Exponentially convergent method for a differential equation with fractional derivative and unbounded operator coefficient in Banach space

• Date: 2022
• Main Authors: Vasylyk, V. B., Gavrilyuk, I. P., Makarov , V. L., Vasylyk, Vitaliy, Василик, В. Б., Гаврилюк, I. П. , Макаров , В. Л.
• Format: Article
• Language: Ukrainian
• Published: Institute of Mathematics, NAS of Ukraine 2022
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/6984
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

UDC 519.62, 519.63 We propose and analyze an exponentially convergent numerical method for solving a differential equation with a right-hand fractional Riemann-Liouville derivative and an unbounded operator coefficient in Banach space. We apply the representation of the solution by the Danford-Cauchy integral on the hyperbola, which covers the spectrum of the operator coefficient with the subsequent application of an exponentially convergent quadrature. To do this, the parameters of the hyperbola are chosen so that the integration function has an analytical extension in the strip around the real axis and then apply the Sinc-quadrature. We show the exponential accuracy of the method and show numerical example that confirms the obtained a priori estimate.