# Reduction of  Boundary Value Problem to Possio Integral Equation in Theoretical  Aeroelasticity

Authors: A. V. Balakrishnan; M. A. Shubov
Journal: Journal of Applied Mathematics
 (2008)

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## Abstract

The present paper is the first in a series of works devoted to the solvability of the
Possio singular integral equation. This equation relates the pressure distribution over a
typical section of a slender wing in subsonic compressible air flow to the normal velocity
of the points of a wing (downwash). In spite of the importance of the Possio equation,
the question of the existence of its solution has not been settled yet. We provide a
rigorous reduction of the initial boundary value problem involving a partial differential
equation for the velocity potential and highly nonstandard boundary conditions to a
singular integral equation, the Possio equation. The question of its solvability will be
addressed in our forthcoming work.


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## Body

## 1. Introduction

The present paper is the first in a series of three works devoted to a systematic study of a specific singular integral
equation that plays a key role in aeroelasticity. This is the Possio integral
equation that relates the pressure distribution over a typical section of a
slender wing in subsonic compressible air flow to the normal velocity of the
points on a wing surface (downwash). First derived by Possio  [1], it is an
essential tool in stability (wing flutter) analysis. In spite of the fact that
there exists an extensive literature on numerical analysis of the Possio (or
modified Possio) equation, its solvability has never been proved rigorously. In
our study, we focus on subsonic compressible flow. The problem of a
pressure distribution around a flying wing can be reduced to a problem of
velocity potential dynamics. Having a pressure distribution over a flying wing,
one can calculate forces and moments exerted on a wing due to the air flow,
which is an extremely important component of a wing modeling. Mathematically,
the problem can be formulated in the time domain in the form of a system of
nonlinear integrodifferential equations in a state space of a system (evolution-convolution
equations). (As illustrating example for the case of an incompressible air flow, see  [2–4].) So, the problem of stability involves as an essential
part a nonlinear equation governing the air flow. Assuming that the flow is
subsonic, compressible, and inviscid, we will work with the linearized
version together with the Neumann-type boundary condition on a part of the
boundary. It can be cast equivalently as an integral equation, which is exactly the Possio integral equation. Our main goal is to prove the solvability
of this equation using only analytical tools. We should mention here that most
of the research is currently done by different numerical methods, for example,
[5, 6], and very few papers on analytical treatment of the Possio equation are
available  [7, 8]. In essence, the partial differential equations are
approximated by ordinary differential equations for both the structural
dynamics and aerodynamics. However, it is important to retain full continuous
models. It is the Possio integral equation that is the bridge between
Lagrangian structural dynamics and Eulerian aerodynamics. In particular, using
the solution of the Possio equation, one can calculate the aerodynamic loading
for the structural equations. As the simplest structural model (Goland model
[9, 10]), let us consider a uniform rectangular beam (see Figure 1) endowed with
two degrees of freedom, plunge and pitch. Let the flow velocity be along the
positive x-axis with x denoting the cord variable, −b⩽x⩽b;
let y be the span or length variable along the y-axis, 0⩽y⩽l.
Let h be the plunge, or bending, along z-axis; let θ be the pitch, or torsion angle, about the
elastic axis located at x=ab,
where 0<a<1 and ab is the distance between the center-line and
the pitch axis; let X(y,t)=(h(y,t),θ(y,t))T (the superscript “T” means the transposition). Then the structural
dynamics equation is(1.1)MXtt(y,t)+KX(y,t)=(𝕃(y,t),𝕄(y,t))T,where M is the mass–inertia matrix and K is the
stiffness differential operator, K=diag{EI(∂4/∂y4),−GJ(∂2/∂y2)},
with EI and GJ being the bending and torsion stiffness,
respectively. 𝕃(y,t) and 𝕄(y,t) are the aerodynamic lift and moment about the
elastic axis. System (1.1) is considered with the boundary conditions h(0,t)=h′(0,t)=θ(0,t)=0;h′′(l,t)=h′′′(l,t)=θ′(l,t)=0.
Certainly the structure model can be and has been extended to several degrees
of freedom (but still linear) or to nonlinear models  [11, 12]. For example, in
[12] complete and general nonlinear theory with particular emphasis given to
the fundamentals of the nonlinear behavior has been developed. The theory is
intended for applications to long, straight, slender, homogeneous isotropic
beams with moderate displacements, and is accurate up to the second order
provided that the bending slopes and twist are small with respect to unity.
Radial nonuniformities (mass, stiffness, twist), cordwise offsets of the mass
centroid and tension axis from the elastic axis, and wrap of the cross-section
are included into the structural equations. The nonlinearities can be important
in determining the dynamic response of cantilever blades, and they are
especially important in determining the aeroelastic stability of torsionally
flexible blades. One can also add to (1.1) a control term as in  [13, 14].
However, the emphasis in the present paper is on aerodynamics—by far the most
complicated part—and the structure interaction, specifically, on force and
moment terms in (1.1).Figure 1
Wing structure beam model.Now we mention that the distinguishing feature of the
aerodynamic model is that the air flow is assumed to be nonviscous. The flutter
instability phenomena are captured already in nonviscous flow. Hence, the
governing field equation is no longer the Navier-Stokes equation but the Euler
full potential equation, and the main assumption is that the entropy is constant.
As a result, the flow is curl-free and can be described in terms of a velocity
potential. The unknown variable in the Possio equation is the velocity
potential. Let us show that the velocity potential yields explicit expressions
for the lift and moment from (1.1) (see, e.g., [9]).The aerodynamic lift and moment (per unit length) are
given by the formulas(1.2)𝕃(y,t)=∫−bbδpdx,𝕄(y,t)=∫−bb(x−a)δpdx,where p is pressure and δp=p(x,y,0+,t)−p(x,y,0−,t),0⩽y⩽l,|x|<b. As already mentioned, the velocity potential φ(x,y,z,t) satisfies the Euler full potential equation
[8]. If one can derive a representation for the potential φ,
then the following formula can be used for pressure
calculation:(1.3)p(x,y,z,t)=p∞(1+γ−1a∞2(U22−∂φ∂t−∇φ⋅∇φ2))γ/(γ−1),where p∞=ρ2a∞2γ−1,a∞ is the far-field speed of sound, ρ is the air density, and γ>1 is the adiabatic constant; U is a speed of a moving wing. So knowing the
velocity potential φ,
we immediately obtain δp and thus lift 𝕃 and moment 𝕄.Any aeroelastic problem breaks into two parts. First,
it is a field equation (for a linearized version of the field equation for a
velocity potentialφ,
see (2.2)) with the boundary conditions, that is, (a) flow tangency condition
(see (2.3)), (b) Kutta-Joukowski condition (see (2.5)), and (c) far-field
conditions (see (2.6)). It is the flow tangency condition that establishes the
connection between the structural and aerodynamical parts since the downwash
function wa of (2.3) is expressed explicitly in terms of
the plunge h and pitch θ.
We mention here that the unique feature of the aerodynamic boundary conditions
is that we have the Neumann condition only on a part of the boundary z=0,
that is, only on the structure, and a related but different condition on the
rest of the boundary, which is purely aerodynamic (not involving the structure
dynamics). The second part involves the structure state variables via flow
tangency condition (2.3) and more importantly the lift and moment.Summarizing all of the above, one can see that the
entire aeroelastic problem (structure and aerodynamics) becomes in general a
nonlinear convolution (due to the expressions for the lift and moment) and/or
evolution equation in terms of the structure state variables, whose stability
with respect to the parameterU is then the “flutter problem” one has to
resolve.Dealing with the “flutter problem,” one cannot
proceed without mentioning that it was an important topic of famous
Th.Theodorsen research (see, e.g.,  [15]). The determination of pressure
distribution and aerodynamic loads on an airfoil exposed to a two-dimensional
stream of incompressible fluid flow was a central problem in aeronautics
of the early 1930's. Flutter was first encountered on tail-planes and wings
during World War I, but rigorous theory for its prediction took many years to
develop. The greatest challenge was to supply aerodynamic terms for the
governing equations. This was the strongest motivation for research on
air-loads experienced by wings and airfoils performing time-dependent motions.
It was recognized that the first step would be to adapt the methods of the
“thin-airfoil theory” so as to account for phenomena such as small
oscillations normal to the directions of flight and impulsive changes of angle
of attack. The main feature of the analytical scheme in Theodorsen's technical
report  [15] (which is now a classical paper) was related to the Joukowski
transformation between parallel-stream flows past a circle and a zero-thickness
flat plate. Even though this looks as oversimplification, the author of TR−496 (see [15]) knew that within the framework
of thin-airfoil theory, the steady-flow problems of thickness and camber could
be rigorously separated from the unsteady case, on which he focused. He
correctly enforced the Kutta condition in the presence of infinite wake of
trailing vortices. The main discovery of this report is a set of complex
frequency-response functions connecting vertical translation (or bending) and
angle of attack (or torsion) as “inputs” with unsteady lift and
pitching moment as “outputs.” The most important result of the entire
investigation was that regardless of the nature of the small oscillation or of
the “output” quantity to be found, only a single transcendental
function appears in their relationship. It is the well-known “Theodorsen
function”(1.4)𝒞(k)=F(k)+iG(k)=H1(2)(k)H1(2)(k)+iH0(2)(k),with H0,1(2)(k) being the Hankel functions of the zero or
first order, respectively, and of the second kind. This exact flutter solution
including results for control surfaces has had a keystone role in the flutter
analysis. It is interesting to note that Theodorsen function 𝒞(k) occurs in the theory of propulsion of birds
and fish as well. With the aerodynamic terms constructed by
adapting the formulas and ideas in TR−496  [15], it becomes possible to predict
critical “flutter boundaries” quite accurately.One of the most complete and important sources related
to contemporary status of aeroelasticity is found in  [16], where theoretical
methods are combined with experimental and numerical results providing better
understanding of the theory and its limitations. As demonstrated in  [16], much
of the theoretical and experimental developments can be applied to different
engineering areas, and a common language can be used for explanation of
different phenomena. Even though historically the entire field of aeroelasticity
has centered in aeronautical applications, now the applications are found in
civil engineering (on flows about bridges and tall buildings, see  [16, Chapter 6] and  [17]); in mechanical engineering (on flows about
turbomachinery blades and fluid flows in flexible pipes, see  [16, Chapters 3, 7, 8, and 12] and  [18]); in
nuclear engineering (on flows about fuel elements and heat exchanger vanes).
Moreover, aeroelasticity plays a crucial role in the development of new
aerospace systems such as unmanned air vehicles (UAVs)  [19, 20].Before we turn to the description of the results of
the present paper, we would like to mention earlier efforts to use integral
transformations in order to calculate lift and pitching moment on a wing of given shape. Venters [21] presents a lifting surface theory for steady
incompressible flows based on a shear flow rather than a potential flow model.
The theory developed in  [21] is intended to account for the boundary layer. The
method of Fourier transforms is used to evaluate the pressure on a surface of
infinite extent and arbitrary contour. The research initiated in  [21] is
continued in  [22, 23], where a general theory of planar disturbances of inviscid
parallel shear flows has been developed. This theory has been successfully
applied to such problems as the generation of waves at a free surface, the
interaction of a boundary layer with a flexible wall, the flow about a wing in
or near a jet or wake, and the influence of the main boundary layer of a wing
on control surface effectiveness.Now we briefly describe the content of the present
paper. In Section2, we give a rigorous formulation of an initial boundary
value problem for a partial differential equation for a disturbance potential
(see (2.2)). We discuss the Lp-space, 1<p<2,
setting for the equation and the boundary conditions. In Section 3, we present
a new form for the initial boundary value problem by applying two integral
transformations, which yields Laplace transform in time variable and Fourier
transform in x-variable of the unknown function. Such a
double transform allows us to give the first version of the Possio equation
(see (3.22)). Evidently, (3.22) is only the convenient initial point for the
next move, that is, eliminating the Fourier transform representation. Section
4, being just a technical result, is very important. In this section, we
prepare the main equation from (3.22) for Mikhlin multiplier theory
application. In Section 5, we transform (3.22) to a singular integral equation,
which is singular in more than one sense. Brief discussion and conclusions are
given in Section 6.In the conclusion of the introduction, we briefly
outline the original version of the Possio equation  [1], and provide some
justification that a different version of it, derived and studied in the
aforementioned series of our papers, is more convenient for analytical
investigation.The derivation of the original Possio integral
equation  [1], that can be found in  [9], is based on representing the airfoil
with a sheet of acceleration potential doublets along the projection of the
airfoil. The doublet is obtained from a simple solution of (6.80) in  [9] (which
coincides with (2.8) of the present paper) known as a source pulse. Following the
steps presented in  [9], one obtains a sinusoidally pulsating doublet of some
frequency ω.
After nontrivial calculations, one arrives at the Possio integral equation of
the form(1.5)wa(x)¯=−ωρ∞u2P.V.∫−bbΔpa¯ℜ(M,k(x−ξ)b)dξ,|x|<b.The problem is to find an
operator, which is inverse to the above integral operator understood as a
Cauchy principal value integral. The kernel function ℜ is extremely complicated and is given
explicitly by the following formula:(1.6)ℜ=141−M2{exp(ikM2(x−ξ)b(1−M2)2)[iM|x−ξ|x−ξH1(2)(kM|x−ξ|b(1−M2))−H0(2)(kM|x−ξ|b(1−M2))]+i(1−M2)exp(ik(x−ξ)b)[2π1−M2ln(1+1−M2M)+∫0k(x−ξ)/b(1−M2)eivH0(2)(M|v|)]},where M is the Mach number, k=ωb/U is the reduced frequency, Hj(2),j=0,1, is the Hankel function of the second kind and
of the zero or first order, respectively. By examination, one can see that
(1.5) is very complicated not only because its kernel (1.6) contains special
functions with nonstandard arguments, but also because it is really difficult
to describe explicitly the nature of singularities. Contrary to (1.5), in the
equation that we study in our series of papers, we can precisely formulate the
nature of singularities. Namely, in our version, the Possio integral equation
is singular by two reasons: (a) it contains singular integral operations, that
is, the finite Hilbert transformation and its specific “inverse;” (b)
one integral equation contains integral operators being different by their
nature, that is, the finite Hilbert transformation, Volterra integral operator,
and an integral operator with the degenerate kernel. Each type of integral
operation requires its own approach, which makes the problem such a challenge.
Regarding the Possio integral equation in its original form, C. Possio himself
attempted to solve the equation using numerical integration technique. He
presented an unknown function in the form of a series: Δpa¯=A0cot(θ/2)+Σn⩾1Ansin(nθ),ξ/β=cosθ.
However, this substitution does not permit a straightforward inversion, and
also convergence of the series is not always rapid. Many attempts have been
made to develop a good numerical scheme for inversion of the integral in the
Possio equation, but rigorous proof of the solvability of the equation has
never been produced. In our series of papers, we are addressing the problem of
the unique solvability using rigorous analytical tools.


## 2. Aerodynamic Field Equation

In this
section, we start with the initial boundary value problem for a partial
differential equation, which is known in the literature as the “small
disturbance potential field equation” for subsonic inviscid compressible flow
[8]. We assume that the air flow is around a large aspect-ratio planar wing,
which means that the dependence on the span variable along the wing is
neglected. The wing is then reduced to a “typical section” or a “chord.”We will use the following notations:U is the free-stream velocity; a∞ is the sound speed; M=U/a∞ is the Mach number 0<M⩽1.The velocity potential of the airflow is given by the
expression(2.1)Ux+φ(x,z,t),−∞<x<∞,0⩽z<∞,t>0,where Ux is the free-stream velocity potential, and φ is a small perturbation of the velocity
potential. The disturbance potential φ satisfies the following linearized field
equation  [7]:(2.2)φtt(x,z,t)+2Ma∞φtx(x,z,t)=a∞2(1−M2)φxx(x,z,t)+a∞2φzx(x,z,t),−∞<x<∞,0⩽z<∞,t>0. Together with this equation, we introduce the
following boundary conditions.(1)The flow
tangency (or nonseparable flow) condition is(2.3)∂∂zφ(x,z,t)|z=0=wa(x,t),|x|<b,where wa(x,t) is the given normal velocity of the wing (the
downwash); b is a size of a “half chord.” We note
that condition (2.3) is a nonhomogeneous Neumann condition prescribed only on a part of the boundary, z=0.(2)The
Kutta-Joukowski conditions. To formulate these conditions, we need one more
function, that is, the acceleration potential defined by(2.4)ψ(x,z,t)=φt(x,z,t)+Uφx(x,z,t).The conditions below reflect the
following physical situation: pressure off the wing and at the trailing edge
must be equal to zero. These conditions are(2.5)ψ(x,0,t)=0for|x|>b,limx→b−0ψ(x,0,t)=0.(3)Far-field
conditions. The disturbance potential and velocity tend to zero at a large
distance from the wing:(2.6)φ(x,z,t)→0,∇φ(x,z,t)→0,as|x|→∞orz→∞.Now we present a functional-analytic reformulation of
problem (2.2)–(2.6) that will be used in the rest of the paper. We will
consider the boundary value problem (2.2)–(2.6) in Lp(−∞,∞),1<p<2, assuming that the initial conditions are
trivial, that is, φ(x,z,0)=φt(x,z,0)=0. It will be clear from the analysis below that
it is essential to use p≠2.Now we describe
the function space for our future solution. We assume that the functionφ(x,z,t) is absolutely continuous with absolutely
continuous first derivatives with respect to the variables z and t.
Regarding the properties of φ as a function of x,
we make the following assumption. Let Dx be a closed linear operator in Lp(−∞,∞) corresponding to the partial derivative ∂/∂x.
Then, we require that(2.7)φ(⋅,z,t)∈𝒟(Dx2),with 𝒟 being the domain of  Dx2, and Dxφ(⋅,z,t) being absolutely continuous with respect to t.We require that (2.2) be
satisfied in the sense that(2.8)∂2∂t2φ(⋅,z,t)+2Ma∞∂∂tDxφ(⋅,z,t)=(a∞2(1−M2)Dx2+a∞2∂2∂z2)φ(⋅,z,t).The flow tangency condition (2.3) will be understood in the sense that(2.9)∫−bb|∂∂zφ(x,z,t)−wa(x,t)|pdx→0asz→0.The Kutta-Joukowski
conditions (2.5) will be written in integral form as well. We note that due
to our assumptions on φ,
the acceleration potential is well defined. Conditions (2.5) will be replaced
with the following requirements:(2.10)∫−∞∞|ψ(x,z,t)−ψ(x,0,t)|pdx→0asz→0,∫−∞−b|ψ(x,0,t)|pdx+∫b∞|ψ(x,0,t)|pdx=0.Regarding the initial
conditions, we require(2.11)∥φ(⋅,z,t)∥Lp(−∞,∞)→0,∥∂∂tφ(⋅,z,t)∥Lp(−∞,∞)→0ast→0,0<z<∞.In the next sections, we reduce
the initial boundary value problem (2.8)–(2.11) to a specific singular integral
equation,  the Possio integral equation. This derivation is quite
nontrivial.


## 3. Modification of (2.8) Using Fourier and Laplace Transformations

Letf(⋅,z,t)∈Lp(−∞,∞),z>0,t>0, be a function that has the following
property:(3.1)∫0∞∥f(⋅,z,t)∥Lp(−∞,∞)e−σtdt<∞forσ⩾σa>0.In the future, we will omit the
subindex Lp(−∞,∞) using the notation ∥⋅∥ for the Lp norm. Due to (3.1), we can define the Laplace
transform of f with respect to the time variable. We denote
the Laplace transform of a function f by the same letter capitalized (F). So, if φ(⋅,z,t) belongs to the class of functions satisfying
(3.1), then the Laplace transform of φ(⋅,z,t) is(3.2)Φ(⋅,z,λ)=∫0∞φ(⋅,z,t)e−λtdt,ℜλ⩾σa>0,0<z<∞.It is clear that from
(2.8) for φ(⋅,z,t) we obtain a new equation for Φ(⋅,z,λ):(3.3)λ2Φ(⋅,z,λ)+2Ma∞λDxΦ(⋅,z,λ)=a∞2(1−M2)Dx2Φ(⋅,z,λ)+a∞2Φzz(⋅,z,λ).In the next step, we apply the
Fourier transformation with respect to the variable x to (3.3), and have (“overhat” means the
result of the Fourier transformation)(3.4)Φ^(ω,z,λ)=∫−∞∞Φ(x,z,λ)e−iωxdx,−∞<ω<∞.As it is well known, the Fourier
transformation is a bounded linear operator from Lp(−∞,∞) into Lq(−∞,∞)(p−1+q−1=1),1<p<2. Applying the spatial Fourier transformation to
(3.3), we obtain a new equation for Φ^:(3.5)(λ2+2iωλMa∞)Φ^(ω,z,λ)=−a∞2(1−M2)ω2Φ^(ω,z,λ)+a∞2Φ^zz(ω,z,λ).Rearranging terms in this
equation, we obtain(3.6)Φ^zz(ω,z,λ)=(1−M2)ω2Φ^(ω,z,λ)+(λ2a∞2+2iωλMa∞)Φ^(ω,z,λ).Recalling that U/a∞=M and denoting λ˜=λ/U,
we rewrite (3.6) as(3.7)Φ^zz(ω,z,λ)=(λ˜2M2+2iωλ˜M2+ω2(1−M2))Φ^(ω,z,λ).The quadratic polynomial (λ˜2M2+2iωλ˜M2+ω2(1−M2)) will play an important role in the sequel. We
make the following agreement: let us keep the notation λ for λ˜ and use the notation λ′ for the original λ;
that is, with this agreement, (3.7) obtains the following form:(3.8)Φ^zz(ω,z,λ′)=(λ2M2+2iωλM2+ω2(1−M2))Φ^(ω,z,λ′),λ′=λU.Let(3.9)𝒟(ω,λ)=λ2M2+2iωλM2+ω2(1−M2).Let us show that |𝒟(ω,λ)| is bounded below when ℜλ⩾σa>0. Indeed,(3.10)|𝒟(ω,λ)|=|M2(iω+λ)2+ω2|⩾M2|λ+iω|2⩾M2|ℜλ|2⩾M2σa2U2>0.From this estimate, it follows
that we can define the “positive” square root that we denote by 𝒟(ω,λ) such that(3.11)ℜ𝒟(ω,λ)>0.Thus, from (3.11) it follows
that the differential equation (3.8) has a unique solution satisfying the Far-field
conditions, and this solution is(3.12)Φ^(ω,z,λ′)=Φ^(ω,0,λ′)exp(−z𝒟(ω,λ)).Now we have to satisfy the Kutta-Joukowski
conditions and the flow tangency condition.To write theflow tangency condition, we need a
derivative with respect to z.
Let(3.13)ν^(ω,z,λ′)=ddzΦ^(ω,z,λ′)=−𝒟(ω,λ)Φ^(ω,z,λ′).In terms of ν^, (3.12) can be written as(3.14)Φ^(ω,z,λ′)=−ν^(ω,0,λ′)𝒟(ω,λ)exp(−z𝒟(ω,λ)).We notice that the Laplace
transform of the acceleration potential is(3.15)Ψ(ω,z,λ′)=λ′Φ(x,z,λ′)+UΦ(x,z,λ′)and, therefore, using (3.14), we
obtain(3.16)Ψ^(ω,z,λ′)=(λ′+iωU)Φ^(ω,z,λ′)=−(λ′+iωU)ν^(ω,0,λ′)𝒟(ω,λ)exp(−z𝒟(ω,λ)).From (3.16), we obtain
that(3.17)Ψ^(ω,0,λ′)=−(λ′+iωU)ν^(ω,0,λ′)𝒟(ω,λ).At this point, we introduce the
standard notation, that is,(3.18)𝒜(x,t)=−2Uψ(x,0,t).We will denote the Laplace
transform of 𝒜 with respect to the time variable by A.
Multiplying (3.17) by 2/U,
we have(3.19)A^(ω,λ′)=2(λ+iω)𝒟(ω,λ)ν^(ω,0,λ′),or in another
form(3.20)ν^(ω,0,λ′)=𝒟(ω,λ)2(λ+iω)A^(ω,λ′),with ν^ being defined in (3.13).We notice that if we knowA^(ω,λ′), then we know the velocity potential, that
is,(3.21)Φ^(ω,z,λ′)=−12(λ+iω)exp(−z𝒟(ω,λ))A^(ω,λ′).Finally, we obtain the following
problem for A^:(3.22)ν^(ω,0,λ′)=𝒟(ω,λ)2(λ+iω)A^(ω,λ′),ν(x,0,λ′)=Wa(x,λ′),|x|<b,A(x,λ′)=0,|x|>b.System (3.22) is a nonstandard
boundary value problem. Indeed, the right-hand side of the first equation in
(3.22) has a product of the two functions and, therefore, this product
corresponds to a convolution of the function of x corresponding to 𝒟(ω,λ)/2(λ+iω) and 𝒜(x,λ′). On the other hand, if we restrict x to the interval [−b,b], we know the left-hand side in x-representation (it is Wa(x,λ′)). Therefore, the first equation from (3.22)
is indeed an integral equation for 𝒜(x,λ′) provided that we can restore a function
corresponding to the multiple 𝒟(ω,λ′)/2(λ+iω). In the next section, we obtain the first main
result of the paper. We present a careful derivation of the function of x whose Fourier transformation is given by the
aforementioned multiple. This result yields the Possio integral equation (see
Theorem 5.13), which will be derived in Section 5.


## 4. Main Technical Result for Reconstruction of the Inverse Fourier Transform of(3.20)

The main
purpose of this section is representing the right-hand side of (3.20) in the
form of a function of x depending parametrically on λ. As a consequence, incorporating second and
third conditions from (3.22) we immediately obtain the desired integral
equation. Let(4.1)𝔻(ω,λ)=𝒟(ω,λ)λ+iω.By direct calculations we
obtain(4.2)𝔻(ω,λ)=|ω|1−M2λ+iω1+λM21−M2λ+2iωω2≡|ω|iω1−M2(1+𝒬(ω,λ)),where(4.3)𝒬(ω,λ)=iωλ+iω1+λM21−M2λ+2iωω2−1.The following result is valid
for the function 𝒬(ω,λ).Theorem 4.1.
𝒬(⋅,λ)is a Fourier transform of  a functionq∈Lp(−∞,∞),p>1, given explicitly by the following formulas:(4.4)q(x,λ)={−λe−λx1−M2−λπ∫0α1e−λsxa(x)ds,x>0,λπ∫0α2e−λ|x|sa(−s)ds,x<0,where(4.5)α1=M1+M,α2=M1−M,a(s)=(α1−s)(α2+s)1−s,−α2<s<α2.Proof.
To prove (4.4), we calculate Fourier transform of q directly and show that it is equal to 𝒬(⋅,λ) given in (4.3). So, for the Fourier transform,
denoted by F[q],
we have(4.6)F[q]≡∫−∞∞e−iωxq(x,λ)dx=λπ∫−∞0e−iωxdx∫0α2e−λ|x|sa(−s)ds−λ1−M2∫0∞e−λxe−iωxdx−λπ∫0∞e−iωxdx∫0α1e−λsxa(s)ds=I1+I2+I3.Taking into account that ℜλ⩾σa≫1, we evaluate I2 and have(4.7)I2=−λ1−M2∫0∞e−(λ+iω)xdx=−λ1−M2(λ+iω).Using (4.5), we evaluate I3 and have(4.8)I3=−λπ∫0α1a(s)ds∫0∞e−(λs+iω)xdx=−λπ∫0α1(α1−s)(α2+s)(1−s)(iω+λx)ds.Evaluating I1, we obtain(4.9)I1=λπ∫0α2a(−s)ds∫0∞e(iω−λs)xdx=−λπ∫0α2a(−s)dsiω−λs.Combining together I1 and I3, we obtain(4.10)I1+I3=−λπ∫−α2α1a(s)dsiω+λs.To modify (4.10), let us
introduce a new variable of integration: (4.11)ξ=s+α1α2=s+M(M(1−M2))=s+Mβ,β=M(1−M2).The following relations can be
easily verified: (4.12)α1−s=β−ξ,α2+s=M1−M+ξ−M21−M2=β+ξ;1−s=ξ+11−M2,iωλ+s=iωλ+ξ−Mβ.Using (4.12) and setting z=(iω/λ),
we rewrite (4.10) as (4.13)I1+I3=−1π∫−ββ(β−ξ)(β+ξ)dξ(1−ξ+Mβ)((iω/λ)+ξ−Mβ)=−1π∫−βββ2−ξ2dξ(1−ξ+Mβ)(z+ξ−Mβ).Making transformation ξ↦−ξ in (4.13) and taking into account that (1/β)+M=1/M,
we get (4.14)I1+I3=−1π∫−βββ2−ξ2dξ(1+ξ+Mβ)(z−ξ−Mβ)=1π∫−ββ1−(ξ/β)2dξ((1/β)+M+(ξ/β))((ξ/β)+M−(z/β))β=1π∫−ββ1−(ξ/β)2dξβ((1/M)+(ξ/β))((ξ/β)+M−(z/β)).Let η=ξ/β, then (4.15)I1+I3=1π∫−111−η2dη(η+(1/M))(η+M−(z/β))=βπ(1+z)(i1−i2),where (4.16)i1=∫−111−η2dη(η+M−(z/β)),i2=∫−111−η2dηη+(1/M).To evaluate the second
integral of (4.16), we need the statement below, which will be proved after the
proof of Theorem 4.1.
Lemma 4.2.
The
following formula holds for|a|>1: (4.17)∫−111−x2x+adx=π(a−sign(ℜa)a2−1).
To evaluate i1 from (4.16), we can derive a formula similar to
(4.17). However, we notice that the integrand of i1 is of the form 1−η2/(η+ϰ),
where ϰ is a complex number with a nonvanishing
imaginary part (except for the one point where ϰ=M<1). For such ϰ,
one can reconstruct the proof of the formula similar to (4.17) and have (4.18)i1=π(M−zβ+(M−zβ)2−1).Substituting 1/M for a in (4.17), we obtain (4.19)i1−i2π=M−zβ=(M−zβ)2−1−1M−1M2−1=−1+zβ+1−M2M+1βz2−β2(1−M2)−2zMβ=1+zβ[−1+11+z11−M2+z2−2zγ−γ1+z],where γ=βM and β is defined in (4.11).
Returning to (4.15), we obtain (4.20)I1+I3=−1+1(1+z)1−M2+11+zz2−2zγ−γ.Now we can finalize the
expression for 𝒬 combining (4.7) with (4.20) to have (4.21)𝒬(x,λ)=−λ1−M2(λ+iω)+1(1+z)1−M2+z1+z1−2γz−γz2−1.Recalling that z=iω/λ and noticing that (2γ/z)+(γ/z2)=(2Mβλ/iω)−(Mβλ2/ω2), we obtain from (4.21) that(4.22)𝒬(x,λ)=iωλ+iω1−2Mβλiω+Mβλ2ω2−1=iωλ+iω1+2Mβλiω+Mβλ2ω2−1=iωλ+iω1+λMβω2(λ+2iω)−1.This formula coincides with
(4.3). To complete the proof of the theorem, it remains to show that q∈Lp(−∞,∞). As the first observation, we notice that it
suffices to prove the result for the case λ=1. Indeed, from (4.4) we obtain(4.23)|q(x,λ)|⩽|λ||q(x,σ)|,σ=ℜλ⩾σa>0.Thus,(4.24)∫−∞∞|q(x,λ)|pdx⩽|λ|p∫−∞∞|q(x,σ)|pdx=|λ|p∫−∞∞|q(σx,1)|pdx=|λ|pσ∫−∞∞|q(ξ,1)|pdξ.Taking into account |a(s)|⩽C<∞ for −α2<s<α1, we get the following estimates: (4.25)|∫0α1e−xsa(s)ds|⩽C1−e−α1x1+x,x>0,|∫0α2e−|x|sa(−s)ds|⩽C1−e−α|x|1+|x|,x<0.From estimates (4.25) we obtain(4.26)∫−∞∞|q(x,1)|pdx⩽C[∫−∞0(1−e−α2|x|1+|x|)pdx+∫0∞(1−e−α1x1+|x|)pdx]<∞forp>1.This estimate means that q∈Lp(−∞,∞).
The theorem is completely shown.Now we complete the proof of the lemma.Proof of Lemma4.2.
We consider the following integral:(4.27)I=∫−111−x2x+adx,|a|>1.Let x=cosθ,
then (4.28)I=∫0πsin2θcosθ+adθ=12[∫0πdθcosθ+a−∫0πcos2θcosθ+adθ]≡12(j1−j2).Evaluating j1,
we have (4.29)j1=∫−ππdθeiθ+e−iθ+2a.Integrating in the complex
plane, we assume that z=eiθ,
and thus linear integral can be represented as a contour integral along a unit
circle centered at the origin: (4.30)j1=1i∮C1(0)dzz(z+z−1+2a)=1i∮C1(0)dzz2+2za+1.The roots of the denominator are z1=−a+a2−1,z2=z1−1=−a−a2−1. First, we show that for |a|>1, neither of the roots lie on the circumference.
Indeed, assume that |z1|=1, then |z2|=1. Therefore, (4.31)z1=−a+a2−1=eiψ,z2=−a−a2−1=e−iψ.From (4.31), it follows that a=−cosψ which contradicts our assumption, |a|>1. Thus, one of the roots is inside the circle C1(0)={z:|z|=1} and the second one is outside. Let |z1|=|−a+a2−1|<1, then by the residue theorem we have (4.32)j1=1i∮dz(z−z1)(z−z2)=πa2−1.Finally, we evaluate j2 from (4.28): (4.33)j2=12∫−ππcos2θcosθ+adθ=12∫−ππe2iθ+e−2iθeiθ+e−iθ+adθ=12i∮C1(0)(z2+z−2)dzz2+2az+1=12i∮C1(0)dz(z4+1)z2(z−z1)(z−z2).We have to evaluate the residues
at the points z=0 and z≡z1=−a+a2−1. We have(4.34)(a)Res[f;0]=π(z4+1z2+2az+1)′|z=0=−2πa,(b)Res[f;z1]=πz2+z−22z2−1|z=−a+a2−1=π(a−a2−1)2+(a+a2−1)22a2−1=π2a2−1a2−1.Using (4.34), we get (4.35)j2=Res[f;0]+Res[f;z1]=πa2−1(2a2−1−2aa2−1)=πa2−1(a−(a2−1))2,which together with (4.32)
yields (4.36)j1−j2=πa2−1(1−(a−a2−1)2)=πa2−1(1−a2+aa2−1)=2π(a−a2−1).Inserting (4.36) into (4.28), we
obtain (4.17).
The lemma is completely shown.Remark 4.3.
Now we outline how the results obtained in this section will be used for
rewriting the first equation of (3.22) in the form of an integral equation with
respect to the unknown function A(⋅,λ′). If we consider that integral equation for |x|<b, then the left-hand side will be given
explicitly as the Laplace transform of the given downwash function Wa(⋅,λ′). The right-hand side will be the convolution of
the inverse Fourier transforms of the functions from the right-hand side of
(3.22). In other words, the right-hand side will be given as an integral
convolution operation over unknown function A(⋅,λ′).To achieve our goal, we will use (4.4), which gives an explicit function q whose Fourier transform is (4.3). It turns out
that (4.3) is not a convenient formula to work with. Namely, using (4.4), we
obtain the following Fourier representation for 𝒬: (4.37)𝒬(ω,λ)=−λ1−M21λ+iω−λ∫0α1a(s)λs+iωds+λ∫0α2a(−s)λ−iωds,ℜλ⩾δa>0.In terms of (4.37), we obtain
that our main multiplier 𝔻(ω,λ) (from (4.2)) can be represented in the
form (4.38)11−M2𝔻(ω,λ)=|ω|iω(1+𝒬(ω,λ))=|ω|iω−|ω|iω11−M2λλ+iω−|ω|iωλ∫0α1a(s)dsλs+iω+|ω|iωλ∫0α2a(−s)dsλs−iω,ℜλ⩾δa>0.In the next section, we prove
that each function at the right-hand side of (4.38) is the so-called Mikhlin
multiplier. Evidently each individual multiplier looks simpler than 𝔻(⋅,λ),
which allows us to construct an operator in Lp(−∞,∞) which corresponds to the entire multiplier 𝔻. In particular, we derive the formulas for the
operators corresponding to all the multipliers of (4.38) and then sum them up.


## 5. Mikhlin Multipliers

In this section, we derive the desired form of the
Possio equation as an integral equation, and show that the integral operator in
this equation is bounded inLp(−∞,∞),1<p<2.
The main tool in this derivation is the notion of Mikhlin multipliers  [24].Deffinition 5.1.
Letg and f be two functions from Lp(−∞,∞),p>1, and let G and F be their Fourier transforms. Let there exist a
function μ that relates G and F by the rule (5.1)G(ω)=μ(ω)F(ω).If μ is a continuously differentiable function (with
one possible exception at ω=0) such that (5.2)|μ(ω)|+|ω||μ′(ω)|⩽C<∞,then μ is called a Mikhlin multiplier.Proposition 5.2 (See [24] for the Proof).
Letfandgbe the functions from Definition5.1,and
letμbe a Mikhlin multiplier. Then, there exists
a bounded linear operatorℚinLp(−∞,∞)that relatesfandgby the rule(5.3)g=ℚ[f].The norm of the operatorℚcan be estimated as follows:(5.4)∥ℚ∥Lp(−∞,∞)⩽CMp,whereCis a constant from (5.2), andMpis a constant depending only onp.Our first result is related to the function from (3.20), that is, 𝒟(ω,λ)/2(λ+iω). As pointed out, if we can identify a function
from Lp(−∞,∞) whose Fourier transform coincides with the
right-hand side of (3.20), then we will be able to rewrite (3.20) in a standard
form, that is, as an integral convolution equation. Our first statement is the
following lemma.Lemma 5.3.
The
function(5.5)𝔻(ω,λ)=𝒟(ω,λ)λ+iωis a Mikhlin multiplier.Proof.
Since𝒟(ω,λ)=M2λ2+2iωλM2+(1−M2)ω2, we have (5.6)𝔻′(ω,λ)=iλM2+ω(1−M2)(λ+iω)𝒟(ω,λ)−i𝒟(ω,λ)(λ+iω)2.From (5.6), it can be readily
seen that (5.7)|ω𝔻′(ω,λ)|⩽C<∞.We note that estimate (5.7) is
valid for all λ including zero (though it is not our case).
Indeed, we have 𝔻′(ω,0)=(1−M2/iω)+(i1−M2/ω)=0. Therefore, the function 𝔻(ω,λ) is a Mikhlin multiplier.
The lemma is shown.The next statement is related to the function(λ+iω)−1 which is a part of the main multiplier (5.5)
(see (4.38)).Lemma 5.4.
(1)The function(λ+iω)−1is a Mikhlin multiplier.
(2)Letf∈Lp(−∞,∞),p>1,and letgbe related tof through the operator𝒬1by the following formula:(5.8)g(x,λ)=(𝒬1f)(x)=∫−∞xe−λ(x−η)f(η)dη.Then, the Fourier transforms
are related through the multiplier(5.9)G(ω,λ)=1λ+iωF(ω).
(3)
𝒬1is a bounded operator inLp(−∞,∞)for eachλsuch thatℜλ⩾σa.Proof.
(1)
To check that(λ+iω)−1 is a Mikhlin multiplier, it suffices to verify
that for ℜλ⩾σa>0, the following estimate holds: (5.10)|(λ+iω)−1|+|ω(λ+iω)−2|⩽C<∞,|ω|<∞,which is clearly the case.
(2) Let us evaluate the Fourier transform of both
parts of representation (5.8) and have (5.11)G(ω,λ)=∫−∞∞eiωxdx∫−∞∞e−λ(x−η)f(η)dη=∫−∞∞f(η)eληdη∫η∞e(λ+iω)xdx=1λ+iω∫−∞∞f(η)eiωηdη1λ+iωF(ω).Equation (5.11) means that if 𝒬1 is defined by (5.8), then the corresponding
multiplier is (λ+iω)−1.
(3) It remains to prove that𝒬1 is a bounded operator in Lp(−∞,∞). We have (5.12)∥𝒬1[f]∥Lp(−∞,∞)=(∫−∞∞dx|∫−∞xf(η)e−λ(x−η)dη|p)1/p⩽(∫−∞∞dx|∫−∞x|f(η)|e−σ(x−η)dη|p)1/p,where σ=ℜλ⩾σa>0. Let us use y instead of η in (5.12), that is, y=x−η,
and have (5.13)∥𝒬1[f]∥Lp(−∞,∞)⩽(∫−∞∞dx|∫0∞f(x−y)e−σydy|p)1/p⩽∫0∞e−σydy(∫−∞∞|f(x−y)|pdx)1/p⩽1σa∥f∥Lp(−∞,∞).In the last step, we have used
the Minkowski inequality.
The lemma is
shown.In our next
statement, we give only the formulation of the result since the proof can be
easily reconstructed from the proof of Lemma5.4.Lemma 5.5.
(1)The function(λ−iω)−1is a Mikhlin multiplier.
(2)Letf∈Lp(−∞,∞),p>1,and letgbe related tof through the operator𝒬3by the following formula:(5.14)g(x,λ)=(𝒬3f)(x)=∫x∞e−λ(ξ−x)f(ξ)dξ.Then, the Fourier transforms
are related through the multiplier(5.15)G(ω,λ)=1λ−iωF(ω).
(3)
𝒬3is a bounded operator inLp(−∞,∞)for eachλsuch thatℜλ⩾σa.To present our next result, we have to introduce
the following notations.(i)ℰ1(z) is “the exponential integral”  [25] defined
by the formula (5.16)ℰ1(z)=1π∫z∞e−ττdτ,ℜz>0.The following properties of ℰ1 will be needed in the sequel. From (5.16), we
get (5.17)ℰ1′(z)=−e−zπz=−1πz+1+e−zπz,which yields another
representation for ℰ1(z):(5.18)ℰ1(z)=−1πlnz+∫1z1−e−ττdτ.Equation (5.18) means that ℰ1 is an analytic function correctly defined on
the complex plane with the branch cut along the negative real semiaxis
including zero. From (5.18), it also follows that ℰ1 belongs to Lp(−b,b) for any p⩾1.(ii) Let us introduce a new function by the
formula(5.19)g−(λ,x)=e−iλxℰ1(λ(b−x)),|x|<b.For each x,g− is an analytic function of λ on the complex plane with the branch cut along
the negative real semiaxis and (5.20)∫−bb|g−(λ,x)|pdx<∞,p⩾1.Below we define several linear
operators that we need in the future.(iii) Let𝒫 be a projection, that is, (5.21)(𝒫f)(x)=f(x)for|x|<b,(𝒫f)(x)=0for|x|>b.(iv) Letℋ be the Hilbert transformation defined by
[26, 27] (5.22)(ℋf)(x)=1π∫−∞∞*f(ξ)x−ξdξ,where “∗” (star) means that the integral is
understood as a Cauchy principal value integral. Regarding the operator ℋ, we need the following result.For anyf∈Lp(−∞,∞), (5.22) defines a functionF˜=ℋ[f]∈Lp(−∞,∞).In addition,(5.23)ℋ[ℋ[f]]=−f.The Mikhlin multiplier,
corresponding to the Hilbert transformation, is given explicitly by  [26] (5.24)Γ(ω)=|ω|iω,−∞<ω<∞.(v) Letℋb be a “finite Hilbert transformation”
defined by (5.25)(ℋbf)(x)=1π∫−bb*f(ξ)x−ξdξ.(vi) Let𝒬1 be a Volterra integral operator defined in
(5.8): (5.26)(𝒬1f)(x)=∫−∞xe−λ(x−ξ)f(ξ)dξ,f∈Lp(−∞,∞).(vii) Letℒ(λ) be the following operator: (5.27)ℒ(λ)f=∫−bxe−λ(x−ξ)f(ξ)dξ=(𝒬1𝒫)f,f∈Lp(−∞,∞),|x|<b.(viii) LetL(λ,f) be the following linear functional: (5.28)L(λ,f)=∫−bbeληf(η)dη.We notice that L defines a bounded linear functional in Lp(−b,b),p⩾1.Now we are in a position to present the next result.Lemma 5.6.
The
following formula is valid for the linear mapping𝒫ℋ𝒬1𝒫inLp(−∞,∞):(5.29)(𝒫ℋ𝒬1𝒫)[f]=(ℋbℒ(λ))[f]−g−(λ,x)L(λ,f)for|x|<b,(𝒫ℋ𝒬1𝒫)[f]=0for|x|>b.Proof.
Using
(5.21), (5.22), and (5.26), we get for f∈Lp(−∞,∞) and |x|<b, (5.30)(ℋ𝒬1𝒫)[f]=ℋ[∫−∞xe−λ(x−ξ)(𝒫f)(ξ)dξ]=1π∫−∞∞∗dξx−ξ∫−bξe−λ(ξ−η)(𝒫f)(η)dη=1π[∫−∞−b*+∫−bb*+∫b∞*]dξx−ξ∫−bξe−λ(ξ−η)(𝒫f)(η)dη≡i1+i2+i3.For the integral i1, we obtain the following result: if ξ<−b, then η<ξ<−b,
and thus for such η,𝒫[f](η)=0, which yields i1=0.
Using definitions (5.25) and (5.27), we obtain that i2 can be given as (5.31)i2=1π∫−bb∗dξx−ξ∫−bξe−λ(ξ−η)f(η)dη=(ℋbℒ(λ)𝒫)[f],|x|<b.Finally, we consider i3 and have (5.32)i3=1π∫−b∞∗e−λξx−ξdξ∫−bbeληf(η)dη=1π∫−b∞∗e−λξx−ξdξL(λ,f),|x|<b,with L(λ,f) being defined as (5.28).
Consider an integral from (5.32) (we are interested
only in the case |x|<b): (5.33)1π∫b∞e−λξx−ξdξ=−1π∫b−x∞e−λ(τ+x)τdτ=−e−λxπ∫b−x∞e−λττdτ=−e−λxπ∫λ(b−x)∞e−ττdτ=−e−λxℰ1(λ(b−x)).In (5.33), we have used
definition (5.16). Based on (5.19), we obtain (5.29).
The lemma is
completely shown.Corollary 5.7.
The following formula holds for anyf∈Lp(−∞,∞):(5.34)(𝒫ℋ𝒬2𝒫)[f]=∫0α1a(s)[(ℋbℒ(λs))[f]+g−(λs,x)L(λs,f)]ds,|x|<b,where𝒬2is an operator corresponding to a Mikhlin multiplier∫0α1a(s)ds/(λs+iω)withg−,ℒ(λ),andLbeing defined in (5.19), (5.27), and
(5.28), respectively.Proof.
First,
we notice that both functions(λs+iω)−1 and (λs−iω)−1 for −α2⩽s⩽α1,ℜλ⩾σa>0 are the Mikhlin multipliers. We want to
identify the operator that corresponds to the multiplier (5.35)ϰ1(ω)=∫0α1a(s)dsλs+iω(orϰ2(ω)=∫0α2a(−s)dsλs−iω),with α1 being defined in (4.5). By Lemma 5.4, the
operator corresponding to the multiplier (λs+iω)−1 can be defined as follows: (5.36)ifG(λ,s)=1λs+iωF(λ),theng(x,s)=∫−∞xe−λs(x−ξ)f(ξ)dξ.Applying (5.36) to f∈Lp(−b,b),
multiplying by a,
and integrating, we obtain (5.37)∫0α1g(x,s)a(s)ds=∫0α1a(s)ds∫−bxe−λs(x−ξ)f(ξ)dξ.Taking into account that the
factor |ω|(iω)−1 corresponds to the application of the Hilbert
transformation, we get (5.38)ℋ[∫0α1g(x,s)a(s)ds]=∫0α1a(s)ℋ[∫−bxe−λs(x−ξ)f(ξ)dξ]ds.In (5.38), we have changed the
order of the two integrals; one is a standard integral and another one is “the
star” integral. This step is justified in our case since f∈Lp(−b,b) and a is a continuous function  [27]. Therefore, if
we denote by 𝒬2 the operator corresponding to the multiplier ∫0α1a(s)(ds/(λs+iω)),
the following formula holds: (5.39)(𝒫ℋ𝒬2𝒫)[f]=∫0α1a(s)[(ℋbℒ(λs))[f]+g−(λs,x)L(λs,f)]ds,|x|<b,with g−,ℒ(λ),andL being defined in (5.19), (5.27), and (5.28),
respectively.
The proof of the corollary is complete.Lemma 5.8.
The
following formula is valid for the mapping𝒫ℋQ3𝒫inLp(−∞,∞): (5.40)(𝒫ℋ𝒬3𝒫)[f]=(ℋbℒ*(λ))[f]+g+(λ,x)L(−λ,f)for|x|<b,(𝒫ℋ𝒬3𝒫)[f]=0for|x|>b.Here𝒬3is an operator corresponding to the
multiplier(λ−iω)−1(see Lemma5.5); a linear operatorℒ∗is defined by(5.41)ifℒ∗f=g,theng(x)=∫xbe−λ(ξ−x)f(ξ)dξ,|x|<b.The functiong+(λ,⋅)is defined by the formula(5.42)g+(λ,x)=eλxℰ1(λ(b+x)),|x|<b,whereℰ1is an “exponential integral” from (5.16).Proof.
We discuss only the case|x|<b since the opposite case for |x|>b is obvious. For f∈Lp(−∞,∞), we have (5.43)(ℋ𝒬3𝒫)[f]=ℋ[∫x∞e−λ(ξ−x)(𝒫f)(ξ)dξ]=ℋ[∫xbe−λ(ξ−x)f(ξ)dξ]=1π∫−∞∞∗dξx−ξ∫ξbe−λ(η−ξ)(𝒫f)(η)dη=1π[∫−∞−b∗+∫−bb∗+∫b∞∗]dξx−ξ∫ξbe−λ(η−ξ)(𝒫f)(η)dη≡i1′+i2′+i3′.The integral i3′=0 since (𝒫f)(η)=0 for η>b.
Consider the second integral, then we have (5.44)i2′=1π∫−bb∗dξx−ξ∫ξbeλ(η−ξ)f(η)dη=(ℋbℒ∗(λ))[f],|x|<b,where ℒ∗ is given in (5.41). Finally, for i1′ we have when |x|<b that (5.45)i1′=1π∫−∞−b∗eλξdξx−ξ∫ξbe−ληf(η)dη=1π∫−∞−beλξdξx−ξ∫−bbe−ληf(η)dη=1π∫−∞−beλξdξx−ξL(−λ,f).In (5.45), we have taken into
account that |x|<b and definition (5.28). Finally, we consider an
integral term from (5.45) and transform it as (5.46)1π∫−∞−beλξx−ξdξ=1π∫−b∞e−ληx+ξdξ=eλx1π∫x+b∞e−λ(τ)τdτ=eλxℰ1(λ(x+b))=g+(λ,x),with ℰ1 and g+ being defined in (5.16) and (5.42),
respectively. Collecting together (5.43) and (5.45), we get (5.39).
The lemma is
shown.Corollary 5.9.
The operator corresponding to the multiplierϰ2from (5.35) can be given in the following
form:(5.47)∫0α2a(−s)[(ℋbℒ∗(λs))[f]+g+(λs,x)L(−λs,f)]ds,|x|<b.Collecting together the results of Lemmas5.4–5.8 and
Corollaries 5.7 and 5.9, we arrive at an important result that we formulate as
a theorem below.Theorem 5.10.
The Possio equation for0<M<1can be expressed in the form(5.48)Wa(x,λ˜)=1−M22×{ℋb[F(⋅,λ˜)]−11−M2[(λℋbℒ(λ))[F(⋅,λ˜)]−λg−(x,λ)L(λ,F(⋅,λ˜))]−∫0α1a(s)[(λℋbℒ(λs))[F(⋅,λ˜)]−λg−(x,λs)L(λs,F(⋅,λ˜))]ds+∫0α2a(−s)[(λℋbℒ∗(λs))[F(⋅,λ˜)]−λg+(x,λs)L(−λs,F(⋅,λ˜))]ds},|x|<b.We will modify (5.48) to a more
conventional form using Tricomi-Söhngen result  [26, 27].Proposition 5.11.
Letρ(x)=(b−x)/(b+x),|x|<b, and letL2,ρ(−b,b)andL2,ρ−1(−b,b)be the two weighted Hilbert spaces with the
weightsρandρ−1, respectively. Then, the “airfoil equation” can be given in
the form
(5.49)f(x)=1π∫−bb*w(ξ)x−ξdξ=1π∫−bb*v(ξ)x−ξρ(ξ)dξ≡ℍ[v].The operatorℍdefined in (5.49) is a Hilbert space
isomorphism ofL2,ρ(−b,b)ontoL2,ρ−1(−b,b). The inverse operatorℍ−1is given by(5.50)v(x)=ℍ−1[f](x)=1π∫−bb*f(ξ)x−ξρ−1(ξ)dξ,f∈L2,ρ−1(−b,b).Equation (5.50) represents the classical So¨hngen inversion formula for the airfoil
equation in the presence of the Kutta condition. Multiplying both sides of
(5.50) byρ, one obtains(5.51)w(x)=v(x)ρ(x)=1πb−xb+x∫−bb*b+ξb−ξf(ξ)x−ξdξ≡𝒯[f],f∈L2,ρ−1(−b,b).Our goal is to apply the operator𝒯 to both sides of (5.48) in order to reduce it
to a form convenient to deal with.Lemma 5.12.
The
operator𝒯can be applied to each term of the
right-hand side of (5.48).Proof.
To
prove this lemma, it suffices to show that (5.48) can be represented in the
form (5.52)21−M2Wa(x,λ)=ℋb[G1]+G(x,λ),with G∈Lp(−b,b),p>4/3, and G1∈Lp(−b,b),1<p<2. Let us consider the sum of the first, second,
fourth, and sixth terms from the right-hand side of (5.48) and have (5.53)F1=ℋb[F]+11−M2ℋb[λℒ(λ)F]−∫0α1a(s)ℋb[ℒ(λs)F]ds+∫0α2a(−s)ℋb[λℒ∗(λs)F]ds.Let us show that we can change
the order of integration and the Hilbert transformation in the last two terms
of (5.53). We discuss only the first integral; the second can be treated in a
similar way. In the expression ∫0α1a(s)ℋb[ℒ(λs)F]ds there are two integrals with the inner
integral being integral with “the star” and the outer integral being
standard. Since a is a continuous function and F∈Lp(−b,b), that is, both integrals go over finite
intervals, one can change the order of integration  [27]. Thus, (5.54)∫0α1a(s)ℋb[ℒ(λs)F]ds=ℋb[∫0α1a(s)ℒ(λs)F]ds.Therefore, (5.55)F1=ℋb[F−11−M2λℒ(λ)F−∫0α1a(s)ℒ(λs)Fds+∫0α2a(−s)ℒ∗(λs)Fds].It can be easily shown that the
Hilbert transform ℋb in (5.55) is applied to a function from Lp(−∞,∞) as long as F∈Lp(−b,b).
Now we consider the third, fifth, and seventh terms in
the right-hand side of (5.48). Since L(⋅,F) is a functional over F,
to verify that the sum of those three terms belongs to Lp(−b,b), it suffices to check that the two functions g−(⋅,λ) and g+(⋅,λ) belong to Lp(−b,b). The latter fact follows immediately from the
definitions (5.19) and (5.42), and from the properties of “exponential
integral” ℰ1 of (5.16).
The lemma is completely shown.In what
follows, we will apply the operator𝒯 to both sides of (5.48). In particular, we
will need the expressions and the properties for the functions h− and h+ defined below as (5.56)h−(x,λ)=𝒯[ebλg−(⋅,λ)],h+(x,λ)=𝒯[ebλg+(⋅,λ)].Evaluating h−, we have (5.57)h−(x,λ)=1πb−xb+x∫−bb∗b+ξb−ξg−(ξ,λ)ebλξ−xdξ=1πb−xb+x∫−bb∗b+ξb−ξdξξ−xe(b−ξ)λ∫1∞e−(b−ξ)λttdt=1πb−xb+x∫−bb∗b+ξb−ξdξξ−x∫1∞e−(b−ξ)λ(t−1)tdt.Change of the variable of
integration in (5.57), σ=(b−ξ)(t−1), yields (5.58)h−(x,λ)=1πb−xb+x∫−bb∗b+ξb−ξdξξ−x∫0∞e−λσb−ξ+σdσ=1πb−xb+x∫0∞1b−x+δ2b+σσeλσdσ,|x|<b.Similarly, it can be shown that(5.59)h+(x,λ)=1πb−xb+x∫0∞1b+x+σσ2b+σeλσdσ.The following properties for h− and h+ hold:(5.60)limx→b−0h−(x,λ)=1,limx→b−0h+(x,λ)=0.Finally, we present the desired
modification of the integral equation (5.48).Theorem 5.13.
The Possio integral equation can be written in the following form:(5.61)21−M2𝒯[Wa(⋅,λ′)]=[1−11−M2λℒ(λ)]F(⋅,λ′)+λ1−M2h−(x,λ)e−bλL(λ,F(⋅,λ′))−∫0α1a(s)(λℒ(λs))[F]ds+∫0α1a(s)λh−(⋅,λs)e−bλsL(λs,F(⋅,λ′))ds+∫0α2a(−s)(λℒ∗(λs))[F]ds+∫0α2a(−s)λh+(⋅,λs)e−bλsL(−λs,F(⋅,λ′))ds.


## 6. Concluding Remarks

We would like to emphasize that (5.61) is a
complicated singular integral equation. In our forthcoming work, we will prove
a unique solvability of (5.61) using a two-step procedure. In the first step,
we prove that as |λ|→∞, the right-hand side of the integral operator
of (5.61) splits into two distinct parts: one does not get smaller and another
one asymptotically goes to zero in a Banach space equipped with a specific
norm. Having such a behavior in mind, we consider the equation obtained from
(5.61) if one keeps only the leading asymptotical parts of all terms as |λ|→∞,ℜλ⩾δa≫1. It turns out that the new equation (that we
will call the asymptotical version of the Possio equation) is a singular
integral equation as well. This asymptotic version of the Possio equation is a
singular equation in more than one sense. First, it is singular due to the fact
that it contains a specific “inverse operator” to a finite Hilbert
transformation ℋb (ℋb is a singular integral operator in Lp(−b,b),1<p<2). Secondly, the aforementioned equation
contains integral operators being different by their nature: Hilbert
transformation, Volterra integral operator, and an integral operator with the
degenerate kernel  [28]. It is exactly the second reason that makes the problem difficult
and nonstandard.



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