A novel regularization scheme for a backward problem in a sub-diffusion model

Abstract In this paper, an ill-posed problem of multi-term time-fraction diffusion equation with unknown initial values is considered. This is usually done by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>g</m:mi> <m:mo>:=</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo rspace="4.2pt" stretchy="false">(</m:mo> <m:mo rspace="4.2pt">⋅</m:mo> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {g:=u(\,\cdot\,,T)} for some <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>T</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {T&gt;0} to find <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>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {u(x,t)} for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mi>t</m:mi> <m:mo>&lt;</m:mo> <m:mi>T</m:mi> </m:mrow> </m:math> {0\leq t&lt;T} which is so-called backward problem. For convenience, we will label this problem as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>P</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(P_{t})} . Through analysis, we know that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>P</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(P_{t})} is well posed for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>t</m:mi> <m:mo>&lt;</m:mo> <m:mi>T</m:mi> </m:mrow> </m:math> {0&lt;t&lt;T} , but ill-posed for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>t</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {t=0} that is different from classical backward heat conduction problem. Therefore, a novel identification scheme of stable approximate solutions of ill-posed inverse problem <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>P</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(P_{0})} is obtained by regularization family <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msub> <m:mi>R</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>:</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>t</m:mi> <m:mo>&lt;</m:mo> <m:mi>T</m:mi> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> {\{R_{t}:0&lt;t&lt;T\}} , and the error estim