Composite type equations and inverse problems (Q1968814)

Most mathematical problems in science, technology and medicine are inverse problems. Studying such problems is the only complete way of analyzing experimental results. In order to solve an inverse problem one has to master the direct problem both theoretically and numerically. The book deals with some direct and inverse problems relative to the Korteweg-de Vries equation \[ u_t+ u_{xxx}+ uu_x= f \] and further equations of composite type of third-order \[ Lu= \ell Au+ Bu= f(x), \] where \(A\) and \(B\) are differential operators of the second-order and \[ \ell= \alpha^k(x){\partial\over\partial x_k}+ \alpha(x) \] satisfying certain boundary conditions. The author mainly uses the well-known schemes based on obtaining a priori estimates, a `hard analysis' to prove solutions of the corresponding problems. Further, the author studies some special inverse problems of the form \[ Lu=\Delta u+ b(x)u= a(x) q(x)+ f(x). \] Here \(u\) and \(q(x)\) are to be determined by using certain boundary conditions and additional information. The author uses results of the first part of this book. The last paragraphs contain some results on the solvability of nonclassical boundary-value problems for equations of second- and third-order.