In this paper a matrix separation of the variables method for solving initial-boundary value problems related to coupled systems of second-order partial differential equations is presented. In an analogous way to the uncoupled scalar case, an infinite series solution of the problem is given. Error bounds for approximate solutions obtained from truncation of the infinite series and the accuracy of this method are studied in terms of data problems.
Jo´dar, L. (2002). A separation of the variables method for solving coupled systems of second-order partial differential equations: exact, approximate solutions and error bounds. Journal of Computational and Applied Mathematics, 31(2), 243–251. https://doi.org/10.1016/0377-0427(90)90167-x