An algebraic approach to solving boundary value problems

Abstract

Let be polynomials in and let be the real algebraic set associated with Q() and let be a compact subset of the algebraic set 𝒬. We describe an algebraic approach for solving the general boundary value problem (BVP): given partial differential equation (PDE) and a continuous function find so that

We will show how the general technique applies in the case that P(y) is a homogeneous polynomial of degree deg(P(y)) and , where is a polynomial having deg  < deg (P(y)) and prove that the solution is unique in this case. This article brings together ideas from partial differential equations, a generalization of the theory of functions of a complex variable and the theory of commutative algebras.

Keywords:

• boundary value problem
• commutative algebra
• Clifford analysis
• residue class ring

AMS Subject Classifications: :

• 37K20
• 32A20