[74.1.2.1] Consider therefore briefly the problem
of determining λϕ from eq. (3.2)
given εω and μϕ.
[74.1.2.2] A general theoretical discussion can be given
based on the observation that
eq. (3.2) is now linear.
[74.1.2.3] It can be written as

∫01Kω;ϕλϕdϕ=fω, | | (7.1) |

where

Kω;ϕ | =μϕεCω;ϕ-εωεCω;ϕ+2εω-εBω;ϕ-εωεBω;ϕ+2εω, | | (7.2) |

fω | =-∫01εBω;ϕ-εωεBω;ϕ+2εωμϕdϕ. | | (7.3) |

[74.1.2.4] Equation (7.1) is a linear
Fredholm integral equation of the first kind.
[74.1.2.5] Because the kernel Kω;ϕ is not symmetric, define

K1ω,ω′ | =∫01Kω,ϕKω′,ϕdϕ, | |

K2ϕ,ϕ′ | =∫01Kω,ϕKϕ,ϕ′dω. | |

[74.1.2.6] General results can be employed
to solve eq. (7.1)
if fω is continuous
and such that ∫01fω2dω exists,
and if ∫01∫01Kiω;ϕdωdϕ
exists and Kiω;ϕ is piecewise
continuous in 0≤ω≤1, 0≤ϕ≤1.
[74.1.2.7] The Ki are symmetric and have eigenvalues λn2.
[74.1.2.8] The normal modes called Ωni
and Φni can be chosen orthonormal and satisfy

| λn∫01Kω;ϕΩniϕdϕ=Φniω, | |

| λn∫01Kω;ϕΦniωdω=Ωniϕ. | |

[74.2.0.1] These modes are used to solve eq. (7.1).
[74.2.0.2] If (7.1) has any solution,
then the inhomogeneity fω
can be written as

fω=∑n,iΦniω∫01fω′Φniω′dω′. | | (7.4) |

[74.2.0.3] It is assumed that the two sets Ωni and Φni
have been made orthonormal.
[74.2.0.4] Then the solution to eq. (7.1)
is given as

λϕ=∑i,nλnΩniϕ∫01fω′Φniω′dω′. | | (7.5) |

[74.2.0.5] The eigenvalues λn2 are the solutions of Dλ2=0, where

| Dλ2=∑n=0∞κnλ2n, | |

| κ0=1,Ki0x,y=0, | |

| κn=-1n∫01Kinx,xdx, | |

| Kinx,y=∫01Kix,tKin-1t,ydt+κn-1Kix,y. | |

[74.2.0.6] These brief remarks about the inverse problem
are intended to outline the general characteristics
of the problem.
[74.2.0.7] A more detailed discussion must await
the availability of experimentally
observed local porosity distibutions.[27]