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

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.

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 is𝐌𝑋_𝑡𝑡(𝑦,𝑡)+𝐊𝑋(𝑦,𝑡)=(𝕃(𝑦,𝑡),𝕄(𝑦,𝑡))^𝑇, (1.1)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 formulas𝕃(𝑦,𝑡)=^𝑏_−𝑏𝛿𝑝𝑑𝑥,𝕄(𝑦,𝑡)=^𝑏_−𝑏(𝑥−𝑎)𝛿𝑝𝑑𝑥, (1.2)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:𝑝(𝑥,𝑦,𝑧,𝑡)=𝑝_∞(1+𝛾−1𝑎²_∞(𝑈²2−𝜕𝜑−𝜕𝑡∇𝜑⋅∇𝜑2))^{𝛾/(𝛾−1)}, (1.3)where 𝑝_∞=𝜌²𝑎²_∞𝛾⁻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 potential 𝜑, 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 𝑤_𝑎 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 parameter 𝑈 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”𝐻𝒞(𝑘)=𝐹(𝑘)+𝑖𝐺(𝑘)=¹⁽²⁾(𝑘)𝐻₁⁽²⁾(𝑘)+𝑖𝐻₀⁽²⁾(𝑘), (1.4)with 𝐻⁽²⁾_0,1(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 Section 2, 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 𝐿_𝑝-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 form𝑤_𝑎𝜔(𝑥)=−𝜌_∞𝑢²P.V.∫^𝑏_−𝑏Δ𝑝_𝑎ℜ(𝑀,𝑘(𝑥−𝜉)𝑏)𝑑𝜉,|𝑥|<𝑏. (1.5)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:1ℜ=41−𝑀²{exp(𝑖𝑘𝑀²(𝑥−𝜉)𝑏(1−𝑀²)²)[𝑖𝑀|𝑥−𝜉|𝐻𝑥−𝜉¹⁽²⁾(𝑘𝑀|𝑥−𝜉|𝑏(1−𝑀²))−𝐻₀⁽²⁾(𝑘𝑀|𝑥−𝜉|𝑏(1−𝑀²))]+𝑖(1−𝑀²)exp(𝑖𝑘(𝑥−𝜉)𝑏2)[𝜋1−𝑀²√ln(1+1−𝑀²𝑀)+^{𝑘(𝑥−𝜉)/𝑏(1−𝑀2)}₀𝑒^𝑖𝑣𝐻₀⁽²⁾(𝑀|𝑣|)]}, (1.6)where 𝑀 is the Mach number, 𝑘=𝜔𝑏/𝑈 is the reduced frequency, 𝐻_𝑗⁽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: Δ𝑝_𝑎=𝐴₀cot(𝜃/2)+Σ_𝑛⩾1𝐴_𝑛sin(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 expression𝑈𝑥+𝜑(𝑥,𝑧,𝑡),−∞<𝑥<∞,0⩽𝑧<∞,𝑡>0, (2.1)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]:𝜑_𝑡𝑡(𝑥,𝑧,𝑡)+2𝑀𝑎_∞𝜑_𝑡𝑥(𝑥,𝑧,𝑡)=𝑎²_∞(1−𝑀²)𝜑_𝑥𝑥(𝑥,𝑧,𝑡)+𝑎²_∞𝜑_𝑧𝑥(𝑥,𝑧,𝑡),−∞<𝑥<∞,0⩽𝑧<∞,𝑡>0. (2.2) Together with this equation, we introduce the following boundary conditions.

(1) The flow tangency (or nonseparable flow) condition is𝜕𝜕𝑧𝜑(𝑥,𝑧,𝑡)|_𝑧=0=𝑤_𝑎(𝑥,𝑡),|𝑥|<𝑏, (2.3)where 𝑤_𝑎(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 by𝜓(𝑥,𝑧,𝑡)=𝜑_𝑡(𝑥,𝑧,𝑡)+𝑈𝜑_𝑥(𝑥,𝑧,𝑡). (2.4)The conditions below reflect the following physical situation: pressure off the wing and at the trailing edge must be equal to zero. These conditions are𝜓(𝑥,0,𝑡)=0for|𝑥|>𝑏,lim_{𝑥→𝑏−0}𝜓(𝑥,0,𝑡)=0. (2.5)

(3) Far-field conditions. The disturbance potential and velocity tend to zero at a large distance from the wing:𝜑(𝑥,𝑧,𝑡)→0,∇𝜑(𝑥,𝑧,𝑡)→0,as|𝑥|→∞or𝑧→∞. (2.6)

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 𝐿_𝑝(−∞,∞),1<𝑝<2, assuming that the initial conditions are trivial, that is, 𝜑(𝑥,𝑧,0)=𝜑_𝑡(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 function 𝜑(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 𝐷_𝑥 be a closed linear operator in 𝐿_𝑝(−∞,∞) corresponding to the partial derivative 𝜕/𝜕𝑥. Then, we require that𝜑(⋅,𝑧,𝑡)∈𝒟(𝐷²_𝑥), (2.7)with 𝒟 being the domain of 𝐷²_𝑥, and 𝐷_𝑥𝜑(⋅,𝑧,𝑡) being absolutely continuous with respect to t.

We require that (2.2) be satisfied in the sense that𝜕²𝜕𝑡²𝜑(⋅,𝑧,𝑡)+2𝑀𝑎_∞𝜕𝐷𝜕𝑡^𝑥𝜑(⋅,𝑧,𝑡)=(𝑎²_∞(1−𝑀²)𝐷²_𝑥+𝑎²_∞𝜕²𝜕𝑧²)𝜑(⋅,𝑧,𝑡). (2.8)The flow tangency condition (2.3) will be understood in the sense that^𝑏_−𝑏|𝜕𝜕𝑧𝜑(𝑥,𝑧,𝑡)−𝑤_𝑎(𝑥,𝑡)|^𝑝𝑑𝑥→0as𝑧→0. (2.9)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:∫^∞_−∞|𝜓(𝑥,𝑧,𝑡)−𝜓(𝑥,0,𝑡)|^𝑝∫𝑑𝑥→0as𝑧→0,_{−𝑏−∞}|𝜓(𝑥,0,𝑡)|^𝑝𝑑𝑥+∫^∞_𝑏|𝜓(𝑥,0,𝑡)|^𝑝𝑑𝑥=0. (2.10)Regarding the initial conditions, we require‖𝜑(⋅,𝑧,𝑡)‖_𝐿^𝑝_(−∞,∞)𝜕→0,‖𝜕𝑡𝜑(⋅,𝑧,𝑡)‖_𝐿^𝑝_(−∞,∞)→0as𝑡→0,0<𝑧<∞. (2.11)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

Let 𝑓(⋅,𝑧,𝑡)∈𝐿_𝑝(−∞,∞),𝑧>0,𝑡>0, be a function that has the following property:^∞₀‖𝑓(⋅,𝑧,𝑡)‖_{𝐿𝑝(−∞,∞)}𝑒^−𝜎𝑡𝑑𝑡<∞for𝜎⩾𝜎_𝑎>0. (3.1)In the future, we will omit the subindex 𝐿_𝑝(−∞,∞) using the notation ‖⋅‖ for the 𝐿_𝑝 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) isΦ(⋅,𝑧,𝜆)=^∞₀𝜑(⋅,𝑧,𝑡)𝑒^−𝜆𝑡𝑑𝑡,ℜ𝜆⩾𝜎_𝑎>0,0<𝑧<∞. (3.2)It is clear that from (2.8) for 𝜑(⋅,𝑧,𝑡) we obtain a new equation for Φ(⋅,𝑧,𝜆):𝜆²Φ(⋅,𝑧,𝜆)+2𝑀𝑎_∞𝜆𝐷_𝑥Φ(⋅,𝑧,𝜆)=𝑎²_∞(1−𝑀²)𝐷²_𝑥Φ(⋅,𝑧,𝜆)+𝑎²_∞Φ_𝑧𝑧(⋅,𝑧,𝜆). (3.3)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)Φ(𝜔,𝑧,𝜆)=^∞_−∞Φ(𝑥,𝑧,𝜆)𝑒^{−𝑖𝜔𝑥}𝑑𝑥,−∞<𝜔<∞. (3.4)As it is well known, the Fourier transformation is a bounded linear operator from 𝐿_𝑝(−∞,∞) into 𝐿_𝑞(−∞,∞)(𝑝⁻¹+𝑞⁻¹=1),1<𝑝<2. Applying the spatial Fourier transformation to (3.3), we obtain a new equation for Φ:(𝜆²+2𝑖𝜔𝜆𝑀𝑎_∞)Φ(𝜔,𝑧,𝜆)=−𝑎²_∞(1−𝑀²)𝜔²Φ(𝜔,𝑧,𝜆)+𝑎²_∞Φ_𝑧𝑧(𝜔,𝑧,𝜆). (3.5)Rearranging terms in this equation, we obtainΦ_𝑧𝑧(𝜔,𝑧,𝜆)=(1−𝑀²)𝜔²𝜆Φ(𝜔,𝑧,𝜆)+(²𝑎²_∞+2𝑖𝜔𝜆𝑀𝑎_∞)Φ(𝜔,𝑧,𝜆). (3.6)Recalling that 𝑈/𝑎_∞=𝑀 and denoting ^0𝑥0𝜆^02𝑑𝑐=𝜆/𝑈, we rewrite (3.6) asΦ_𝑧𝑧(𝜔,𝑧,𝜆)=(𝜆^{0𝑥002𝑑𝑐2}𝑀²+2𝑖𝜔𝜆^{0𝑥002𝑑𝑐}𝑀²+𝜔²(1−𝑀²))Φ(𝜔,𝑧,𝜆). (3.7)The quadratic polynomial (𝜆^{0𝑥002𝑑𝑐2}𝑀²+2𝑖𝜔𝜆^{0𝑥002𝑑𝑐}𝑀²+𝜔²(1−𝑀²)) will play an important role in the sequel. We make the following agreement: let us keep the notation 𝜆 for ^0𝑥0𝜆^02𝑑𝑐 and use the notation 𝜆^′ for the original 𝜆; that is, with this agreement, (3.7) obtains the following form:Φ_𝑧𝑧(𝜔,𝑧,𝜆^′)=(𝜆²𝑀²+2𝑖𝜔𝜆𝑀²+𝜔²(1−𝑀²))Φ(𝜔,𝑧,𝜆^′),𝜆^′=𝜆𝑈. (3.8)Let𝒟(𝜔,𝜆)=𝜆²𝑀²+2𝑖𝜔𝜆𝑀²+𝜔²(1−𝑀²). (3.9)Let us show that |𝒟(𝜔,𝜆)| is bounded below when ℜ𝜆⩾𝜎_𝑎>0. Indeed,|𝒟(𝜔,𝜆)|=|𝑀²(𝑖𝜔+𝜆)²+𝜔²|⩾𝑀²|𝜆+𝑖𝜔|²⩾𝑀²|ℜ𝜆|²⩾𝑀²𝜎²_𝑎𝑈²>0. (3.10)From this estimate, it follows that we can define the “positive” square root that we denote by 𝒟(𝜔,𝜆) such thatℜ𝒟(𝜔,𝜆)>0. (3.11)Thus, from (3.11) it follows that the differential equation (3.8) has a unique solution satisfying the Far-field conditions, and this solution isΦ(𝜔,𝑧,𝜆^′)=Φ(𝜔,0,𝜆^′)exp(−𝑧𝒟(𝜔,𝜆)). (3.12)Now we have to satisfy the Kutta-Joukowski conditions and the flow tangency condition.

To write the flow tangency condition, we need a derivative with respect to 𝑧. Let𝜈(𝜔,𝑧,𝜆^′𝑑)=𝑑𝑧Φ(𝜔,𝑧,𝜆^′)=−𝒟(𝜔,𝜆)Φ(𝜔,𝑧,𝜆^′). (3.13)In terms of 𝜈, (3.12) can be written asΦ(𝜔,𝑧,𝜆^′)=−𝜈(𝜔,0,𝜆^′)𝒟(𝜔,𝜆)exp(−𝑧𝒟(𝜔,𝜆)). (3.14)We notice that the Laplace transform of the acceleration potential isΨ(𝜔,𝑧,𝜆^′)=𝜆^′Φ(𝑥,𝑧,𝜆^′)+𝑈Φ(𝑥,𝑧,𝜆^′) (3.15)and, therefore, using (3.14), we obtainΨ(𝜔,𝑧,𝜆^′)=(𝜆^′+𝑖𝜔𝑈)Φ(𝜔,𝑧,𝜆^′)=−(𝜆^′+𝑖𝜔𝑈)𝜈(𝜔,0,𝜆^′)𝒟(𝜔,𝜆)exp(−𝑧𝒟(𝜔,𝜆)). (3.16)From (3.16), we obtain thatΨ(𝜔,0,𝜆^′)=−(𝜆^′+𝑖𝜔𝑈)𝜈(𝜔,0,𝜆^′)𝒟(𝜔,𝜆). (3.17)At this point, we introduce the standard notation, that is,2𝒜(𝑥,𝑡)=−𝑈𝜓(𝑥,0,𝑡). (3.18)We will denote the Laplace transform of 𝒜 with respect to the time variable by 𝐴. Multiplying (3.17) by 2/𝑈, we have𝐴(𝜔,𝜆^′)=2(𝜆+𝑖𝜔)𝒟(𝜔,𝜆)𝜈(𝜔,0,𝜆^′), (3.19)or in another form𝜈(𝜔,0,𝜆^′)=𝒟(𝜔,𝜆)2(𝜆+𝑖𝜔)𝐴(𝜔,𝜆^′), (3.20)with 𝜈 being defined in (3.13).

We notice that if we know 𝐴(𝜔,𝜆^′), then we know the velocity potential, that is,Φ(𝜔,𝑧,𝜆^′1)=−2(𝜆+𝑖𝜔)exp(−𝑧𝒟(𝜔,𝜆))𝐴(𝜔,𝜆^′). (3.21)Finally, we obtain the following problem for 𝐴∶𝜈(𝜔,0,𝜆^′)=𝒟(𝜔,𝜆)2(𝜆+𝑖𝜔)𝐴(𝜔,𝜆^′),𝜈(𝑥,0,𝜆^′)=𝑊_𝑎(𝑥,𝜆^′),|𝑥|<𝑏,𝐴(𝑥,𝜆^′)=0,|𝑥|>𝑏. (3.22)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 𝑊_𝑎(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. Let𝔻(𝜔,𝜆)=𝒟(𝜔,𝜆)𝜆+𝑖𝜔. (4.1)By direct calculations we obtain𝔻(𝜔,𝜆)=|𝜔|1−𝑀²𝜆+𝑖𝜔1+𝜆𝑀²1−𝑀²𝜆+2𝑖𝜔𝜔²≡|𝜔|𝑖𝜔1−𝑀²(1+𝒬(𝜔,𝜆)), (4.2)where𝒬(𝜔,𝜆)=𝑖𝜔𝜆+𝑖𝜔1+𝜆𝑀²1−𝑀²𝜆+2𝑖𝜔𝜔²−1. (4.3)The following result is valid for the function 𝒬(𝜔,𝜆).

Theorem 4.1. 𝒬(⋅,𝜆)is a Fourier transform of a function 𝐪∈𝐿_𝑝(−∞,∞),𝑝>1, given explicitly by the following formulas:𝑒𝐪(𝑥,𝜆)=−𝜆^−𝜆𝑥1−𝑀²−𝜆𝜋^𝛼1₀𝑒^{−𝜆𝑠𝑥}𝜆𝑎(𝑥)𝑑𝑠,𝑥>0,𝜋^𝛼2₀𝑒^{−𝜆|𝑥|𝑠}𝑎(−𝑠)𝑑𝑠,𝑥<0, (4.4) where𝛼₁=𝑀1+𝑀,𝛼₂=𝑀1−𝑀,𝑎(𝑠)=(𝛼₁−𝑠)(𝛼₂+𝑠)1−𝑠,−𝛼₂<𝑠<𝛼₂. (4.5)

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 have𝐅[𝐪]≡∫^∞_−∞𝑒^{−𝑖𝜔𝑥}=𝜆𝑞(𝑥,𝜆)𝑑𝑥𝜋∫⁰_−∞𝑒^{−𝑖𝜔𝑥}𝑑𝑥∫^𝛼2₀𝑒^{−𝜆|𝑥|𝑠}𝜆𝑎(−𝑠)𝑑𝑠−1−𝑀²∫^∞₀𝑒^−𝜆𝑥𝑒^{−𝑖𝜔𝑥}−𝜆𝑑𝑥𝜋∫^∞₀𝑒^{−𝑖𝜔𝑥}𝑑𝑥∫^𝛼1₀𝑒^{−𝜆𝑠𝑥}𝑎(𝑠)𝑑𝑠=𝐼₁+𝐼₂+𝐼₃. (4.6)Taking into account that ℜ𝜆⩾𝜎_𝑎≫1, we evaluate 𝐼₂ and have𝐼₂𝜆=−1−𝑀^2∞0𝑒^{−(𝜆+𝑖𝜔)𝑥}𝜆𝑑𝑥=−1−𝑀²(𝜆+𝑖𝜔). (4.7)Using (4.5), we evaluate 𝐼₃ and have𝐼₃𝜆=−𝜋^𝛼1₀𝑎(𝑠)𝑑𝑠^∞₀𝑒^{−(𝜆𝑠+𝑖𝜔)𝑥}𝜆𝑑𝑥=−𝜋^𝛼1₀(𝛼₁−𝑠)(𝛼₂+𝑠)(1−𝑠)(𝑖𝜔+𝜆𝑥)𝑑𝑠. (4.8)Evaluating 𝐼₁, we obtain𝐼₁=𝜆𝜋^𝛼2₀𝑎(−𝑠)𝑑𝑠^∞₀𝑒^{(𝑖𝜔−𝜆𝑠)𝑥}𝜆𝑑𝑥=−𝜋^𝛼2₀𝑎(−𝑠)𝑑𝑠𝑖𝜔−𝜆𝑠. (4.9)Combining together 𝐼₁ and 𝐼₃, we obtain𝐼₁+𝐼₃𝜆=−𝜋^𝛼1_−𝛼2𝑎(𝑠)𝑑𝑠𝑖𝜔+𝜆𝑠. (4.10)To modify (4.10), let us introduce a new variable of integration: 𝜉=𝑠+𝛼₁𝛼₂𝑀=𝑠+𝑀((1−𝑀²)𝑀)=𝑠+𝑀𝛽,𝛽=(1−𝑀²). (4.11)The following relations can be easily verified: 𝛼₁−𝑠=𝛽−𝜉,𝛼₂𝑀+𝑠=𝑀1−𝑀+𝜉−²1−𝑀²1=𝛽+𝜉;1−𝑠=𝜉+1−𝑀²,𝑖𝜔𝜆+𝑠=𝑖𝜔𝜆+𝜉−𝑀𝛽. (4.12)Using (4.12) and setting 𝑧=(𝑖𝜔/𝜆), we rewrite (4.10) as 𝐼₁+𝐼₃1=−𝜋^𝛽_−𝛽(𝛽−𝜉)(𝛽+𝜉)𝑑𝜉1(1−𝜉+𝑀𝛽)((𝑖𝜔/𝜆)+𝜉−𝑀𝛽)=−𝜋^𝛽_−𝛽𝛽²−𝜉²𝑑𝜉(1−𝜉+𝑀𝛽)(𝑧+𝜉−𝑀𝛽). (4.13)Making transformation 𝜉↦−𝜉 in (4.13) and taking into account that (1/𝛽)+𝑀=1/𝑀, we get 𝐼₁+𝐼₃1=−𝜋^𝛽_−𝛽𝛽²−𝜉²𝑑𝜉=1(1+𝜉+𝑀𝛽)(𝑧−𝜉−𝑀𝛽)𝜋^𝛽_−𝛽1−(𝜉/𝛽)²𝑑𝜉=1((1/𝛽)+𝑀+(𝜉/𝛽))((𝜉/𝛽)+𝑀−(𝑧/𝛽))𝛽𝜋^𝛽_−𝛽1−(𝜉/𝛽)²𝑑𝜉.𝛽((1/𝑀)+(𝜉/𝛽))((𝜉/𝛽)+𝑀−(𝑧/𝛽)) (4.14)Let 𝜂=𝜉/𝛽, then 𝐼₁+𝐼₃=1𝜋¹₋₁1−𝜂²𝑑𝜂=𝛽(𝜂+(1/𝑀))(𝜂+𝑀−(𝑧/𝛽))𝜋(1+𝑧)(𝑖₁−𝑖₂), (4.15)where 𝑖₁=¹₋₁1−𝜂²𝑑𝜂(𝜂+𝑀−(𝑧/𝛽)),𝑖₂=¹₋₁1−𝜂²𝑑𝜂𝜂+(1/𝑀). (4.16)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: ¹₋₁√1−𝑥²√𝑥+𝑎𝑑𝑥=𝜋(𝑎−sign(ℜ𝑎)𝑎²−1). (4.17) 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 𝑖₁𝑧=𝜋(𝑀−𝛽+𝑧(𝑀−𝛽)²−1). (4.18)Substituting 1/𝑀 for 𝑎 in (4.17), we obtain 𝑖₁−𝑖₂𝜋𝑧=𝑀−𝛽=𝑧(𝑀−𝛽)²1−1−𝑀−1𝑀²−1=−1+𝑧𝛽+1−𝑀²𝑀+1𝛽𝑧²−𝛽²(1−𝑀²=)−2𝑧𝑀𝛽1+𝑧𝛽1[−1+11+𝑧1−𝑀²+𝑧²−2𝑧𝛾−𝛾1+𝑧], (4.19)where 𝛾=𝛽𝑀 and 𝛽 is defined in (4.11).
Returning to (4.15), we obtain 𝐼₁+𝐼₃1=−1+(1+𝑧)1−𝑀²+11+𝑧𝑧²−2𝑧𝛾−𝛾. (4.20)Now we can finalize the expression for 𝒬 combining (4.7) with (4.20) to have 𝜆𝒬(𝑥,𝜆)=−1−𝑀²+1(𝜆+𝑖𝜔)(1+𝑧)1−𝑀²+𝑧1+𝑧1−2𝛾𝑧−𝛾𝑧²−1. (4.21)Recalling that 𝑧=𝑖𝜔/𝜆 and noticing that (2𝛾/𝑧)+(𝛾/𝑧²)=(2𝑀𝛽𝜆/𝑖𝜔)−(𝑀𝛽𝜆²/𝜔²), we obtain from (4.21) that𝒬(𝑥,𝜆)=𝑖𝜔𝜆+𝑖𝜔1−2𝑀𝛽𝜆+𝑖𝜔𝑀𝛽𝜆²𝜔²=−1𝑖𝜔𝜆+𝑖𝜔1+2𝑀𝛽𝜆𝑖𝜔+𝑀𝛽𝜆²𝜔²=−1𝑖𝜔𝜆+𝑖𝜔1+𝜆𝑀𝛽𝜔²(𝜆+2𝑖𝜔)−1. (4.22)This formula coincides with (4.3). To complete the proof of the theorem, it remains to show that 𝐪∈𝐿_𝑝(−∞,∞). As the first observation, we notice that it suffices to prove the result for the case 𝜆=1. Indeed, from (4.4) we obtain|𝐪(𝑥,𝜆)|⩽|𝜆||𝑞(𝑥,𝜎)|,𝜎=ℜ𝜆⩾𝜎_𝑎>0. (4.23)Thus,^∞_−∞|𝐪(𝑥,𝜆)|^𝑝𝑑𝑥⩽|𝜆|^𝑝∞_−∞|𝐪(𝑥,𝜎)|^𝑝𝑑𝑥=|𝜆|^𝑝∞_−∞|𝐪(𝜎𝑥,1)|^𝑝𝑑𝑥=|𝜆|^𝑝𝜎^∞_−∞|𝐪(𝜉,1)|^𝑝𝑑𝜉. (4.24)Taking into account |𝑎(𝑠)|⩽𝐶<∞ for −𝛼₂<𝑠<𝛼₁, we get the following estimates: |^𝛼1₀𝑒^−𝑥𝑠𝑎(𝑠)𝑑𝑠|⩽𝐶1−𝑒^−𝛼1𝑥|1+𝑥,𝑥>0,_𝛼20𝑒^{−|𝑥|𝑠}𝑎(−𝑠)𝑑𝑠|⩽𝐶1−𝑒^{−𝛼|𝑥|}1+|𝑥|,𝑥<0. (4.25)From estimates (4.25) we obtain^∞_−∞|𝑞(𝑥,1)|^𝑝𝑑𝑥⩽𝐶[⁰_−∞(1−𝑒^{−𝛼2|𝑥|})1+|𝑥|^𝑝𝑑𝑥+^∞₀(1−𝑒^−𝛼1𝑥)1+|𝑥|^𝑝𝑑𝑥]<∞for𝑝>1. (4.26)This estimate means that 𝐪∈𝐿_𝑝(−∞,∞).
The theorem is completely shown.

Now we complete the proof of the lemma.

Proof of Lemma 4.2. We consider the following integral: 𝐼=¹₋₁√1−𝑥²𝑥+𝑎𝑑𝑥,|𝑎|>1. (4.27)Let 𝑥=cos𝜃, then 𝐼=^𝜋₀sin²𝜃1cos𝜃+𝑎𝑑𝜃=2[^𝜋₀𝑑𝜃−cos𝜃+𝑎^𝜋₀cos2𝜃1cos𝜃+𝑎𝑑𝜃]≡2(𝑗₁−𝑗₂). (4.28)Evaluating 𝑗₁, we have 𝑗₁=^𝜋_−𝜋𝑑𝜃𝑒^𝑖𝜃+𝑒^−𝑖𝜃+2𝑎. (4.29)Integrating in the complex plane, we assume that 𝑧=𝑒^𝑖𝜃, and thus linear integral can be represented as a contour integral along a unit circle centered at the origin: 𝑗₁=1𝑖_𝐶1(0)𝑑𝑧𝑧(𝑧+𝑧⁻¹=1+2𝑎)𝑖_𝐶1(0)𝑑𝑧𝑧²+2𝑧𝑎+1. (4.30)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, 𝑧₁=−𝑎+√𝑎²−1=𝑒^𝑖𝜓,𝑧₂=−𝑎−√𝑎²−1=𝑒^−𝑖𝜓. (4.31)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 𝑗₁=1𝑖𝑑𝑧(𝑧−𝑧₁)(𝑧−𝑧₂)=𝜋√𝑎²−1. (4.32)Finally, we evaluate 𝑗₂ from (4.28): 𝑗₂=12^𝜋_−𝜋cos2𝜃1cos𝜃+𝑎𝑑𝜃=2^𝜋_−𝜋𝑒^2𝑖𝜃+𝑒^−2𝑖𝜃𝑒^𝑖𝜃+𝑒^−𝑖𝜃=1+𝑎𝑑𝜃2𝑖_𝐶1(0)(𝑧²+𝑧⁻2)𝑑𝑧𝑧²=1+2𝑎𝑧+12𝑖_𝐶1(0)𝑑𝑧(𝑧⁴+1)𝑧²(𝑧−𝑧₁)(𝑧−𝑧₂). (4.33)We have to evaluate the residues at the points 𝑧=0 and 𝑧≡𝑧₁=−𝑎+√𝑎²−1. We have𝑧(a)Res[𝑓;0]=𝜋(⁴+1𝑧²)+2𝑎𝑧+1^′|_𝑧=0=−2𝜋𝑎,(b)Res[𝑓;𝑧₁𝑧]=𝜋²+𝑧⁻²2√𝑧²|−1^{√𝑧=−𝑎+𝑎2−1}√=𝜋(𝑎−𝑎²−1)²+(𝑎+√𝑎²−1)²2√𝑎²−1=𝜋2𝑎²−1√𝑎².−1 (4.34)Using (4.34), we get 𝑗₂=Res[𝑓;0]+Res[𝑓;𝑧₁𝜋]=√𝑎²−1(2𝑎²−1−2𝑎√𝑎²𝜋−1)=√𝑎²√−1(𝑎−(𝑎²−1))², (4.35)which together with (4.32) yields 𝑗₁−𝑗₂=𝜋√𝑎²√−1(1−(𝑎−𝑎²−1)²𝜋)=√𝑎²−1(1−𝑎²+𝑎√𝑎²−1)=2𝜋(𝑎−√𝑎²−1). (4.36)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 𝑊_𝑎(⋅,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: 𝜆𝒬(𝜔,𝜆)=−1−𝑀²1𝜆+𝑖𝜔−𝜆^𝛼1₀𝑎(𝑠)𝜆𝑠+𝑖𝜔𝑑𝑠+𝜆^𝛼2₀𝑎(−𝑠)𝜆−𝑖𝜔𝑑𝑠,ℜ𝜆⩾𝛿_𝑎>0. (4.37)In terms of (4.37), we obtain that our main multiplier 𝔻(𝜔,𝜆) (from (4.2)) can be represented in the form 11−𝑀²𝔻(𝜔,𝜆)=|𝜔|=𝑖𝜔(1+𝒬(𝜔,𝜆))|𝜔|−𝑖𝜔|𝜔|1𝑖𝜔1−𝑀²𝜆−𝜆+𝑖𝜔|𝜔|𝜆𝑖𝜔^𝛼1₀𝑎(𝑠)𝑑𝑠+𝜆𝑠+𝑖𝜔|𝜔|𝜆𝑖𝜔^𝛼2₀𝑎(−𝑠)𝑑𝑠𝜆𝑠−𝑖𝜔,ℜ𝜆⩾𝛿_𝑎>0. (4.38)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 𝐿_𝑝(−∞,∞) 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 in 𝐿_𝑝(−∞,∞),1<𝑝<2. The main tool in this derivation is the notion of Mikhlin multipliers [24].

Deffinition 5.1. Let 𝑔 and 𝑓 be two functions from 𝐿_𝑝(−∞,∞),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 𝐺(𝜔)=𝜇(𝜔)𝐹(𝜔). (5.1)If 𝜇 is a continuously differentiable function (with one possible exception at 𝜔=0) such that |𝜇(𝜔)|+|𝜔||𝜇^′(𝜔)|⩽𝐶<∞, (5.2)then 𝜇 is called a Mikhlin multiplier.

Proposition 5.2 (See [24] for the Proof). Let 𝑓 and 𝑔 be the functions from Definition 5.1, and let 𝜇 be a Mikhlin multiplier. Then, there exists a bounded linear operator ℚ in 𝐿_𝑝(−∞,∞) that relates 𝑓 and 𝑔 by the rule 𝑔=ℚ[𝑓]. (5.3) The norm of the operator ℚ can be estimated as follows: ‖ℚ‖_{𝐿𝑝(−∞,∞)}⩽𝐶𝑀_𝑝, (5.4) where 𝐶 is a constant from (5.2), and 𝑀_𝑝 is a constant depending only on 𝑝.

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 𝐿_𝑝(−∞,∞) 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 𝔻(𝜔,𝜆)=𝒟(𝜔,𝜆)𝜆+𝑖𝜔 (5.5)is a Mikhlin multiplier.

Proof. Since 𝒟(𝜔,𝜆)=𝑀²𝜆²+2𝑖𝜔𝜆𝑀²+(1−𝑀²)𝜔², we have 𝔻^′(𝜔,𝜆)=𝑖𝜆𝑀²+𝜔(1−𝑀²)(𝜆+𝑖𝜔)−𝑖𝒟(𝜔,𝜆)𝒟(𝜔,𝜆)(𝜆+𝑖𝜔)². (5.6)From (5.6), it can be readily seen that |𝜔𝔻^′(𝜔,𝜆)|⩽𝐶<∞. (5.7)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∈𝐿_𝑝(−∞,∞),𝑝>1, and let𝑔be related to 𝑓 through the operator 𝒬₁by the following formula:𝑔(𝑥,𝜆)=(𝒬₁𝑓)(𝑥)=^𝑥_−∞𝑒^{−𝜆(𝑥−𝜂)}𝑓(𝜂)𝑑𝜂. (5.8) Then, the Fourier transforms are related through the multiplier 1𝐺(𝜔,𝜆)=𝜆+𝑖𝜔𝐹(𝜔). (5.9) (3) 𝒬₁ is a bounded operator in 𝐿_𝑝(−∞,∞) for each 𝜆such thatℜ𝜆⩾𝜎_𝑎.

Proof. (1) To check that (𝜆+𝑖𝜔)⁻¹ is a Mikhlin multiplier, it suffices to verify that for ℜ𝜆⩾𝜎_𝑎>0, the following estimate holds: |(𝜆+𝑖𝜔)⁻¹|+|𝜔(𝜆+𝑖𝜔)⁻²|⩽𝐶<∞,|𝜔|<∞, (5.10)which is clearly the case.
(2) Let us evaluate the Fourier transform of both parts of representation (5.8) and have 𝐺(𝜔,𝜆)=^∞_−∞𝑒^𝑖𝜔𝑥𝑑𝑥^∞_−∞𝑒^{−𝜆(𝑥−𝜂)}𝑓(𝜂)𝑑𝜂=^∞_−∞𝑓(𝜂)𝑒^𝜆𝜂𝑑𝜂^∞_𝜂𝑒^{(𝜆+𝑖𝜔)𝑥}=1𝑑𝑥𝜆+𝑖𝜔^∞_−∞𝑓(𝜂)𝑒^𝑖𝜔𝜂1𝑑𝜂𝜆+𝑖𝜔𝐹(𝜔). (5.11)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 that 𝒬₁ is a bounded operator in 𝐿_𝑝(−∞,∞). We have ‖𝒬₁[𝑓]‖_{𝐿𝑝(−∞,∞)}=(^∞_−∞𝑑𝑥|^𝑥_−∞𝑓(𝜂)𝑒^{−𝜆(𝑥−𝜂)}𝑑𝜂|^𝑝)^1/𝑝⩽(^∞_−∞𝑑𝑥|^𝑥_−∞|𝑓(𝜂)|𝑒^{−𝜎(𝑥−𝜂)}𝑑𝜂|^𝑝)^1/𝑝, (5.12)where 𝜎=ℜ𝜆⩾𝜎_𝑎>0. Let us use 𝑦 instead of 𝜂 in (5.12), that is, 𝑦=𝑥−𝜂, and have ‖𝒬₁[𝑓]‖_{𝐿𝑝(−∞,∞)}⩽(^∞_−∞𝑑𝑥|^∞₀𝑓(𝑥−𝑦)𝑒^−𝜎𝑦𝑑𝑦|^𝑝)^1/𝑝⩽^∞₀𝑒^−𝜎𝑦𝑑𝑦(^∞_−∞|𝑓(𝑥−𝑦)|^𝑝𝑑𝑥)^1/𝑝⩽1𝜎_𝑎‖𝑓‖_{𝐿𝑝(−∞,∞)}. (5.13)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 Lemma 5.4.

Lemma 5.5. (1) The function(𝜆−𝑖𝜔)⁻¹is a Mikhlin multiplier.
(2) Let 𝑓∈𝐿_𝑝(−∞,∞),𝑝>1, and let 𝑔be related to 𝑓 through the operator 𝒬₃ by the following formula:𝑔(𝑥,𝜆)=(𝒬₃𝑓)(𝑥)=^∞_𝑥𝑒^{−𝜆(𝜉−𝑥)}𝑓(𝜉)𝑑𝜉. (5.14) Then, the Fourier transforms are related through the multiplier 1𝐺(𝜔,𝜆)=𝜆−𝑖𝜔𝐹(𝜔). (5.15) (3) 𝒬₃ is a bounded operator in 𝐿_𝑝(−∞,∞) for each 𝜆 such that ℜ𝜆⩾𝜎_𝑎.

To present our next result, we have to introduce the following notations.

(i) ℰ₁(𝑧) is “the exponential integral” [25] defined by the formula ℰ₁1(𝑧)=𝜋^∞_𝑧𝑒^−𝜏𝜏𝑑𝜏,ℜ𝑧>0. (5.16)The following properties of ℰ₁ will be needed in the sequel. From (5.16), we get ℰ^′₁𝑒(𝑧)=−^−𝑧1𝜋𝑧=−+𝜋𝑧1+𝑒^−𝑧𝜋𝑧, (5.17)which yields another representation for ℰ₁(𝑧)∶ℰ₁1(𝑧)=−𝜋ln𝑧+^𝑧₁1−𝑒^−𝜏𝜏𝑑𝜏. (5.18)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 𝐿_𝑝(−𝑏,𝑏) for any 𝑝⩾1.

(ii) Let us introduce a new function by the formula 𝑔₋(𝜆,𝑥)=𝑒^{−𝑖𝜆𝑥}ℰ₁(𝜆(𝑏−𝑥)),|𝑥|<𝑏. (5.19)For each 𝑥,𝑔₋ is an analytic function of 𝜆 on the complex plane with the branch cut along the negative real semiaxis and ^𝑏_−𝑏|𝑔₋(𝜆,𝑥)|^𝑝𝑑𝑥<∞,𝑝⩾1. (5.20)Below we define several linear operators that we need in the future.

(iii) Let 𝒫 be a projection, that is, (𝒫𝑓)(𝑥)=𝑓(𝑥)for|𝑥|<𝑏,(𝒫𝑓)(𝑥)=0for|𝑥|>𝑏. (5.21)

(iv) Let ℋ be the Hilbert transformation defined by [26, 27] 1(ℋ𝑓)(𝑥)=𝜋^∞∗_−∞𝑓(𝜉)𝑥−𝜉𝑑𝜉, (5.22)where “∗” (star) means that the integral is understood as a Cauchy principal value integral. Regarding the operator ℋ, we need the following result.

For any 𝑓∈𝐿_𝑝(−∞,∞), (5.22) defines a function ^0𝑥0𝐹^02𝑑𝑐=ℋ[𝑓]∈𝐿_𝑝(−∞,∞). In addition,ℋ[ℋ[𝑓]]=−𝑓. (5.23)The Mikhlin multiplier, corresponding to the Hilbert transformation, is given explicitly by [26] Γ(𝜔)=|𝜔|𝑖𝜔,−∞<𝜔<∞. (5.24)

(v) Let ℋ_𝑏 be a “finite Hilbert transformation” defined by (ℋ_𝑏1𝑓)(𝑥)=𝜋^𝑏∗_−𝑏𝑓(𝜉)𝑥−𝜉𝑑𝜉. (5.25)

(vi) Let 𝒬₁ be a Volterra integral operator defined in (5.8): (𝒬₁𝑓)(𝑥)=^𝑥_−∞𝑒^{−𝜆(𝑥−𝜉)}𝑓(𝜉)𝑑𝜉,𝑓∈𝐿_𝑝(−∞,∞). (5.26)

(vii) Let ℒ(𝜆) be the following operator: ℒ(𝜆)𝑓=^𝑥_−𝑏𝑒^{−𝜆(𝑥−𝜉)}𝑓(𝜉)𝑑𝜉=(𝒬₁𝒫)𝑓,𝑓∈𝐿_𝑝(−∞,∞),|𝑥|<𝑏. (5.27)

(viii) Let 𝐿(𝜆,𝑓) be the following linear functional: 𝐿(𝜆,𝑓)=^𝑏_−𝑏𝑒^𝜆𝜂𝑓(𝜂)𝑑𝜂. (5.28)We notice that 𝐿 defines a bounded linear functional in 𝐿_𝑝(−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 mapping 𝒫ℋ𝒬₁𝒫 in 𝐿_𝑝(−∞,∞)∶(𝒫ℋ𝒬₁𝒫)[𝑓]=(ℋ_𝑏ℒ(𝜆))[𝑓]−𝑔₋(𝜆,𝑥)𝐿(𝜆,𝑓)for|𝑥|<𝑏,(𝒫ℋ𝒬₁𝒫)[𝑓]=0for|𝑥|>𝑏. (5.29)

Proof. Using (5.21), (5.22), and (5.26), we get for 𝑓∈𝐿_𝑝(−∞,∞) and |𝑥|<𝑏, (ℋ𝒬₁𝒫)[𝑓]=ℋ[^𝑥_−∞𝑒^{−𝜆(𝑥−𝜉)}=1(𝒫𝑓)(𝜉)𝑑𝜉]𝜋^∞∗_−∞𝑑𝜉𝑥−𝜉^𝜉_−𝑏𝑒^{−𝜆(𝜉−𝜂)}=1(𝒫𝑓)(𝜂)𝑑𝜂𝜋[^−𝑏∗_−∞+^𝑏∗_−𝑏+^∞∗_𝑏]𝑑𝜉𝑥−𝜉^𝜉_−𝑏𝑒^{−𝜆(𝜉−𝜂)}(𝒫𝑓)(𝜂)𝑑𝜂≡𝑖₁+𝑖₂+𝑖₃. (5.30)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 𝑖₂=1𝜋^𝑏∗_−𝑏𝑑𝜉𝑥−𝜉^𝜉_−𝑏𝑒^{−𝜆(𝜉−𝜂)}𝑓(𝜂)𝑑𝜂=(ℋ_𝑏ℒ(𝜆)𝒫)[𝑓],|𝑥|<𝑏. (5.31)Finally, we consider 𝑖₃ and have 𝑖₃=1𝜋^∞∗_−𝑏𝑒^−𝜆𝜉𝑥−𝜉𝑑𝜉^𝑏_−𝑏𝑒^𝜆𝜂1𝑓(𝜂)𝑑𝜂=𝜋^∞∗_−𝑏𝑒^−𝜆𝜉𝑥−𝜉𝑑𝜉𝐿(𝜆,𝑓),|𝑥|<𝑏, (5.32)with 𝐿(𝜆,𝑓) being defined as (5.28).
Consider an integral from (5.32) (we are interested only in the case |𝑥|<𝑏): 1𝜋^∞_𝑏𝑒^−𝜆𝜉1𝑥−𝜉𝑑𝜉=−𝜋^∞_𝑏−𝑥𝑒^{−𝜆(𝜏+𝑥)}𝜏𝑒𝑑𝜏=−^−𝜆𝑥𝜋^∞_𝑏−𝑥𝑒^−𝜆𝜏𝜏𝑒𝑑𝜏=−^−𝜆𝑥𝜋^∞_{𝜆(𝑏−𝑥)}𝑒^−𝜏𝜏𝑑𝜏=−𝑒^−𝜆𝑥ℰ₁(𝜆(𝑏−𝑥)). (5.33)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 any 𝑓∈𝐿_𝑝(−∞,∞)∶(𝒫ℋ𝒬₂𝒫)[𝑓]=^𝛼1₀𝑎(𝑠)[(ℋ_𝑏ℒ(𝜆𝑠))[𝑓]+𝑔₋(𝜆𝑠,𝑥)𝐿(𝜆𝑠,𝑓)]𝑑𝑠,|𝑥|<𝑏, (5.34) where 𝒬₂ is an operator corresponding to a Mikhlin multiplier ∫^𝛼1₀𝑎(𝑠)𝑑𝑠/(𝜆𝑠+𝑖𝜔) with 𝑔₋,ℒ(𝜆),and𝐿 being 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⩾𝜎_𝑎>0 are the Mikhlin multipliers. We want to identify the operator that corresponds to the multiplier 𝜘₁(𝜔)=^𝛼1₀𝑎(𝑠)𝑑𝑠𝜆𝑠+𝑖𝜔(or𝜘₂(𝜔)=^𝛼2₀𝑎(−𝑠)𝑑𝑠𝜆𝑠−𝑖𝜔), (5.35)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: 1if𝐺(𝜆,𝑠)=𝜆𝑠+𝑖𝜔𝐹(𝜆),then𝑔(𝑥,𝑠)=^𝑥_−∞𝑒^{−𝜆𝑠(𝑥−𝜉)}𝑓(𝜉)𝑑𝜉. (5.36)Applying (5.36) to 𝑓∈𝐿_𝑝(−𝑏,𝑏), multiplying by 𝑎, and integrating, we obtain ^𝛼1₀𝑔(𝑥,𝑠)𝑎(𝑠)𝑑𝑠=^𝛼1₀𝑎(𝑠)𝑑𝑠^𝑥_−𝑏𝑒^{−𝜆𝑠(𝑥−𝜉)}𝑓(𝜉)𝑑𝜉. (5.37)Taking into account that the factor |𝜔|(𝑖𝜔)⁻¹ corresponds to the application of the Hilbert transformation, we get ℋ[^𝛼1₀𝑔(𝑥,𝑠)𝑎(𝑠)𝑑𝑠]=^𝛼1₀𝑎(𝑠)ℋ[^𝑥_−𝑏𝑒^{−𝜆𝑠(𝑥−𝜉)}𝑓(𝜉)𝑑𝜉]𝑑𝑠. (5.38)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∈𝐿_𝑝(−𝑏,𝑏) and 𝑎 is a continuous function [27]. Therefore, if we denote by 𝒬₂ the operator corresponding to the multiplier ∫^𝛼1₀𝑎(𝑠)(𝑑𝑠/(𝜆𝑠+𝑖𝜔)), the following formula holds: (𝒫ℋ𝒬₂𝒫)[𝑓]=^𝛼1₀𝑎(𝑠)[(ℋ_𝑏ℒ(𝜆𝑠))[𝑓]+𝑔₋(𝜆𝑠,𝑥)𝐿(𝜆𝑠,𝑓)]𝑑𝑠,|𝑥|<𝑏, (5.39)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 mapping𝒫ℋ𝑄₃𝒫 in 𝐿_𝑝(−∞,∞): (𝒫ℋ𝒬₃𝒫)[𝑓]=(ℋ_𝑏ℒ^∗(𝜆))[𝑓]+𝑔₊(𝜆,𝑥)𝐿(−𝜆,𝑓)for|𝑥|<𝑏,(𝒫ℋ𝒬₃𝒫)[𝑓]=0for|𝑥|>𝑏. (5.40) Here 𝒬₃ is an operator corresponding to the multiplier (𝜆−𝑖𝜔)⁻¹ (see Lemma 5.5); a linear operator ℒ^∗ is defined by ifℒ^∗𝑓=𝑔,then𝑔(𝑥)=^𝑏_𝑥𝑒^{−𝜆(𝜉−𝑥)}𝑓(𝜉)𝑑𝜉,|𝑥|<𝑏. (5.41) The function 𝑔₊(𝜆,⋅) is defined by the formula 𝑔₊(𝜆,𝑥)=𝑒^𝜆𝑥ℰ₁(𝜆(𝑏+𝑥)),|𝑥|<𝑏, (5.42) where ℰ₁ is 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∈𝐿_𝑝(−∞,∞), we have (ℋ𝒬₃𝒫)[𝑓]=ℋ[^∞_𝑥𝑒^{−𝜆(𝜉−𝑥)}(𝒫𝑓)(𝜉)𝑑𝜉]=ℋ[^𝑏_𝑥𝑒^{−𝜆(𝜉−𝑥)}1𝑓(𝜉)𝑑𝜉]=𝜋^∞∗_−∞𝑑𝜉𝑥−𝜉^𝑏_𝜉𝑒^{−𝜆(𝜂−𝜉)}=1(𝒫𝑓)(𝜂)𝑑𝜂𝜋[^−𝑏∗_−∞+^𝑏∗_−𝑏+^∞∗_𝑏]𝑑𝜉𝑥−𝜉^𝑏_𝜉𝑒^{−𝜆(𝜂−𝜉)}(𝒫𝑓)(𝜂)𝑑𝜂≡𝑖^′₁+𝑖^′₂+𝑖^′₃. (5.43)The integral 𝑖^′₃=0 since (𝒫𝑓)(𝜂)=0 for 𝜂>𝑏. Consider the second integral, then we have 𝑖^′₂=1𝜋^𝑏∗_−𝑏𝑑𝜉𝑥−𝜉^𝑏_𝜉𝑒^{𝜆(𝜂−𝜉)}𝑓(𝜂)𝑑𝜂=(ℋ_𝑏ℒ^∗(𝜆))[𝑓],|𝑥|<𝑏, (5.44)where ℒ^∗ is given in (5.41). Finally, for 𝑖^′₁ we have when |𝑥|<𝑏 that 𝑖^′₁=1𝜋^−𝑏∗_−∞𝑒^𝜆𝜉𝑑𝜉𝑥−𝜉^𝑏_𝜉𝑒^−𝜆𝜂1𝑓(𝜂)𝑑𝜂=𝜋^−𝑏_−∞𝑒^𝜆𝜉𝑑𝜉𝑥−𝜉^𝑏_−𝑏𝑒^−𝜆𝜂1𝑓(𝜂)𝑑𝜂=𝜋^−𝑏_−∞𝑒^𝜆𝜉𝑑𝜉𝑥−𝜉𝐿(−𝜆,𝑓). (5.45)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 1𝜋∫^−𝑏_−∞𝑒^𝜆𝜉1𝑥−𝜉𝑑𝜉=𝜋∫^∞_−𝑏𝑒^−𝜆𝜂𝑥+𝜉𝑑𝜉=𝑒^𝜆𝑥1𝜋∫^∞_𝑥+𝑏𝑒^{−𝜆(𝜏)}𝜏𝑑𝜏=𝑒^𝜆𝑥ℰ₁(𝜆(𝑥+𝑏))=𝑔₊(𝜆,𝑥), (5.46)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 multiplier 𝜘₂ from (5.35) can be given in the following form: ^𝛼2₀𝑎(−𝑠)[(ℋ_𝑏ℒ^∗(𝜆𝑠))[𝑓]+𝑔₊(𝜆𝑠,𝑥)𝐿(−𝜆𝑠,𝑓)]𝑑𝑠,|𝑥|<𝑏. (5.47)

Collecting together the results of Lemmas 5.4–5.8 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 for 0<𝑀<1 can be expressed in the form 𝑊_𝑎(𝑥,𝜆^{0𝑥002𝑑𝑐})=1−𝑀²2×{ℋ_𝑏[𝐹(⋅,^{0𝑥002𝑑𝑐}1𝜆)]−1−𝑀²[(𝜆ℋ_𝑏ℒ(𝜆))[𝐹(⋅,𝜆^{0𝑥002𝑑𝑐})]−𝜆𝑔₋(𝑥,𝜆)𝐿(𝜆,𝐹(⋅,^{0𝑥002𝑑𝑐}−𝜆))]_𝛼10𝑎(𝑠)[(𝜆ℋ_𝑏ℒ(𝜆𝑠))[𝐹(⋅,𝜆^{0𝑥002𝑑𝑐})]−𝜆𝑔₋(𝑥,𝜆𝑠)𝐿(𝜆𝑠,𝐹(⋅,^{0𝑥002𝑑𝑐}+𝜆))]𝑑𝑠_𝛼20𝑎(−𝑠)[(𝜆ℋ_𝑏ℒ^∗(𝜆𝑠))[𝐹(⋅,𝜆^{0𝑥002𝑑𝑐})]−𝜆𝑔₊(𝑥,𝜆𝑠)𝐿(−𝜆𝑠,𝐹(⋅,𝜆^{0𝑥002𝑑𝑐}))]𝑑𝑠},|𝑥|<𝑏. (5.48)

We will modify (5.48) to a more conventional form using Tricomi-Söhngen result [26, 27].

Proposition 5.11. Let 𝜌(𝑥)=(𝑏−𝑥)/(𝑏+𝑥),|𝑥|<𝑏, and let 𝐿_2,𝜌(−𝑏,𝑏) and 𝐿_2,𝜌−1(−𝑏,𝑏) be the two weighted Hilbert spaces with the weights 𝜌 and 𝜌⁻¹, respectively . Then, the “airfoil equation” can be given in the form
1𝑓(𝑥)=𝜋^𝑏∗_−𝑏𝑤(𝜉)1𝑥−𝜉𝑑𝜉=𝜋^𝑏∗_−𝑏𝑣(𝜉)𝑥−𝜉𝜌(𝜉)𝑑𝜉≡ℍ[𝑣]. (5.49) The operator ℍ defined in (5.49) is a Hilbert space isomorphism of 𝐿_2,𝜌(−𝑏,𝑏) onto 𝐿_2,𝜌−1(−𝑏,𝑏). The inverse operator ℍ⁻¹ is given by 𝑣(𝑥)=ℍ⁻¹1[𝑓](𝑥)=𝜋^𝑏∗_−𝑏𝑓(𝜉)𝜌𝑥−𝜉⁻¹(𝜉)𝑑𝜉,𝑓∈𝐿_2,𝜌−1(−𝑏,𝑏). (5.50) Equation (5.50) represents the classical S𝑜̈hngen inversion formula for the airfoil equation in the presence of the Kutta condition. Multiplying both sides of (5.50) by 𝜌, one obtains 1𝑤(𝑥)=𝑣(𝑥)𝜌(𝑥)=𝜋𝑏−𝑥𝑏+𝑥^𝑏∗_−𝑏𝑏+𝜉𝑏−𝜉𝑓(𝜉)𝑥−𝜉𝑑𝜉≡𝒯[𝑓],𝑓∈𝐿_2,𝜌−1(−𝑏,𝑏). (5.51)

Our goal is to apply the operator 𝒯 to both sides of (5.48) in order to reduce it to a form convenient to deal with.

Lemma 5.12. The operator 𝒯 can 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 21−𝑀²𝑊_𝑎(𝑥,𝜆)=ℋ_𝑏[𝐺₁]+𝐺(𝑥,𝜆), (5.52)with 𝐺∈𝐿_𝑝(−𝑏,𝑏),𝑝>4/3, and 𝐺₁∈𝐿_𝑝(−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 𝐹₁=ℋ_𝑏1[𝐹]+1−𝑀²ℋ_𝑏[𝜆ℒ(𝜆)𝐹]−∫^𝛼1₀𝑎(𝑠)ℋ_𝑏[ℒ(𝜆𝑠)𝐹]𝑑𝑠+∫^𝛼2₀𝑎(−𝑠)ℋ_𝑏[𝜆ℒ^∗(𝜆𝑠)𝐹]𝑑𝑠. (5.53)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 ∫^𝛼1₀𝑎(𝑠)ℋ_𝑏[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∈𝐿_𝑝(−𝑏,𝑏), that is, both integrals go over finite intervals, one can change the order of integration [27]. Thus, ^𝛼1₀𝑎(𝑠)ℋ_𝑏[ℒ(𝜆𝑠)𝐹]𝑑𝑠=ℋ_𝑏[^𝛼1₀𝑎(𝑠)ℒ(𝜆𝑠)𝐹]𝑑𝑠. (5.54)Therefore, 𝐹₁=ℋ_𝑏1[𝐹−1−𝑀²𝜆ℒ(𝜆)𝐹−^𝛼1₀𝑎(𝑠)ℒ(𝜆𝑠)𝐹𝑑𝑠+^𝛼2₀𝑎(−𝑠)ℒ^∗(𝜆𝑠)𝐹𝑑𝑠]. (5.55)It can be easily shown that the Hilbert transform ℋ_𝑏 in (5.55) is applied to a function from 𝐿_𝑝(−∞,∞) as long as 𝐹∈𝐿_𝑝(−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 𝐿_𝑝(−𝑏,𝑏), it suffices to check that the two functions 𝑔₋(⋅,𝜆) and 𝑔₊(⋅,𝜆) belong to 𝐿_𝑝(−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 operator 𝒯 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 ℎ₋(𝑥,𝜆)=𝒯[𝑒^𝑏𝜆𝑔₋(⋅,𝜆)],ℎ₊(𝑥,𝜆)=𝒯[𝑒^𝑏𝜆𝑔₊(⋅,𝜆)]. (5.56)Evaluating ℎ₋, we have ℎ₋1(𝑥,𝜆)=𝜋𝑏−𝑥𝑏+𝑥^𝑏∗_−𝑏𝑏+𝜉𝑔𝑏−𝜉⁻(𝜉,𝜆)𝑒^𝑏𝜆=1𝜉−𝑥𝑑𝜉𝜋𝑏−𝑥𝑏+𝑥^𝑏∗_−𝑏𝑏+𝜉𝑏−𝜉𝑑𝜉𝑒𝜉−𝑥^{(𝑏−𝜉)𝜆∞}₁𝑒^{−(𝑏−𝜉)𝜆𝑡}𝑡=1𝑑𝑡𝜋𝑏−𝑥𝑏+𝑥^𝑏∗_−𝑏𝑏+𝜉𝑏−𝜉𝑑𝜉𝜉−𝑥^∞₁𝑒^{−(𝑏−𝜉)𝜆(𝑡−1)}𝑡𝑑𝑡. (5.57)Change of the variable of integration in (5.57), 𝜎=(𝑏−𝜉)(𝑡−1), yields ℎ₋1(𝑥,𝜆)=𝜋𝑏−𝑥𝑏+𝑥^𝑏∗_−𝑏𝑏+𝜉𝑏−𝜉𝑑𝜉𝜉−𝑥^∞₀𝑒^−𝜆𝜎=1𝑏−𝜉+𝜎𝑑𝜎𝜋𝑏−𝑥𝑏+𝑥^∞₀1𝑏−𝑥+𝛿2𝑏+𝜎𝜎𝑒^𝜆𝜎𝑑𝜎,|𝑥|<𝑏. (5.58)Similarly, it can be shown thatℎ₊1(𝑥,𝜆)=𝜋𝑏−𝑥𝑏+𝑥^∞₀1𝑏+𝑥+𝜎𝜎𝑒2𝑏+𝜎^𝜆𝜎𝑑𝜎. (5.59)The following properties for ℎ₋ and ℎ₊ hold:lim_{𝑥→𝑏−0}ℎ₋(𝑥,𝜆)=1,lim_{𝑥→𝑏−0}ℎ₊(𝑥,𝜆)=0. (5.60)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: 21−𝑀²𝒯[𝑊_𝑎(⋅,𝜆^′1)]=[1−1−𝑀²𝜆ℒ(𝜆)]𝐹(⋅,𝜆^′𝜆)+1−𝑀²ℎ₋(𝑥,𝜆)𝑒^−𝑏𝜆𝐿(𝜆,𝐹(⋅,𝜆^′−))_𝛼10𝑎(𝑠)(𝜆ℒ(𝜆𝑠))[𝐹]𝑑𝑠+^𝛼1₀𝑎(𝑠)𝜆ℎ₋(⋅,𝜆𝑠)𝑒^{−𝑏𝜆𝑠}𝐿(𝜆𝑠,𝐹(⋅,𝜆^′+))𝑑𝑠_𝛼20𝑎(−𝑠)(𝜆ℒ^∗(𝜆𝑠))[𝐹]𝑑𝑠+^𝛼2₀𝑎(−𝑠)𝜆ℎ₊(⋅,𝜆𝑠)𝑒^{−𝑏𝜆𝑠}𝐿(−𝜆𝑠,𝐹(⋅,𝜆^′))𝑑𝑠. (5.61)

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⩾𝛿_𝑎≫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 ℋ_𝑏 (ℋ_𝑏 is a singular integral operator in 𝐿_𝑝(−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.

Acknowledgment

The authors highly appreciate the partial support of the National Science Foundation (Grants nos. ESC-0400730 and DMS-0604842).