| Summary: | 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.
|
|---|