The dimensional reduction method for solving boundary value problems of Helmholtz's equation in domain Ωd := Ω x (-d , d) ⊂ ℝn+1 by replacing them with systems of equations in n-dimensional space are investigated. It is proved that the existence and uniqueness for the exact solution u and the dimensionally reduced solution uN of the boundary value problem if the input data on the faces are in some class of functions. In addition, the difference between u and uN in H1(Ωd) is estimated as d and N are fixed. Finally, some numerical experiments in a domain Ω = (0, 1) x (0, 1) are given in order to compare theretical results.
All Science Journal Classification (ASJC) codes
- Computational Mathematics
- Applied Mathematics