A filter regularization for recovering wave amplitude in viscoelastic media governed by fractional diffusion-wave systems

In this paper, we consider a final value problem for the following time-fractional diffusion system: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mtable columnspacing="0pt" displaystyle="true" rowspacing="0pt"> <m:mtr> <m:mtd columnalign="right"> <m:mrow> <m:mrow> <m:msubsup> <m:mi>D</m:mi> <m:mi>t</m:mi> <m:mi>α</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi mathvariant="script">𝒜</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mrow> <m:mi/> <m:mo>=</m:mo> <m:mrow> <m:mi>F</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="right"> <m:mrow> <m:mrow> <m:msubsup> <m:mi>D</m:mi> <m:mi>t</m:mi> <m:mi>α</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi mathvariant="script">ℬ</m:mi> <m:mo>⁢</m:mo> <m:mi>v</m:mi> </m:mrow> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mrow> <m:mi/> <m:mo>=</m:mo> <m:mrow> <m:mi>G</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:math> \{{aligned} {}Dₜαu+Au&% =F(u,v),\\ Dₜαv+Bv&=G(u,v),{aligned}. where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒜</m:mi> </m:math> {A} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℬ</m:mi> </m:math> {B} are symmetric uniformly elliptic operators defined on a bounded domain Ω in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>d</m:mi> </m:msup> </m:math> {Rᵈ} with sufficiently smooth boundary and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>D</m:mi> <m:mi>t</m:mi> <m:mi>α</m:mi> </m:msubsup> </m:math> {Dₜα} refers to the Caputo fractional derivative of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>α</m:mi> <m:mo><</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {1<α<2} . Under suitable assumptions, we establish the existence and uniqueness of the solution to the problem and prove that the problem is ill-posed in the sense of Hadamard. A filter regularization method is proposed to approximate the solution and estimate the error between the regularized solution and the exact solution.