An inverse polynomial method for the identification of the leading coefficient in the Sturm-Liouville operator from boundary measurements
An inverse polynomial method of determining the unknown leading coefficient k=k(x) of the linear Sturm-Liouville operator Au=-(k(x)u^'(x))^'+q(x)u(x), x@?(0,1), is presented. As an additional condition only two measured data at the boundary (x=0,x=1) are used. In absence of a singular point (u^'(x) 0,u^''(x) 0,@?x@?[0,1]) the inverse problem is classified as a well-conditioned . If there exists at least one singular point, then the inverse problem is classified as moderately ill-conditioned (u^'(x"0)=0,x"0@?(0,1);u^'(x) 0,@?x x"0;u^''(x) 0,@?x@?[0,1]) and severely ill-conditioned (u^'(x"0)=u^''(x"0)=0,x"0@?(0,1);u^'(x) 0,u^''(x) 0,@?x x"0). For each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a Cauchy problem for to the first order differential equation with respect the unknown function k=k(x). An approximate analytical solution of the each Cauchy problems are derived in explicit form. Numerical simulations all the above cases are given for noise free and noisy data. An accuracy of the presented approach is demonstrated on numerical test solutions.
Determination of leading coefficients in Sturm-Liouville operator from boundary measurements. II. Unicity and an engineering approach
We study the problem of determining the leading coefficient k=k(x) of the Sturm-Liouville operator Au=-(k(x)u^'(x))^'+q(x)u(x),x@?(a,b), from a measured data given in the form of Dirichlet or/and Neumann type (additional) boundary conditions. The contribution of the each measured data to the unicity of the inverse problem solution is analysed. Based on the analysis the problem is treated as a Cauchy problem for the first-order ordinary differential equation with respect to k(x). For the case when the direct problem solution u=u(x) belongs to the class of the second-order polynomials and u^'(x) 0, we present an analytical formula for an approximate solution of the inverse problem. Numerical examples related to accuracy of the solution and to essential features of the inverse problem are demonstrated.
Inhomogeneous Boundary Value Problems for Compressible Navier-Stokes Equations: Well-Posedness and Sensitivity Analysis
In this paper compressible, stationary Navier–Stokes equations are considered. A framework for analysis of such equations is established. In particular, the well-posedness for inhomogeneous boundary value problems of elliptic-hyperbolic type is shown. Analysis is performed for small perturbations of the so-called approximate solutions that take form (1.12). The approximate solutions are determined from Stokes problem (1.11). The small perturbations are given by solutions to (1.13). The uniqueness of solutions for problem (1.13) is proved, and, in addition, the differentiability of solutions with respect to the coefficients of differential operators is shown. The results on the well-posedness of nonlinear problems are interesting on their own and are used to obtain the shape differentiability of the drag functional for incompressible Navier–Stokes equations. The shape gradient of the drag functional is derived in the classical and useful for computations form; an appropriate adjoint state is introduced to ...