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Suppose that p and q are arbitrary polynomials in k variables, \(k\in \mathbb N.\) Denote by \(n_0\) the degree of the nonzero lower homogeneous component of p and denote by n the (total) degree of q. In this paper we prove that the partial differential equation \(p(D)f=q\) has a polynomial solution. Moreover, the solution we find has the smallest possible degree, which is \(n_0+n.\) We present a method to solve the PDE.

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On Polynomial Solutions of PDE with Constant Coefficients

  • 会议论文

  • 发表日期：

    2025年

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  • Anna Gharibyan,
  • Hakop Hakopian

摘要

Suppose that p and q are arbitrary polynomials in k variables, \(k\in \mathbb N.\) Denote by \(n_0\) the degree of the nonzero lower homogeneous component of p and denote by n the (total) degree of q. In this paper we prove that the partial differential equation \(p(D)f=q\) has a polynomial solution. Moreover, the solution we find has the smallest possible degree, which is \(n_0+n.\) We present a method to solve the PDE.