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.
Thermal diffusivity identification based on an iterative regularization method
This article deals with the identification of a space and time dependent material thermal diffusivity. Such parameter is involved in heat transfers described by partial differential equations. An iterative regularization method based on a conjugate gradient algorithm is implemented. Such approach is attractive in order to efficiently deal with measurement noises and model errors. Numerical results are illustrated according to several simulations.
Athenian Democracy at War
Classical Athens perfected direct democracy. The plays of this ancient Greek state are still staged today. These achievements are rightly revered. Less well known is the other side of this success story. Democratic Athens completely transformed warfare and became a superpower. The Athenian armed forces were unmatched in size and professionalism. This book explores the major reasons behind this military success. It shows how democracy helped the Athenians to be better soldiers. For the first time David M. Pritchard studies, together, all four branches of the armed forces. He focuses on the background of those who fought Athens' wars and on what they thought about doing so. His book reveals the common practices that Athens used right across the armed forces and shows how Athens' pro-war culture had a big impact on civilian life. The book puts the study of Athenian democracy at war on an entirely new footing.