</mo></mrow></semantics></math></inline-formula> The numerical procedure is based on approximating <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>f</mi><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></msup></semantics></math></inline-formula> by the <i>m</i>-th derivative of a Lagrange polynomial, interpolating <i>f</i> at Jacobi zeros and some additional nodes suitably chosen to have corresponding logarithmically diverging Lebsegue constants. Error estimates in a uniform norm are provided, showing that the rate of convergence is related to the smoothness of the function <i>f</i> according to the best polynomial approximation error and depending on order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. As an application, we approximate the solution of a Volterra integral equation, which is equivalent in some sense to the Bagley–Torvik initial value problem, using a Nyström-type method. Finally, some numerical tests are presented to assess the performance of the proposed procedure.https://www.mdpi.com/2075-1680/13/11/750Caputo’s derivativesLagrange interpolationJacobi polynomialsproduct integration rulesfractional differential equations |
| spellingShingle | Maria Carmela De Bonis Donatella Occorsio A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik Equation Axioms Caputo’s derivatives Lagrange interpolation Jacobi polynomials product integration rules fractional differential equations |
| title | A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik Equation |
| title_full | A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik Equation |
| title_fullStr | A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik Equation |
| title_full_unstemmed | A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik Equation |
| title_short | A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik Equation |
| title_sort | global method for approximating caputo fractional derivatives an application to the bagley torvik equation |
| topic | Caputo’s derivatives Lagrange interpolation Jacobi polynomials product integration rules fractional differential equations |
| url | https://www.mdpi.com/2075-1680/13/11/750 |
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