Abstract
Abstract 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.
Highlights
An improvement set behaves just like a cone and in essence generalizes the notion of a cone
The notion of cones, introduced by Giannessi [4, 5], is the cornerstone of the theory of optimization and almost every optimal notion uses this concept to introduce the optimal elements associated with that optimal notion
The improvement sets are the basis of the E-optimal points for a set and many authors [3, 7,8,9, 11, 12] have employed this kind of optimal notion in vector optimization which generalizes the notion of efficiency and weak efficiency
Summary
An improvement set behaves just like a cone and in essence generalizes the notion of a cone. The improvement sets are the basis of the E-optimal points for a set and many authors [3, 7,8,9, 11, 12] have employed this kind of optimal notion in vector optimization which generalizes the notion of efficiency and weak efficiency. We consider À E optimal points instead of E-optimal points for a set, as this approach is very common in the literature, in locally convex spaces. For appropriately chosen, the set of all À E optimal points is greater (or even strictly greater) than the set of all E-optimal points for any set.
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