Solving the Cauchy Problem Related to the Helmholtz Equation

through a Genetic Algorithm

JAMAL DAOUDI1, CHAKIR TAJANI2∗

1,2SMAD Team, Department of Mathematics

Polydisciplinary Faculty of Larache

Abdelmalek Essaadi University

MOROCCO

Abstract: - The Cauchy problem associated with the Helmholtz equation is an ill-posed inverse problem

that is challenging to solve due to its instability and sensitivity to noise. In this paper, we propose a

metaheuristic approach to solve this problem using Genetic Algorithms in conjunction with Tikhonov

regularization. Our approach is able to produce stable, convergent, and accurate solutions for the Cauchy

problem, even in the presence of noise. Numerical results on both regular and irregular domains show the

eectiveness and accuracy of our approach.

Key-Words: - Inverse Problem, Helmholtz Equation, Tikhonov Regularization, Optimization, Genetic

Algorithms

Received: May 19, 2023. Revised: August 21, 2023. Accepted: September 27, 2023. Published: October 9, 2023.

1 Introduction

Let Ωbe an open and bounded domain in R2with

a smooth boundary Γ. We divide the boundary

into two disjoint parts, Γ = Γi∪Γc, where Γi∩Γc=

∅and mes(Γc)6= 0.

Mathematical formulation of the Cauchy prob-

lem for the Helmholtz equation can be expressed

as:

(P) : 









−∆u+κ2u= 0 in Ω

u=fon Γc

∂nu=gon Γc

(1)

where ∆is the Laplacian operator, ∂ndenotes the

outward normal derivative, κis a complex num-

ber (wave number), fand gare the Cauchy data

available on the accessible boundary Γc.

This problem arises in many important physi-

cal applications related to wave propagation and

vibration phenomena see, [1], [2], [3], [4].

The Helmholtz equation is a fundamental equa-

tion in physics that describes the propagation of

waves. In the past century, extensive research

has been carried out on the direct problem of

the Helmholtz equation, which involves nding

the solution to the equation given boundary data

(Dirichlet, Neumann and Dirichlet-Neumann).

However, in practical situations, it is often not

possible to obtain boundary data for the entire

boundary. Instead, we may only have access to

noisy data related to a specic section of the

boundary or some points within the domain. This

leads to inverse problems, in which the goal is to

nd the solution to the equation given incomplete

or noisy data.

The Cauchy problem for the Helmholtz equa-

tion is an example of an inverse problem that is ill-

posed. This means that small perturbations in the

given data can result in signicant changes to the

solution, and the solution does not continuously

depend on the given Cauchy data. This makes it

dicult to nd accurate solutions to the Cauchy

problem, and special methods are often required.

References, [5], and, [6], discuss the ill-posedness

of the Cauchy problem for the Helmholtz equation

in more detail. Conventional numerical methods

are not sucient for solving the problem being

investigated. To remedy this, numerous numeri-

cal methods have been suggested in order to solve

the Cauchy problem for the Helmholtz equation,

such as the method of fundamental solution, [7],

the method of plane waves, [8], the Landweber ap-

proach, [9], the method of Fourier regularization,

[10], [11], conjugate gradient method, [12], [13],

the boundary element minimal error method, [14],

the method of spherical wave expansion, [15], the

method of boundary knot, [16].

For solving the Cauchy problem associated

with the Helmholtz equation, two main ap-

proaches are commonly used: iterative methods

and direct methods. Iterative methods start with

an initial guess of the solution and then iteratively

improve the guess by minimizing a cost function,

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such as the error between the calculated and mea-

sured data. This process can be computationally

intensive, as the problem must be solved at each

iteration. However, iterative methods are often

more robust than direct methods and can be used

to solve problems that are ill-posed. On the other

hand, direct methods require less computational

time as the problem is discretized only once, but

they may be susceptible to numerical instability.

It should be noted that the aforementioned

methods are deterministic techniques, However,

deterministic approach has limitations, particu-

larly when dealing with complex systems that are

inuenced by many variables and factors. In such

cases, deterministic models may not be able to

account for all the variables and uncertainties in-

volved, leading to inaccuracies and incomplete un-

derstanding of the system. Beside deterministic

techniques, there is a second class called stochastic

techniques. Stochastic techniques refer to a class

of mathematical methods that deal with random-

ness, uncertainty, and probability.

Metaheuristic algorithms draw inspiration from

natural processes such as biological evolution,

swarm intelligence, and other phenomena. For

instance, genetic algorithms, [17], emulate natu-

ral selection and evolution, while particle swarm

optimization, [18], mimics the collective behavior

of ocks of birds or swarms of insects. Similarly,

ant colony optimization, [19], is based on the be-

havior of real ant colonies, and the bat algorithm,

[20], imitates the echolocation behavior of bats.

These algorithms are designed to eciently ex-

plore a vast search space by iteratively generating

and evaluating candidate solutions, with the goal

of nding an optimal or near-optimal solution.

This study proposes a new computational al-

gorithm for solving the Cauchy problem related

to the Helmholtz equation. The method is based

on a genetic algorithms coupled with Tikhonov

Regularization, and considers the solution on the

underspecied Γiboundary as a control in a di-

rect mixed well-posed problem. The proposed ap-

proach aims at accurately tting the Cauchy data

on the overspecied boundary Γcby minimizing

a cost function that measuring the discrepancies

between the available data and the corresponding

calculated values.

The rest of this paper is outlined as follows.

Section 2 introduces the formulation of the inverse

problem under consideration. In Section 3, we of-

fer a concise overview of genetic algorithms and

explore the capabilities of the real-coded genetic

algorithms, which has been tailored to solve the

inverse problem under consideration. To demon-

strate the accuracy and eciency of the proposed

method, Section 4 presents two numerical exam-

ples featuring regular and irregular domains. Fi-

nally, Section 5 summarizes the key ndings of the

research and oers concluding remarks.

2 Formulation of the problem as an

optimization problem

2.1 Optimization problem

The purpose of this paper is the use an adapted

genetic algorithm with real coded combined with

nite element method to estimate the Cauchy data

on the inaccessible part of the boundary Γifrom

the available data fand gon Γc.

Since the φand φ0on the boundary Γiis to be

determined, two direct problems are considered:

(PD) : 









−∆u+κ2u= 0 in Ω

u=φon Γi

∂nu=gon Γc

(2)

(PN) : 









−∆u+κ2u= 0 in Ω

u=fon Γc

∂nu=φ0on Γi

(3)

It should be noted that if φ∈H1/2(Γi)and

g∈H−1/2(Γc)(resp f∈H1/2(Γc)and φ0∈

H−1/2(Γc)), then there is a unique solution u(φ, g)

(resp u(φ0, f)) of the direct problem Eq.(2) (resp

Eq.(3)) see, [21], and we are looking for φ(resp φ0

) such that:

(u(φ, g) = fon Γc

∂nu(φ0, f) = gon Γc

(4)

which leads to minimize the least-squares func-

tional JDand JNdened by:

JD(φ) = 1

2ku(φ, g)−fk2

L2(Γc)(5)

and

JN(φ0) = 1

2ku(φ0, f)−gk2

L2(Γc)(6)

2.2 Tikhonov regularization

In an inverse problem, the observed data is typ-

ically aected by noise and measurement errors,

which can lead to instability and poor accuracy

in the estimation of the unknown parameters.

Tikhonov regularization helps to overcome these

issues by introducing a regularization term.

In this case the Tikhonov regularization method is

used to convert the tow objective functions Eq.(5)

and Eq.(6) to the well-posed form given as follows:

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JDR(φ) = 1

2ku(φ, g)−fk2

L2(Γc)+α

2kφk2

L2(Γi)(7)

and

JN R(φ0) = 1

2ku(φ0, f)−gk2

L2(Γc)+β

2kφ0k2

L2(Γi)

(8)

where αand βare the regularization parameters,

2kφk2and β

2kφ0k2are the well-known Tikhonov

regularization terms. In the literature, there

are various eective techniques recommended for

choosing the most suitable value for the regular-

ization parameter, including the L-curve method,

[22], and the discrepancy principle, [23]. These

approaches avoid the need to use excessively small

or large positive values of α(resp β) to assure the

stability of the solution.

3 Application of genetic algorithms

to inverse problem

3.1 Overview of genetic algorithms

Genetic algorithms, [24], have proven to be eec-

tive in solving a variety of optimization problems.

They are based on the principles of biological evo-

lution and operate as a searching method. A pop-

ulation of chromosomes is used to represent po-

tential solutions and genetic operators are applied

to progressively improve each chromosome, which

becomes the basis for the next generation. This

process continues until the desired number of gen-

erations has been completed or a predened stop-

ping criteria value has been reached.

Genetic algorithms oer a number of advan-

tages over other optimization approaches. First,

they search from a population of solutions instead

of just one. Second, they can use any tness func-

tion, even if it is not continuous. Third, they

use random operators to generate new solutions.

Fourth, they do not need to know anything about

the problem to nd a good solution.

Genetic algorithms typically consist of the fol-

lowing basic elements:

1. Initialization: The genetic algorithm begins

by creating a population of potential solutions

to the problem being solved. This is typically

done by randomly generating a set of indi-

viduals, where each individual is a potential

solution represented as a set of genes or chro-

mosomes.

2. Fitness function: The tness function is used

to evaluate each individual in the population

and assign a tness score based on how well

it solves the problem being considered. The

tness score is used to select individuals for

reproduction in the next generation.

3. Selection: The selection process involves

choosing the ttest individuals from the cur-

rent generation to be parents for the next gen-

eration. The individuals are selected using

various techniques such as roulette wheel se-

lection or tournament selection.

4. Crossover: Crossover is the process of com-

bining genetic material from two parents to

create a new individual in the next genera-

tion. This is typically done by selecting two

parents based on their tness score and swap-

ping genetic material between them to create

a new individual.

5. Mutation: Mutation introduces random

changes to the genetic material of an individ-

ual, leading to potentially new and improved

solutions. It is typically applied to a small

fraction of individuals in the population to

maintain genetic diversity.

6. Termination: The algorithm terminates when

a stopping criterion is met, such as reaching

a desired tness score or running for a certain

number of generations.

We can summarize these steps in the following di-

agram, Fig.1.

Figure 1: Flowchart of genetic algorithms.

These elements work together to produce a pop-

ulation of increasingly t individuals that can be

used to nd optimal solutions to a wide range of

problems.

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3.2 Genetic operators

In order to address the Cauchy problem associ-

ated with the Helmholtz equation, we consider a

real coded (oating-point) GAs (RCGA), which

perform better than binary coded GA, where the

chromosome corresponds to a vector of real pa-

rameters, the gene corresponds to a real number,

and the allele corresponds to a real value.

3.2.1 Crossover

This operator is the most important operator in

the genetic process between two individuals se-

lected according to a probability pc, [25], [26], pro-

ducing new ospring. Several crossover operators

have been developed, adapted to the type of en-

coding used. In this study, we consider the arith-

metic crossover operator with real encoding.

Typically, parents are denoted as:

P ar(1) =P ar(1)

1, . . . ., P ar(1)

n

P ar(2) =P ar(2)

1, . . . , P ar(2)

n(9)

The representation of ospring is given by:

Off(1) =Off(1)

1, . . . , Off(1)

n

Off(2) =Off(2)

1, . . . , Off(2)

n(10)

where, Off(i)and P ar(j)represent the ith o-

spring and jth parent, respectively. The variable

ndenotes the number of genes on each individual.

In arithmetic crossover two parents produce

two ospring which can be expressed using Eq.(11)

given as follows:

Off(1)

i=αiP ar(1)

i+ (1 −αi)P ar(2)

Off(2)

i=αiP ar(2)

i+ (1 −αi)P ar(1)

(11)

where, αirepresents uniformly distributed random

numbers.

It is important to mention that αican be gener-

ated at each generation; in this case we talk about

non-uniform arithmetic crossover, [27].

3.2.2 Mutation

The mutation operator is applied to specic ele-

ments of selected chromosomes. If we consider the

selected chromosome at the kth generation, in the

following form:

Off = (Off1, . . . , Of fi, . . . , Offn)

the form of the obtained chromosome, knowing

that Offiis the element to be mutated, is given

by:

Off0= (Off1, . . . , Of f 0

i, . . . , Offn)

Non-uniform mutation is one of the commonly

used mutation operators in real-coded genetic al-

gorithms (RCGAs), [28], [29]. It is dened as fol-

lows:

Off0

i=Offi+δ(k, ui−Offi),if τ= 0

Offi−δ(k, Offi−li),if τ= 1

(12)

where, τis a random digit which takes either the

value 0or 1and, liand uiare the upper and

lower bounds of Offi. The function δ(k, y)yields

a value within the range [0, y]and it is designed

such that the likelihood of the value being close to

0 becomes higher as kincreases. The value of the

function δis given as follows:

δ(k, y) = y1−η1−k

Tb(13)

where,

•ηis a uniformly distributed random number

in the interval [0,1],

•kis the current generation,

•Tis the maximal generation number,

•bis a system parameter determining the de-

gree of non-uniformity.

3.3 Computation procedure for the GA

Optimization

In this section, we describe the steps involved

in using the proposed genetic algorithm (GA) to

solve the Cauchy problem for the Helmholtz equa-

tion.

The dierent steps of the proposed procedure are

given by:

•Step 1 : Parameter setting:

–N:Population size

–pc:Probability of Crossover

–pm:Probability of Mutation

–MaxGen :Maximum number of Genera-

tion

•Step 2 : Random generation of initial popu-

lation φ(0)

pwith p= 0,· · · , N.

•Step 3 : Solve the direct problem Eq.(14) be-

low, for each given φ(0)

pby the nite element

method.

(PGA)p:









−∆u+κ2u= 0 in Ω

u=φ(0)

pon Γi

∂nu=gon Γc

(14)

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•Step 4 : Compute the tness value for each

individual using JDR φ(0)

p(Eq.(7)).

•Step 5 : Create the next generation φ(1)

pusing

the GA process given by:

φ(1)

p=Mu.Cr.Se(φ(0)

where:

–Se:Random selection,

–Cr:Arithmetic Crossover,

–Mu:Non-uniform mutation.

•Step 6 : Return to step 3 and replace φ(0)

with φ(1)

•Step 7 : The genetic process continue for φ(m)

m= 1,2,· · · ,MaxGen.

The purpose of this procedure is to establish

the Dirichlet condition on Γi. However, if we want

to determine the Neumann condition instead, we

can modify the procedure by implementing certain

adjustments. Specically, in step 2, we should re-

place φ(0)

pwith (φ(0)

p)0and in step 3 (PGA)pby

(PGA)0

psuch that:

(PGA)0

p:









−∆u+κ2u= 0 in Ω

u=fon Γc

∂nu= (φ(0)

p)0on Γi

(15)

Finally, in step 4 and step 5, we need to substitute

Eq.(8) for Eq.(7).

4 Numerical results and discussion

The aim of this study is to nd an approximation

of the missing Dirichlet and Neumann boundary

conditions. Since we do not know the exact

form of the solution, we will use the polynomial

approximation. In order to illustrate the conver-

gence and the stability of the proposed numerical

method, we solve the Cauchy problem for the

Helmholtz equation by considering two cases of

domains in 2D.

The genetic algorithm used for evolving each

individual population employed the following ge-

netic operators and parameters:

• Number of Generations: MaxGen = 200,

• Population size: npop = 60,

• Crossover operator: Arithmetic Crossover,

with pc= 0.9,

• Mutation operator: Non-uniform Mutation,

with pm= 0.01,

• Insertion: We consider the principle of elitism

to conserve the best solution in the next gen-

eration.

The experiments were conducted on a machine

with an Intel(R) Core(TM) i7-8565U CPU @

1.80GHz 1.99 GHz. The implementation of the al-

gorithm was done using the software FreeFem++,

[30], which is a free software for solving partial

dierential equations (PDEs) in R2and R3using

nite element method. It is worth noting that the

FreeFem++ language enables the rapid specica-

tion of the EDP (direct problem resulting from

the considered optimization problem) by writing

its variational formulation.

We investigate also the stability of the proposed

algorithm by perturbing the Cauchy data fand g

as follows:

(fper, gper) = (1 + νθ)(f, g)(16)

where νdenotes the noise level and θis a ran-

dom number in the range [−1,1] sampled using a

uniform distribution.

4.1 First case:

In the rst case, the numerical tests are made on

a unit square domain Ω =]0,1[2(Fig.2), where the

boundary Γ = ∂Ωis divided into two parts:

Γi={(0, y) : 0 < y < 1}

Γc= Γ\Γi

and the exact solution of the problem Eq.(1) with

k2= 5 is given by:

uex(x, y) = exp(2x−y)

Figure 2: Unit square with mesh.

Fig.3 presents the analytical solution in the

whole domain.

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Figure 3: Analytical solution.

4.1.1 Choice of regulation parameter

Table.[1] reveals that as the value of αand βde-

crease from 1e−01 to 1e−08, the JDR(φ)and

JN R(φ0)cost functions also decrease, indicating

a better t to the data. However, the rate of de-

crease slows down as we move towards smaller val-

ues of αand β, balancing the accuracy of the t

with the complexity of the solution. The results

indicate that a small amount of regularization is

sucient to prevent overtting for both JDR(φ)

and JN R(φ0), as the lowest cost is obtained for

α=β= 1e−05.

Fig.4 and Fig.6 illustrate the progressive con-

vergence of the numerical solution towards the an-

alytical solution throughout the iterative process.

Initially, the numerical solution exhibits substan-

tial deviation from the exact solution, but this dis-

crepancy diminishes rapidly with each iteration.

Such behavior highlights the ecacy of the iter-

ative method in eectively resolving the inverse

problem.

Figure 4: Trace of uon Γi.

Figure 5: Objective function JDR(φ).

Figure 6: The derivative of u(∂nu)on Γi.

Figure 7: Objective function JN R(φ0)for dierent

iterations.

Furthermore, Fig.5 and Fig.7 demonstrate the

signicant reduction in the objective functions

JDR(φ)and JN R(φ0)during the initial iterations.

As the iterations progress, the convergence rate of

the objective function gradually slows down, al-

though it ultimately reaches a low value by itera-

tion k= 200.

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αand β1e-01 1e-02 1e-03 1e-04 1e-05 1e-06 1e-07 1e-08

JDR(φ)1.72e-02 2.09e-03 5.9e-04 1.41e-04 2.56e-05 1.31e-04 3.52e-05 6.103e-04

JN R(φ0)7.108e-02 9.31e-03 1.36e-03 6.604e-04 5.10466e-04 5.19e-04 5.201e-04 5.48e-04

Table 1: Cost function JDR(φ)(resp. JN R(φ0)) for various values of α(resp. β).

4.1.2 Stability of the Proposed Method

Fig.8 and Fig.10 illustrate a comparison between

the numerical solution and the analytical solution

across varying levels of noise in the measurement

data. The numerical solution exhibits a slight de-

viation from the exact solution for low level of

noise level. However, this disparity increase, when

the noise level reaches a high values.

Figure 8: u/Γifor various levels of noise.

Figure 9: JDR(φ)for various levels of noise.

In Fig.9 and Fig.11, the cost function is

displayed for dierent noise levels, specically

ν= 1%,3%,5%,7%. The gures indicate that

as the noise level increases, the cost function

also increases, indicating a less precise t to the

data. Nevertheless, for low noise levels, the cost

function remains relatively low, suggesting that

the numerical solution still oers a satisfactory t

to the data.

Figure 10: ∂nu/Γifor various levels of noise.

Figure 11: JN R(φ0)for various levels of noise.

4.2 Second case:

In the second case, the numerical tests are per-

formed on an unit disc Fig.12 where the boundary

of this domain is divided into two parts:

Γi=n(x, y) : x2+y2= 1, y > 0, x > 0o

Γc= Γ\Γi

and the exact solution of the problem Eq.(1) with

k2=−2is given by:

uex(x, y) = sin(x) sin(y)

Fig.13 shows the analytical solution in the whole

domain.

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Figure 12: Unit disc with mesh.

Figure 13: Analytical solution.

4.2.1 Choice of regulation parameter

Table.[2] demonstrates that reducing the value of

αand βfrom 1e−01 to 1e−08 leads to a de-

crease in the cost functions JDR(φ)and JN R(φ0),

indicating an improved t to the data. However,

as αand βapproache smaller values, the rate of

decrease in the cost functions slows down, striking

a balance between t accuracy and solution com-

plexity. These results suggest that a small degree

of regularization eectively prevents overtting for

both JDR(φ)and JN R(φ0), with the lowest cost

achieved when α= 1e−04 and β= 1e−07.

Figure 14: Trace of uon Γi.

Fig.14 and Fig.16 illustrate the progressive con-

vergence of the numerical solution towards the an-

alytical solution throughout the iterative process.

Initially, the numerical solution exhibits substan-

tial deviation from the exact solution, but this dis-

crepancy diminishes rapidly with each iteration.

Such behavior highlights the ecacy of the iter-

ative method in eectively resolving the inverse

problem.

Figure 15: Objective function JDR(φ).

Furthermore, Fig.15 and Fig.17 demonstrate

the signicant reduction in the objective functions

JDR(φ)and JN R(φ0)during the initial iterations.

As the iterations progress, the convergence rate of

the objective function gradually slows down, al-

though it ultimately reaches a low value by itera-

tion k= 200. These observations indicate that the

obtained solution is highly precise and represents

an excellent t to the available data.

Figure 16: The derivative of u(∂nu)on Γi.

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αand β1e-01 1e-02 1e-03 1e-04 1e-05 1e-06 1e-07 1e-08

JDR(φ)4.65e-03 6.69e-04 1.354e-04 2.19e-05 5.12e-05 3.574e-04 8.63e-004 5.42e-04

JN R(φ0)1.76e-02 2.12e-03 4.18e-04 6.308e-04 1.31e-03 1.07e-04 1.94e-05 1.82e-03

Table 2: Cost function JDR(φ)(resp. JN R(φ0)) for various values of α(resp. β).

Figure 17: Objective function JN R(φ0)for dier-

ent iterations.

4.2.2 Stability of the Proposed Method

Fig.18 and Fig.20 illustrate a comparison between

the numerical solution and the analytical solution

across varying levels of noise in the measurement

data. For a low noise levels, the numerical so-

lution exhibits a slight deviation from the exact

solution. However, this disparity increase for im-

portant noise levels, in particular, when approach-

ing Neumann condition in the extremities of the

domain.

Figure 18: u/Γifor various levels of noise.

Fig.19 and Fig.21 depict the cost func-

tion for various noise levels, specically ν=

1%,2%,3%,5%. These gures show that as the

noise level increases, the cost function also in-

creases, indicating a decreased level of accuracy in

tting the data. However, even with higher noise

levels, the cost function remains relatively low, in-

dicating that the numerical solution still provides

a reasonably good t to the data.

It is important to note that reconstruct-

ing Dirichlet boundary conditions on inaccessible

parts of a boundary is typically more accurate and

reliable than reconstructing Neumann boundary

conditions. This is because Dirichlet boundary

conditions provide more information about the so-

lution than Neumann boundary conditions. The

factors that contribute to this dierence in accu-

racy and reliability can vary, and may depend on

the geometry of the domain being studied or the

regularity of the solution being reconstructed.

Figure 19: JDR(φ)for various levels of noise.

Figure 20: ∂nu/Γifor various levels of noise.

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Jamal Daoudi, Chakir Tajani

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Figure 21: JN R(φ0)for various levels of noise.

5 Conclusion

In this research paper, we address the challeng-

ing ill-posed inverse problem associated with the

Cauchy problem for the Helmholtz equation. Var-

ious optimization methods have been developed

to approximate solutions for such problems. In

this study, we explore the use of genetic algo-

rithms, which have the advantage of not requir-

ing specic regularity assumptions for the under-

lying functional. To achieve this objective, we

propose an optimization formulation that incor-

porates a Tikhonov regularization term. The ef-

fectiveness of our approach is evaluated through

numerical experiments conducted on both regular

and irregular domains. The results demonstrate

the eciency of the real-coded genetic algorithm,

enhanced with adapted genetic operators, in suc-

cessfully solving the Cauchy problem associated

with the Helmholtz equation. However, as with

any other investigation, the present study has lim-

itations related to computational complexity, the

need for parameter tuning, uncertainties in solu-

tion quality, and sensitivity to initial population.

These limitations create opportunities for future

research exploring the utilization of parallel com-

puting and self-adaptive algorithms.

Acknowledgment:

The authors would like to thank the editors and

the anonymous reviewers for their comments and

suggestions.

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Contribution of Individual Authors to the

Creation of a Scientic Article (Ghostwriting

Policy)

The authors equally contributed in the present re-

search, at all stages from the formulation of the

problem to the nal ndings and solution.

Sources of Funding for Research Presented in a

Scientic Article or Scientic Article Itself

No funding was received for conducting this study.

Conicts of Interest

The authors have no conicts of interest to

declare that are relevant to the content of this

article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.79

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