The Tracking of Derivative Discontinuities for Delay Fractional Equations Based on Fitted <i>L</i>1 Method
Dakang Cen, Seakweng Vong
Abstract
Abstract In this paper, the analytic solution of the delay fractional model is derived by the method of steps. The theoretical result implies that the regularity of the solution at <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>s</m:mi> <m:mo>+</m:mo> </m:msup> </m:math> {s^{+}} is better than that at <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mn>0</m:mn> <m:mo>+</m:mo> </m:msup> </m:math> {0^{+}} , where s is a constant time delay. The behavior of derivative discontinuity is also discussed. Then, improved regularity solution is obtained by the decomposition technique and a fitted <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>L</m:mi> <m:mo></m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {L1} numerical scheme is designed for it. For the case of initial singularity, the optimal convergence order is reached on uniform meshes when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mfrac> <m:mn>2</m:mn> <m:mn>3</m:mn> </m:mfrac> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\alpha\in[\frac{2}{3},1)} , α is the order of fractional derivative. Furthermore, an improved fitted <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>L</m:mi> <m:mo></m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {L1} method is proposed and the region of optimal convergence order is larger. For the case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>t</m:mi> <m:mo>></m:mo> <m:mi>s</m:mi> </m:mrow> </m:math> {t>s} , stability and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>min</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo></m:mo> <m:mi>α</m:mi> </m:mrow> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> {\min\{2\alpha,1\}} order convergence of the fitted <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>L</m:mi> <m:mo></m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {L1} scheme are deduced. At last, the numerical tests are carried out and confirm the theoretical result.