Abstract :
Consider the following six coupled Volterra integral equations of the second kind for the unknown functions ϱ+(r, λ) and ϱ−(r, λ) defined in the two-dimensional plane of independent variables r and λ. Define View the MathML source to be the domain in the (r, λ) plane within which the solution to (1) is desired, and ∂View the MathML source to be the boundary about which conditions on the unknown functions ϱ±(r, λ) are prescribed. The boundary conditions on the unknowns are that: (a) along the line r = r0 (constant) the function ϱ+(r0, λ) = 1 for all λ > 0, and (b) along the line r = δ (a small positive constant) the functions are equal, ϱ+(δ, λ) = ϱ−(δ, λ), for all λ. The boundary, ∂View the MathML source, of the domain in the fourth quadrant of the (r, λ) plane is defined by the condition φ(r) = λ, where φ is the solution of Poissonʹs equation: Δφ = 4πϱ. The final boundary condition (c) is that along the curve defined by φ(r) = λ, the functions are again equal: ϱ+(rE, λ) = ϱ−(rE, λ) for all λ ≤ 0, where rE is the solution of the equation φ(r) = λ for a given energy λ.
