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Riemann boundary value problem for triharmonic equation in higher space.

Longfei Gu1

  • 1Department of Mathematics, Linyi University, Linyi, Shandong 276000, China.

Thescientificworldjournal
|August 13, 2014
PubMed
Summary
This summary is machine-generated.

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This study addresses a boundary value problem for triharmonic functions in universal Clifford algebras. A unique solution is proven to exist under specific conditions for this complex mathematical problem.

Area of Science:

  • Mathematical Physics
  • Complex Analysis
  • Clifford Algebra

Background:

  • Boundary value problems are crucial in modeling physical phenomena.
  • Triharmonic functions and Clifford algebras present advanced mathematical challenges.
  • Previous research has explored related problems in simpler algebraic structures.

Purpose of the Study:

  • To investigate a specific boundary value problem for triharmonic functions.
  • To extend the analysis to functions with values in universal Clifford algebras.
  • To establish the existence and uniqueness of solutions for this problem.

Main Methods:

  • Formulation of the boundary value problem involving a triharmonic equation and specific boundary conditions.
  • Utilizing properties of the Dirac operator within Clifford analysis.

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  • Application of Lyapunov surface theory for domain characterization.

    • Main Results:

    • The study focuses on the equation Δ(3)[u](x) = 0 within a universal Clifford algebra Cl(Vn,n).
    • Boundary conditions involve function values and derivatives multiplied by Clifford algebra elements.
    • Existence and uniqueness of a solution are demonstrated under stated hypotheses.

    Conclusions:

    • The boundary value problem for triharmonic functions in universal Clifford algebras is well-posed.
    • The established unique solution provides a significant advancement in Clifford analysis.
    • This work contributes to the theoretical understanding of differential equations in advanced mathematical frameworks.