Alternating algorithms applied to Cauchy problems for a fractional elliptic equation with fractional-order boundary conditions

Abstract The main objective of this work is to investigate the possibility of developing alternating algorithms that preserve the governing fractional differential equation when applied to solving associated Cauchy problems. To this end, we consider an ill-posed Cauchy problem associated to a fractional partial differential equation with fractional-order boundary conditions. We suggest an alternating algorithm inspired by the one of [V. A. Kozlov, V. G. Mazya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations, USSR Comput. Math. Math. Phys. 31 1991, 1, 45–52] developed for the classical Cauchy problem for the elliptic Laplace’s equation. Such algorithm involves two types of problems whose stabilities are proved rigorously by two methods. The first method consists of the establishment of weak formulations for these problems and subsequently the stability can be obtained through the coercivity and continuity of the associated bilinear and linear forms, respectively. The second method is based on the use of the method of separation of variables. Subsequently, we develop several new inequalities based, essentially, on the asymptotic behaviour of Mittag-Leffler functions, to control the solutions in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msubsup> <m:mi>H</m:mi> <m:mn>0</m:mn> <m:mn>1</m:mn> </m:msubsup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {L^{2}(H^{1}_{0})} . Some numerical simulations are presented to support the main key inequalities involved in our analysis. The convergence of the sequence of solutions of the alternating algorithm towards the solution of the ill-posed Cauchy problem is proved to hold in the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msubsup> <m:mi>H</m:mi> <m:mn>0</m:mn> <m:mn>1</m:mn> </m:msubsup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {L^{2}(H^{1}_{0})} -seminorm.