Numerical solution of time fractional Volterra integro-differential equations

In this work, we develop and analyze a finite difference-based numerical scheme for a system of time-fractional Volterra-type integro-differential equations. The fractional derivative of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>α</mml:mi> <mml:mo class="MathClass-rel" stretchy="false">∈</mml:mo> <mml:mo class="MathClass-open" stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo class="MathClass-punc" stretchy="false">,</mml:mo><mml:mn>1</mml:mn><mml:mo class="MathClass-close" stretchy="false">)</mml:mo></mml:mrow></mml:math> is considered in the Caputo sense. We provide sufficient conditions to ensure the existence of a unique solution. To construct the difference scheme, classical L1 discretization is employed for the fractional operator, and the integral part is approximated using the composite trapezoidal rule. The convergence analysis and error estimates are discussed. Having a mild singularity in the solution of the system, the proposed scheme achieves only <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math> order accuracy. In the case of sufficiently smooth solutions, it achieves <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mo class="MathClass-open" stretchy="false">(</mml:mo><mml:mn>2</mml:mn> <mml:mo class="MathClass-bin" stretchy="false">−</mml:mo> <mml:mi>α</mml:mi><mml:mo class="MathClass-close" stretchy="false">)</mml:mo></mml:mrow></mml:math> order of accuracy. Finally, several numerical experiments are presented to support our theoretical findings and validate the proposed scheme.