An approach to solutions of systems of linear partial differential equations with applications

Initial and boundary value problems governed by a system of linear partial differential equations can be solved by using the classical methods. This holds in solving problems which are governed by a unique system of equations over the whole region of interest. But if a problem is governed by a given system of equations over a region and by another system over the complementary one, classical methods may fail in treating this problem. A typical problem is that of evaluating the time-dependent electric field in the conductive half-space (the substratum) as a model in geophysical prospecting. The electric field in the air above the substratum is time independent. This problem has been solved numerically. Here, we solve it analytically. We proceed by presenting an approach for finding the solutions of systems of linear partial differential equations. Eigen operators and fractional powers of matrices of operators have been introduced. The formal solutions obtained are adequate for studying initial and boundary value problems whose solutions are anharmonic ones. They are used to solve the above-mentioned problem.