Boundary Value Problems of Modified Helmholtz Equation

Houde Han () and Dongsheng Yin ()
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Houde Han: Tsinghua University, Department of Mathematical Sciences
Dongsheng Yin: Tsinghua University, Department of Mathematical Sciences

Chapter Chapter 4 in Mathematical Foundation of the Boundary Integro-Differential Equation Method, 2026, pp 97-127 from Springer

Abstract: Abstract In this chapter we discuss the boundary integro-differential equations for the boundary value problems of the modified Helmholtz equation given by $$ -\Delta u({\boldsymbol{x}})+k^2u({\boldsymbol{x}})=0,$$ - Δ u ( x ) + k 2 u ( x ) = 0 , with constant $$k>0$$ k > 0 . One of the sources of the modified Helmholtz equation (4.0.1) is from the convection diffusion equation: $$ -\epsilon ^2 \Delta v({\boldsymbol{x}})+\boldsymbol{a}\cdot \nabla v({\boldsymbol{x}})=0, $$ - ϵ 2 Δ v ( x ) + a · ∇ v ( x ) = 0 , with constant $$\epsilon >0$$ ϵ > 0 and real constant vector $$\boldsymbol{a}=(a_1,a_2,\ldots ,a_n)^\textrm{T}\ne \boldsymbol{0}$$ a = ( a 1 , a 2 , … , a n ) T ≠ 0 .

Date: 2026
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DOI: 10.1007/978-981-95-1088-7_4

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