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In this chapter we discuss the boundary integro-differential equations for the boundary value problems of the modified Helmholtz equation given by \( -\Delta u({\boldsymbol{x}})+k^2u({\boldsymbol{x}})=0,\) with constant \(k>0\) . One of the sources of the modified Helmholtz equation (4.0.1) is from the convection diffusion equation: \( -\epsilon ^2 \Delta v({\boldsymbol{x}})+\boldsymbol{a}\cdot \nabla v({\boldsymbol{x}})=0, \) with constant \(\epsilon >0\) and real constant vector \(\boldsymbol{a}=(a_1,a_2,\ldots ,a_n)^\textrm{T}\ne \boldsymbol{0}\) .

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Boundary Value Problems of Modified Helmholtz Equation

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  • 发表日期：

    2026年

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  • Houde Han,
  • Dongsheng Yin

摘要

In this chapter we discuss the boundary integro-differential equations for the boundary value problems of the modified Helmholtz equation given by \( -\Delta u({\boldsymbol{x}})+k^2u({\boldsymbol{x}})=0,\) with constant \(k>0\) . One of the sources of the modified Helmholtz equation (4.0.1) is from the convection diffusion equation: \( -\epsilon ^2 \Delta v({\boldsymbol{x}})+\boldsymbol{a}\cdot \nabla v({\boldsymbol{x}})=0, \) with constant \(\epsilon >0\) and real constant vector \(\boldsymbol{a}=(a_1,a_2,\ldots ,a_n)^\textrm{T}\ne \boldsymbol{0}\) .