Inverse problem of determining an order of the Caputo time-fractional derivative for a subdiffusion equation
Shavkat Alimov, Ravshan Ashurov
Abstract
Abstract An inverse problem for determining the order of the Caputo time-fractional derivative in a subdiffusion equation with an arbitrary positive self-adjoint operator A with discrete spectrum is considered. By the Fourier method it is proved that the value of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∥</m:mo> <m:mrow> <m:mi>A</m:mi> <m:mo></m:mo> <m:mi>u</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>∥</m:mo> </m:mrow> </m:math> {\|Au(t)\|} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>u</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {u(t)} is the solution of the forward problem, at a fixed time instance recovers uniquely the order of derivative. A list of examples is discussed, including linear systems of fractional differential equations, differential models with involution, fractional Sturm–Liouville operators, and many others.