An analysis on classifications of hyperbolic and elliptic PDEs

Purpose: Our aim in this study is to generate some partial differential equations (PDEs) with variable coefficients by using the PDEs with non-constant coefficients. Methods: Then by applying the single and double convolution products, we produce some new equations having polynomials coefficients. We then classify the new equations on using the classification method for the second order linear partial differential equations. Results: Classification is invariant under single and double convolutions by applying some conditions, that is, we identify some conditions where a hyperbolic equation will be hyperbolic again after single and double convolutions. Conclusions: It is shown that the classifications of the new PDEs are related to the coefficients of polynomials which are considered during the process of convolution product.