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.
Error analysis of a multisingular inverse coefficient problem for the Sturm-Liouville operator based on boundary measurement
In this paper we study the inverse problem 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), from boundary measurements, when u^'(x) or/and u^'^'(x) vanishes at several, called singular, points of the interval (0,1). As a result the considered inverse problem has simultaneously different types (moderate or severely) of ill-conditioned situations in different parts of the interval (0,1). The presented inverse polynomial method permits use of a priori information about singular points either to increase the order of the polynomial approximation in each subinterval or to obtain an artificial Cauchy data for the unknown coefficient. Error estimations for the polynomial approximations are presented for well-conditioned, as well as for ill-conditioned situations. The behaviour of the inverse problem solution with respect to both types (Dirichlet and Neumann) of noisy data is analyzed. The obtained results are illustrated by various numerical examples.
American Insurgents, American Patriots: The Revolution of the People
This is a story of rumour, charity, vengeance, and restraint. "American Insurgents, American Patriots" reminds us that revolutions are violent events. They provoke passion and rage, a willingness to use violence to achieve political ends, a deep sense of betrayal, and a strong religious conviction that God expects an oppressed people to defend their rights. The American Revolution was no exception. A few celebrated figures in the Continental Congress do not make for a revolution. It requires tens of thousands of ordinary men and women willing to sacrifice, kill, and be killed. Breen not only tells the history of these ordinary Americans but, drawing upon a wealth of rarely seen documents, restores their primacy to America's road to independence. Mobilizing two years before the Declaration of Independence, American insurgents in all thirteen colonies concluded that resistance to British oppression required organized violence against the state. They channeled popular rage through elected committees of safety and observation, which before 1776 were the heart of American resistance. "American Insurgents, American Patriots" is the stunning account of their insurgency, without which there would have been no independent republic as we know it.