The Cauchy problem for general elliptic equations of second order is considered. In a previous paper (Berntsson et al. in Inverse Probl Sci Eng 26(7):1062–1078, 2018), it was suggested that the alternating iterative algorithm suggested by Kozlov and Maz’ya can be convergent, even for large wavenumbers $$k^2$$ k 2 , in the Helmholtz equation, if the Neumann boundary conditions are replaced by Robin conditions. In this paper, we provide a proof that shows that the Dirichlet–Robin alternating algorithm is indeed convergent for general elliptic operators provided that the parameters in the Robin conditions are chosen appropriately. We also give numerical experiments intended to investigate the precise behaviour of the algorithm for different values of $$k^2$$ k 2 in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

The Cauchy problem for Helmholtz equation, for moderate wave number $k^{2}$, is considered. In the previous paper of Achieng et al. (2020, Analysis of Dirichlet–Robin iterations for solving the Cauchy problem for elliptic equations. Bull. Iran. Math. Soc.), a proof of convergence for the Dirichlet–Robin alternating algorithm was given for general elliptic operators of second order, provided that appropriate Robin parameters were used. Also, it has been noted that the rate of convergence for the alternating iterative algorithm is quite slow. Thus, we reformulate the Cauchy problem as an operator equation and implement iterative methods based on Krylov subspaces. The aim is to achieve faster convergence. In particular, we consider the Landweber method, the conjugate gradient method and the generalized minimal residual method. The numerical results show that all the methods work well. In this work, we discuss also how one can approach non-symmetric differential operators by using similar operator equations and model problems which are used for symmetric differential operators.

In this thesis, we study the Cauchy problem for an elliptic equation. We use Dirichlet-Robin iterations for solving the Cauchy problem. This allows us to include in our consideration elliptic equations with variable coefficient as well as Helmholtz type equations. The algorithm consists of solving mixed boundary value problems, which include the Dirichlet and Robin boundary conditions. Convergence is achieved by choice of parameters in the Robin conditions. i To my wonderful and loving parents, Mr. and Mrs. David Mutai and my siblings, your compassion, strength, and unconditional love are what carry me throughout my research. Thank you for being there by my side when times are tough. To my lovely son, Asier Kiptoo thank you very much for your patiences and endurances during my studies. I also take this opportunity to thank the entire family of Sing'oei for taking good care of my son.Finally, my most sincere appreciation go to ISP and the Eastern African Universities Mathematics Programme (EAUMP) for the financial support. I offer my sincere thanks to my coordinators; Lief Abrahamson, Patrick G. Weke, Jared Ongaro and the entire team working at ISP who have tirelessly ensure my studies and stay in Sweden is comfortable.