Cauchy problem of the Delta(square(k)) operator related to the Diamond operator and the Laplace operator iterated k times
Given the Laplace operator Delta, is defined by Delta = partial derivative(2)/partial derivative x(1)(2) + partial derivative(2)/partial derivative x(2)(2) + ... + partial derivative(2)/partial derivative x(n)(2), (0.1) the Ultra-hyperbolic operator iterated k times square(k) is defined by square k = (partial derivative(2)/partial derivative x(2)(2) + ... + partial derivative(2)/partial derivative x(1p)(2) - partial derivative(2)/partial derivative x(p+1)(2) - ... - partial derivative(2)/partial derivative x(p+q)(2))(k) , (0.2) where p + q = n is the dimension of the Euclidean space R-n. In this paper, we study Cauchy problem and fundamental solution of the Delta(square(k)) operator by using Green's identity, In particular, Delta(D-k) reduces to the Diamond operator if k = 1. Moreover, for q = 0 the ultra-hyperbolic operator 0 reduces to Delta, and Delta(square(k-1)) reduces to the Laplace operator Delta(k) iterated k times.
cauchy problem fundamental solution distribution