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></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.