In this paper, we present a survey of the inverse eigenvalue problem for a Laplacian equation based on available Cauchy data on a known part

In the paper, we consider the mathematical modeling employed in electrostatic imaging during nondestructive testing. This analysis engenders an inverse boundary value problem for a Laplace equation. This problem entails the identification of an unknown domain,

We suppose that

The inverse problem we are concerned with is to determine the unknown part,

My thesis (please consult [

Here, we aim to compare the results obtained using conformal mapping and the fixed-point method and observe the variations in the shape of

The conformal mapping method was presented in [

This paper is primarily aimed at presenting iterative methods to solve the inverse problem

However, in our approach, we consider a spectral problem on the simply connected domain

The family of holomorphic functions corresponding to the eigenvalues of the spectrum was identified via the fixed-point method and Nyström method.

The proposed reconstruction of the boundary curve

Further,

We can normalize the mapping

It is clear that the values of

Therefore, the operator

The Cauchy-Riemann equations give rise to nonlinear ordinary differential equations via Bessel equations for the values on this part of the boundary

(

The inverse problem

Using the polar coordinates of the solution

Therefore, by

A fixed point is a solution of equation

The determination of

General methods of construction of approximate conformal maps can be found in survey works by [

In this section, we attempt to determinate

Let

The boundary traces on

However, we have the nonlocal differential equation for the conformal map

The boundary correspondence function,

Based on the available Cauchy data,

Let us discuss some definitions and properties related to fixed points before employing the fixed-point methods to determinate

(Fixed-Point theorem of Banach)

Let

Consider any point

Then,

If

Hence

Hence,

The following is the version of the Banach fixed-point theorem and its corollary that is used in the subsequent proofs.

Let

Let

For any

The following estimate holds

Let

We assume that

Further, we introduce an operator

By integrating (

Using the boundary conditions of

It is easy to verify that

Now, we use the following theorem (see [

Let

Then, in terms of the holomorphic map

We obtain a fixed point defined by successive iteration of

The solution of the Cauchy problem is required to identify the boundary function

Using

The following convergence result justifies the procedure of iteration (

Let

Theorem

We calculate the derivate of the nonlinear operator

Therefore, we have,

Using a continuity argument to estimate the norm of

Thus,

The approximation of

We now determinate a numerical approximation of the conformal map

In this section, we use an ordinary differential equation of the boundary correspondence function to obtain an approximation of the conformal map,

Let us introduce two boundary correspondence functions,

On

Here, we use the length of arc

By removing the identity,

For

The successive approximations defined by (

When we obtain

So, on

We can express the expansion of

Now, we describe the numerical algorithm that we will implement in this paper. We use a method involving integral equations on

We give ourselves a regular mesh,

The approximations

The iterations allow approximations of the normal derivatives via trigonometric interpolation.

The integration interpolation polynomial on

Beginning with identity, we have

We substitute

This method, called the quadrature method, is the application of numerical methods of calculating integrals to obtain a linear system. We write

Explicitly, we can express (

Therefore, we can set

If

After fixing

The estimate of

We use

The traces of the curves corresponding to the equations defined below provide an approximate domain.

We attempt to determine

We know that (

After calculation, we have

We set

For

The Picard theorem ensures the convergence of

We can write

Thus, we have

Finally, we can deduce the following result:

Let

If

Then,

The Picard theorem on the convergence of

We can write

So,

This yields the result.

Further, we set

It is evident that

Therefore, according to (

For all

In the following argument, we pose

Hence, for all

For all

Consider the problem

Then,

It suffices to determine the integral and decimal parts of

We determine the approximate shape of the unknown part of the boundary corresponding to the two first eigenvalues.

Let

We put

First, we determine

The coefficients

Therefore, the calculation yields that corresponding to

Corresponding to the first eigenvalue

Therefore, we obtain the result for

Figure

N5 Alpha0.5.

Figure

N5 Alpha1.

It is to be noted that, in each case (

Corresponding to the second eigenvalue

Thus, we conclude the result for

Figure

N5 Alpha1.

Here too,

The result for the first eigenvalue obtained via the minimization functional method for

In Figure

Both methods yielded a spiral shape for

Results for

The following diagram allows us to compare the shapes of

In Figure

Variations of

All previous figures are obtained using “GeoGebra Geometry.”

Figures

So, the simulations yield the following results:

The shape of

However, the shape of

Figure

Therefore, the result obtained using the fixed-point method is more precise and further identifies

In summary,

Therefore,

This result can be applied to the one-dimensional case, but it is more interesting to study this phenomenon in dimensions three and four.

In an inverse problem with similar conditions, it is more relevant to use conformal mapping and the fixed-point method to yield interesting numerical results. The general case that

The data used to support the findings of this study are available from the corresponding author upon request.

The author declares that there is no conflict of interest.

We would like to thank Editage (