On finite products of convolutions and classifications of hyperbolic and elliptic equations

In this paper we consider linear second order partial differential equations with constant coefficients; then by using the single and double convolution products we produce some new equations with variable coefficients and we classify the new equations. It is shown that the classifications of the new equations are similar to the original equations that is, if the original equation is a hyperbolic then the new classification after convolution product is also a hyperbolic similarly, if an elliptic then the new classification is an elliptic equation. Thus we prove that the classifications of new equations are invariant after finite convolution products.