Vavrychuk V. Iterative methods based on integral equations for numerical solution of a Cauchy problem for the parabolic equation

Cauchy problem for parabolic equation with overdetermined data given on the part of the domain boundary is a classical example of ill-posed in Hadamard sense problem and is well-understood from theoretical point of view. But from the numerical standpoint currently more developed is the case of stationary differential equations, and there are only a few publications devoted to numerical solution of the parabolic Cauchy problem with the given initial condition and Cauchy boundary conditions on the part of the domain boundary in more than one dimension in space. Moreover some of existing publications have no numerical results and some have numerical results only for concentric circles or rectangle. Additional overview of the current state of the problem is given in the first chapter. In this dissertation Cauchy problem was numerically solved in a wide class of domains using developed boundary integral equations technique. Of course radically different variants of the problem domain require method which takes into account their specific features. The simplest case is sufficiently smooth doubly connected domain with Cauchy data given on the one of the closed contours, and this case is considered in the second chapter. Cauchy problem is formulated in the form of the operator equation of the first kind with a compact injective operator with dense range. Traditionally such equations are numerically solved using regularization methods, for example Landweber method. This compact operator can be represented via Dirichlet-Neumann or Neumann-Dirichlet initial boundary problem, and it is approximated using combination of Rothe and integral equations method. It is possible to get first or second order of approximation by time and exponential or super-algebraic convergence by the space variables. In the third chapter the same iterative regularization technique is extended to the case of semi-infinite canonical domains with smooth inclusion. Sequence of Green functions of the Neumann problem for elliptic equations sequence is introduced to efficiently implement numerical algorithm in this case. Forth chapter is devoted to the case when domain has corners or common points between parts of the boundary with different boundary conditions. Then mixed problems may lack strong solution, which was overcome by use of weighted spaces. In fifth chapter weight spaces approach was extended to the case of the cut. If there is a priori information about solution having same values of the both sides of the cut, then it can be used via some projection operator which leads to convergence speed improvements. Generally, corresponding boundary integral representation enable reducing of the original problem to the set of one-dimensional integral equation of the first or second kind with kernels with logarithmic or hyper- singularities and densities with square-root or algebraic singularities. Singularities in kernels are extracted via several kinds of weight functions or removed via Laplace transform when it is known analytically. Singularities in densities are reduced using cos-transform or nonlinear mesh grading near by corner points. Well-posedness of the integral equations of the first kind is shown in appropriate spaces.