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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. [ABSTRACT FROM AUTHOR]

The railroad blocking problem (RBP) is an important decision for freight railroad companies. The objective of this problem is to minimize the costs of delivering all commodities by deciding which interyard blocks to build and specifying the assignment of commodities to these blocks. In this paper, a mathematical model is presented for the RBP on Iran Railways. Its decision variables identify blocking scheme and demand assignment to these blocks. The RBP in medium and large sizes is not solvable with any commercial software available in the market. Therefore, a solution method based on tabu search algorithm is proposed for the suggested model. For evaluating the proposed algorithm, several simulated test problems are randomly generated and solved. The obtained results on the test problems are compared with those of solutions generated by CPLEX software. The comparison shows high efficiency and effectiveness of the proposed algorithm. The proposed model and solution method are applied to build a blocking plan for the Iranian railway. [ABSTRACT FROM AUTHOR]