In this paper we study the compound equation N i=1 M j=1 (B + a 2) i (2 B + b 2) j u(x) = f (x), (1) where x = (x 1 , x 2 ,. .. , x n) ∈ R n + = {x ∈ R n | x i > 0}, a and b are nonzero constants. u(x) is unknown function and f (x) is a given distribution. (B + a 2) i is the Bessel-Helmholtz operator iterated i−times and (2 B + b 2) j is the Bessel-Klein-Gordon operator iterated j−times. The existence and the uniqueness solution of (1) is proven.

This paper presents an alternative methodology for finding the solution of the boundary value problem (BVP) for the linear partial differential operator. We are particularly interested in the linear operator ⊕ k , where ⊕ k = ♥ k ♦ k , ♥ k is the biharmonic operator iterated k-times and ♦ k is the diamond operator iterated k-times. The solution is built on the Green's identity of the operators ♥ k and ⊕ k , in which their derivations are also provided. To illustrate our findings, the example with prescribed boundary conditions is exhibited.

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In this paper, we find a distribution solution of some PDEs of the operator ⊕ k , defined by (1), which is related to the wave equation and the diamond operator. The existence of the solution to these equations are proven.