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.
The determination of the leading coefficient in the monotone potential Sturm-Liouville operator from boundary measurements
This paper deals with the problem of determining the leading coefficient k=k((u')2) of the nonlinear (monotone potential) Sturm-Liouville operator Au=-(k((u')2)u'(x))'+q(x)u(x), x?(a,b). As an additional condition only two measured data at the boundary (x=a, x=b) are used. Solvability and linearization of the corresponding nonlinear direct problem are given. An existence of a quasi-solution of the inverse problem is obtained in a suitable compact class of admissible coefficients. In the second part of the paper an approximate analytical solution for the inverse problem is derived. The approach presented permits to analyze well-posed, as well as, all ill-posed situations for the inverse coefficient problem. Numerical examples corresponding to the all considered situations are presented.
Numerical Simulation of Thermal Sensitivity for Explosives
In order to compare the thermal sensitivity of several typical explosives, this article solved numerically one-dimensional transient heat conduction equation including thermal decomposition for explosives with finite difference method. The explicit difference equations under different boundary conditions were established, and the corresponding stability criterion was derived. The ignition times for typical explosives were calculated, which was used to evaluate thermal sensitivity of explosives.