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<p>In this paper, we introduce a new class of generalized monogenic functions within the framework of quaternionic analysis. These functions are defined as solutions to the equation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\alpha Du + \beta uD + cu = 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mi>D</mi> <mi>u</mi> <mo>+</mo> <mi>β</mi> <mi>u</mi> <mi>D</mi> <mo>+</mo> <mi>c</mi> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, where <i>D</i> denotes the Dirac operator in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {R}^3\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>, and <i>c</i> are elastic constants. This equation arises naturally from a factorization of the time-harmonic elastic wave equation. By employing endomorphisms on the quaternion algebra, we develop both integral and series representation formulas for the solutions.</p>

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Representations of solutions of the equation \(\alpha Du+\beta uD+cu=0\) in quaternionic analysis

  • 期刊论文

  • 发表日期：

    2025年9月12日

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  • Dao Viet Cuong,
  • Doan Cong Dinh

摘要

In this paper, we introduce a new class of generalized monogenic functions within the framework of quaternionic analysis. These functions are defined as solutions to the equation \(\alpha Du + \beta uD + cu = 0\) α D u + β u D + c u = 0 , where D denotes the Dirac operator in \(\mathbb {R}^3\) R 3 , and \(\alpha \) α , \(\beta \) β , and c are elastic constants. This equation arises naturally from a factorization of the time-harmonic elastic wave equation. By employing endomorphisms on the quaternion algebra, we develop both integral and series representation formulas for the solutions.