Abstract
This chapter focuses on the Cauchy problems for elliptic and hyperbolic equations. The Cauchy problem for an elliptic equation is a typical ill-posed problem of Mathematical Physics. The solution to the Cauchy problem for an elliptic equation is unstable with respect to small perturbations of data. Numerical method for solving the problems is also presented. It is based on an iterative process in which the Cauchy problem is posed in the form of an inverse problem in finding a function on the part of the boundary. It is shown by simple numerical examples that internal information can significantly improve numerical solution. In the variational and Dirichlet-to-Neumann approaches for a magnetostatic Cauchy problem are proposed. After computation by both methods, it is concluded that variational algorithm is stable, and the numerical solution is in a good agreement with the exact one, while the Dirichlet-to-Neumann algorithm is better in the cost of computations.
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