In a similar way, a parametrix for a variable coefficient differential operator P(x,D) is a distribution u such that where ω is some C ∞ function with compact support.
[2] Having found the analytic expression of the parametrix, it is possible to compute the solution of the associated fairly general elliptic partial differential equation by solving an associated Fredholm integral equation: also, the structure itself of the parametrix reveals properties of the solution of the problem without even calculating it, like its smoothness[3] and other qualitative properties.
[4] An explicit construction of a parametrix for second order partial differential operators based on power series developments was discovered by Jacques Hadamard.
In the case of the heat equation or the wave equation, where there is a distinguished time parameter t, Hadamard's method consists in taking the fundamental solution of the constant coefficient differential operator obtained freezing the coefficients at a fixed point and seeking a general solution as a product of this solution, as the point varies, by a formal power series in t. The constant term is 1 and the higher coefficients are functions determined recursively as integrals in a single variable.
[5][6] A sufficiently good parametrix can often be used to construct an exact fundamental solution by a convergent iterative procedure as follows (Berger, Gauduchon & Mazet 1971).