Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 949703, 9 pages

http://dx.doi.org/10.1155/2015/949703

## Conditional Optimization and One Inverse Boundary Value Problem

Physics Institute, Kazan Federal University, Kremlyovskaya Street 18, Kazan 420008, Russia

Received 20 April 2015; Accepted 23 June 2015

Academic Editor: Debasish Roy

Copyright © 2015 Pyotr N. Ivanshin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Here we construct different approximate solutions of the plane inverse boundary value problem of aerohydrodynamics. In order to do this we solve some conditional optimization problems in the norms , , and and some of their generalizations. We present the example clarifying the mathematical constructions and show that the supremum norm generalization seems to be the optimal one of all the functionals considered in the paper.

#### 1. Introduction

The formulation and the first attempt in solution of the inverse boundary value problem of aerohydrodynamics are due to Tumashev and Mangler [1, 2].

This problem is one of numerous boundary value problems. The example of the boundary value problem is the Hilbert one: given the real value of the analytic function on the known contour it should be possible to reconstruct this function in the inner domain with respect to the contour [3].

The example of the inverse boundary value problem is the following: Assume that we have two real functions , , , such that and . It should be possible to reconstruct both the contour of length and the analytic in the domain bounded by function whose real and imaginary parts at the contour point with the arc parameter coincide with the given functions , [2].

The inverse boundary value problem solution methods can be applied for problems of aerohydrodynamics [2, 4]. For example, all the wing construction methods I know of reduce to two basic problems [5]. The first of them is the plane wing section reconstruction, and the second is mutual positioning of these profiles. We consider the former of these problems.

Our problem is as follows: we have the air or fluid flow particles velocity distribution along the length of the unknown contour , that is, the function , . So we know only the contour length and not the form of this profile. We also assume that the air or fluid flow is potential. The inverse boundary value problem then is to find the shape of the contour depending only on the information on the particles velocity. Since the velocity is given only in the finite set of points it seems natural to apply a spline (not necessarily linear) in order to define this function for all the contour points. This approach was widely used by the authors of [4].

This problem requires certain resolution conditions which appear only after we find the contour parametrization. So these conditions cannot be satisfied initially. Here we try to find a minimal in some sense initial data deformation so that this deformed data meets these conditions. Note that almost all of the equation systems in the article can be solved only numerically due to their transcendent form.

Let us recall the solution procedure from [2].

*(1) Inverse Boundary Value Problem Solution*. Let the velocity distribution along the unknown contour be given; here is the curve length parameter. We must find the contour form. Assume that the contour length equals , the flow bifurcation happens at the point with the parameter value , and the flow velocity vanishes at , . Let for and for .

We consider the constants and the function Here the number is the flow circulation. If it is greater than then we have a positive lifting force for the wing profile.

Now we have the standard external inverse boundary value problem for the complex potential in the unknown contour exterior. This function has a simple pole and a logarithmic singularity at infinity and meets the boundary conditions We introduce the auxiliary analytic function. First note that the complex potential of the flow over the unit disc with the circulation equal to has the following form: Here we have the relations and for .

Since the velocity is in the critical points (where flow separates into two parts and where it vanishes) we have the following equations on the unknown constants , and two auxiliary angles relative to the flow critical points and : This system is uniquely resolvable [2].

Now we equate the complex flow potential real parts of the contour in the complex plane of coordinate to the complex flow potential of the unit circle in the plane of coordinate : Then we obtain the relation on the parameters of the contours. Finally we reconstruct the function , which maps the unit disc exterior onto the flow domain in the plane solving the Schwartz problem for the function at the unit disc exterior. The main object for our consideration is now the function

*(2) Resolvability Conditions*. The profile possesses the mechanical sense in case the function of the previous section meets certain conditions.

The first condition on the function naturally appears when we reconstruct our contour and equate the residue of at infinity to . Then the profile is a closed curve.

So the contour closeness condition is as follows: the function must meet the complex equalityHere and are known constants. Thus the first condition defines the first Fourier coefficients with and for the function .

We arrive at the second condition on due to purely mechanical reasons. Since the flow velocity is fixed at infinity (this is a flight speed) we equate the value of at infinity to some constant and obtain the following condition on :

Note that the constants , , and values under conditions (8), (9) depend only on the form of Zhukovskii-Mitchell modified function [2, 4].

So in the general case, that is, when the conditions do not hold, the problem does not have a solution and becomes an ill-posed problem.

*(3) The Mathematical Problem*. Ivanov [6] proposed application of quasisolutions to this problem. The notion of “quasisolution” was introduced earlier by Elizarov and Fokin [7, 8] and in our case can be described as follows.

*Definition 1. *Let the inverse boundary problem with the given velocity resolution procedure result in the function such that its first Fourier coefficients do not equal the desired ones. Let one denote the set of functions with appropriate first Fourier coefficients by . Then one says that quasisolution of the problem is the aerodynamic contour for whose construction we apply some function instead of . Here one chooses in the normed space so that

Thus we need to modify the function so that this modified function meets conditions (8), (9). Naturally this modification affects the initial data . So we must change the initial velocity distribution in some way.

This modification is as follows.

We consider the function instead of the constructed . Assume that meets conditions (8), (9). Clearly we need to make the modification as small as possible; that is, we search for the function with the minimal possible norm.

In [6] one can find the quasisolution which in our notation minimizes the function norm in . The solution procedure of [6] gave us in the form of the Fourier polynomial which allowed to meet the desired conditions.

It seems necessary to note that the articles [3, 9] contain function minimization similar to one presented here with respect to the norm . At the same time both the posed problems and the solution technique differ from that given in this paper.

We have purely mathematical problem.

We search for the function minimal in some norm under the following restrictions on its Fourier coefficients: We preserve the former of these relations and we rewrite the two latter relations as follows: and here , .

Without loss of generality we assume that . In the other case we simply consider the shifted variable instead of the initially given .

Finally we have the following problem.

We search for the function minimal in some norm under the following three conditions:

Here we construct the functions which minimize functionals , , and , , and give the solution for norm. We also present the example that compares the approximate solutions constructed with the help of the norms , and the functional .

#### 2. Approximate Solution Which Minimizes

The results of the section were proved in [10].

##### 2.1. Conditions (13): Modification Function Construction

Let us first solve the auxiliary problem. Given the function it is possible to construct the class of functions which meet the conditionWe must find .

Clearly the possible solution space choice yields the following result: and here . This can be easily proved by assuming the negative. Indeed, let us take into consideration a function such that . Then for and for . At the same time . Hence , and this inequality contradicts condition (14).

Let us apply this auxiliary problem solution to the case of . Then

Note that . Thus under conditions (13) the auxiliary function can be found from relation .

Let be the set of functions integrable on meeting conditions (13). Given these conditions it should be possible to find in the extreme function such that

Clearly the function is the solution of the problem for , and is also the solution for the case of . Thus it feels natural to construct the solution similarly to [4] and consider the sum . Nevertheless the solution minimizing the norm under condition of nonzero and differs from the given sum.

Let us find now the constants and so that that is, we search for the solution of the equation system

This nonlinear system has a unique solution. The constant value is the solution of equation . This relation possesses a unique solution in the interval . After we obtain the value of we naturally determine the one and only value of .

Thus existence of and makes it possible to construct the desired function

It is easy to verify that the function meets the last condition of (13).

Proposition 2. *Piecewise constant function of relation (20) is the problem solution for .*

##### 2.2. Quasisolution Which Minimizes

Let be the set of functions with almost everywhere bounded first derivative, meeting conditions (13). Given these conditions it should be possible to find a function minimizing the functional

Function class choice makes us search for the function which meets the equations in subsets of the interval . Solutions of these equations are the functions . The constants and must be chosen so that the solution is the continuous -periodic function on .

For example, for we obtain the following solution:and here can be found from conditions (13).

###### 2.2.1. Conditions (13) Satisfaction

Let us for the sake of simplicity fix .

###### 2.2.2. Problem Formulation

Let be a subclass of -periodic functions which meet conditions (13). Find the function such that

Let us construct the piecewise smooth function glued from the functions of the form .

For the case of we consider the functionThe unknown variables and are the solutions of the system In particular is the solution of the equation Under condition (24) this equation has a unique solution in the interval .

Proposition 3. *The function given by relation (25) is the solution of the problem under condition (24).*

Let us now consider the case of false relation (24). Then the solution is the functionHere the system on and is as follows: This system is uniquely resolvable for . Clearly for we have and .

Proposition 4. *The function given by relation (28) solves the problem for the case of *

*Note that the constant describes the relative impact of the auxiliary function derivative on the approximate solution. The greater this constant is the closer the resulting minimizing function is to a piecewise-linear curve being the solution of the similar problem for the functional (cf. Figure 1).*