An inverse problem for Moore–Gibson–Thompson equation arising in high intensity ultrasound

Abstract

In this article, we study the inverse problem of recovering a space-dependent coefficient of the Moore–Gibson–Thompson (MGT) equation from knowledge of the trace of the solution on some open subset of the boundary. We obtain the Lipschitz stability for this inverse problem, and we design a convergent algorithm for the reconstruction of the unknown coefficient. The techniques used are based on Carleman inequalities for wave equations and properties of the MGT equation.

Identification of the time-dependent lowest term in a fourth order in time partial differential equation

“…B T , m 5 , and m 6 . From the last inequalities it follows that ∥O(ν 1 ) − O(ν 2 )∥ E T ≤ A(T )C(a (1) , y (2) , m 5 , m 6 ) ∥ν 1 − ν 2 ∥ E T where A(T ) = T 1 + and C(a (1) , y (2) , m 5 , m 6 ) = max {C 1 , C 2 , C 3 , C 4 } is the constant depends on the norms a (1) C[0,T ] , y…”

“…B T , m 5 , and m 6 . (1) , y (2) , m 5 , m 6 ) are bounded above. Thus A(T )C(a (1) , y (2) , m 5 , m 6 ) tends to zero as T → 0.…”

“…Consider the third order in time nonlinear partial differential equation model in abstract form ∂τττ y(x, τ ) + α∂ττ y(x, τ ) − c 2 ∂xxy(x, τ ) − b∂xxτ y(x, τ ) = G(x, τ , y, yτ , yττ ) (1) where G(x, t, y, yτ , yττ ) is a non-linear or linear function and α, c, b > 0 are given parameters. This type of model is often called Moore-Gibson-Thompson equation and appeared in many scientific fields such as nonlinear acoustics, medical ultrasound, viscoelasticity and thermoelasticity, [4,6,[10][11][12]20].…”

“…The inverse problems of determining time or space dependent coefficients for the higher order in time (more than 2) PDEs attract many scientists. The inverse problem of recovering the solely space dependent and solely time dependent coefficients for the third order in time PDEs are studied by [1] and [21], respectively. More recently, in [9] authors studied the inverse problem of determining time dependent potential and time dependent force terms from the third order in time partial differential equation by considering the critical parameter equal to zero.…”

Inverse coefficient problem for differential equation in partial derivatives of a fourth order in time with integral over-determination

“…The inverse problems of determining time or space dependent coefficients for the higher order in time (more than 2) PDEs attract many scientists. The inverse problem of recovering the solely space dependent and solely time dependent coefficients for the third order in time PDEs are studied by [15,16], respectively. More recently, in [17] authors studied the inverse problem of determining time dependent potential and time dependent force terms from the third order in time partial differential equation theoretically and numerically by considering the critical parameter equal to zero.…”