Vol. 32, No. 1, 1970

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On an initial value problem in the theory of two-dimensional transonic flow patterns

Stefan Bergman

Vol. 32 (1970), No. 1, 29–46

In the case of the differential equation

       ∂2ψ-  ∂2ψ-    ∂ψ-
L (ψ ) = ∂λ2 + ∂ 𝜃2 + N ∂λ = 0,N ≡ N (λ,𝜃),

where N is an analytic function, the integral operator of the first kind

       ∫ 1                        ∘ -----
P (f) d=ef     E (λ,𝜃,t)f(ζ(1 − t2)∕2)dt∕  1 − t2

transforms analytic functions of a complex variable ζ = λ + i𝜃 into solutions of L(ψ) = 0. Here E is a fixed function which depends only on L, while f(ζ) is an arbitrary analytic function of the complex variable ζ;f is assumed to be regular at ζ = 0. Using this operator, one shows that many theorems valid for analytic functions of the complex variable can be generalized for the solutions ψ of L(ψ) = 0. Continuing ψ(λ,𝜃) to complex values U = λ + iΛ and setting λ = 0, one shows that many theorems in the theorems in the theory of functions of a real variable can be generalized to the case of solutions of

        ∂2ψ   ∂2ψ      ∂ψ
H(ψ) ≡ −∂-Λ2 +-∂𝜃2 − iN ∂Λ-= 0.

By change of the variables,

         2        2
M  (ψ ) ≡ ∂-ψ-+ l(x)∂-ψ-= 0,
∂x2      ∂y2

l(x) > 0 for x < 0,l(x) < 0 for x > 0,l(O) = 0, when considered for x < 0 can be reduced to the equation L(ψ) = 0. The variables can be chosen so that U = 0 corresponds to x = 0. However, in this case the function N(λ) becomes singular at λ = 0. Nevertheless, one can apply the theory of the so-called integral operators of the second kind. If ψ(0,𝜃) = χ1(𝜃) and

  lim  ψM (M, 𝜃) = χ2(𝜃)
M →1−

are given, one can determine the function f. Here M is the Mach number. In this way one can determine from χ1 and χ2 the location and character of singularities of ψ in the subsonic region. When considering ψ in the supersonic region, one can show that some theorems on functions of one real variable can be generalized to the case of certain sets of particular solutions ψν,𝜃)= 1,2, , of H(ψ) = 0.

Mathematical Subject Classification
Primary: 76.35
Received: 10 February 1969
Published: 1 January 1970
Stefan Bergman