Divergent series and generalized mixed problem for a wave equation of the simplest type

Abstract

With the use of the operation of integrating the divergent series of a formal solution in the separating variables method, there are obtained the results concerning a generalized mixed problem (homogeneous and non-homogeneous) for the wave equation. The key moment consists in finding the sum of the divergent series  which corresponds to the simplest mixed problem with a summable initial function. This result helps to get  the solution of the generalized mixed problem for a non-homogeneous equation under the assumption that non-homogeneity is characterized by a locally summable function. As an application, the mixed problem with a non-zero potential is considered, in which the differential equation is treated quite formally but the  mixed problem itself is no longer a generalized one: instead of the formal solution of the separating  variables method we get an integral equation which can be solved by the successive substitutions method. Thus we essentially simplify the arguments.

Keywords

• divergent series
• wave equation
• mixed problem