Approximating Multiple Roots of Applied Mathematical Problems Using Iterative Techniques

Round 1

Reviewer 1 Report

    In this paper, the authors introduced a new iterative techniques which is more stable, derivative free and multi-point.  Some better numerical results of well-known problems were given. Overall, the results of this paper seem valid and interesting. However, the authors should revise their paper according to the following comments.

(1) The language of this manuscript is acceptable, but several typos and grammatical errors are still observed. Please revise the whole manuscript carefully.

(2) In page 3,  inequality (9) is incomplete.

(3) In page 4, inequality (12) is incomplete.

(4) In page 5, inequalities (19) and (20) are incomplete.

(5) In page 9, Table 2 and Table 3 are incomplete.

(6) In page 10, Table 4 and Table 5 are incomplete.    

    In summary, I think that the paper will be acceptable if the authors may responsible to the above-mentioned comments and suggestions.

Author Response

Answer:

1) We have revised the written English in the whole manuscript.
2) Inequalities (9), (12) (19) and (20) have been completed.
3) Tables 2,3,4 and 5 have been revised in order to present the results more clear.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments to the authors: Review of "Approximate solutions of applied mathematical problems by iterative techniques".

I have read the manuscript in a detailed fashion. This paper is interesting and might be a significant addition to the existing literature. However, the paper can be considered after the authors make the following modifications:

1.     The originality of the paper needs to be stated clearly. It is of important to have sufficient results to justify the novelty of a high-quality journal paper. The Introduction should make a compelling case for why the study is useful along with a clear statement of its novelty or originality by providing relevant information and providing answers to basic questions such as: What is already known in the open literature? What is missing (i.e., research gaps)? What needs to be done, why and how? Clear statements of the novelty of the work should also appear briefly in the Abstract and Conclusions sections.

2.     In page 2 before line 35 should be label as Eq. (2). What is p in this equation?

3.     Check Tables 2 to 5 and make necessary corrections. Reduce the size because some columns cannot be read.

4.     Authors should explain the advantages of the present method over the other methods.

5.     The whole manuscript should be checked for typos and grammatical errors. An overall review is needed for fixing the grammatical and typos errors in the manuscript.

6.     The equations in Example 3 and 4 should be full stop not comma.

7.     All references should be checked very carefully that have been cited throughout the text.

Author Response

Answer:

1)We have modi ed some clue lines in the introduction including the important
reasons that have motivated this work, and we also resume these ideas in the abstract of the paper, saying that we found a lack in the approximation of multiple roots of nonlinear equations in the case that the nonlinear operator be non di erentiable. So, we present in this paper iterative methods that do not use the derivative of the non linear operator in their iterative expression. In addition we have changed the title of the work giving to it a more speci c treatment. Moreover, we justify also this fact in the conclusions section.
2) Parameter p refers to the multiplicity of the root that has been denoted before by m, so we have corrected this mistake.
3) Tables 2,3,4 and 5 have been revised in order to present the results more clear.
4) The advantages of the present methods are contrasted in the numerical results as we set in paragraph 5 of the concluding remarks section.
5),6),7) We have revised the written English in the whole manuscript, the  nal full stops for equations and checked the cited in the references included.

Author Response File: Author Response.pdf

Reviewer 3 Report

The Authors propose a new iterative technique, which is claimed to be more stable, derivative-free and multi-point oriented. The Authors use their new approach to get more stable numerical results of Planck’s radiation, van der Waals, Beam designing and Isothermal continuous stirred tank-reactor problems.

The manuscript contains original results and it is clearly written.

The research design is appropriate and the methodology employed is adequately described. 

It meets the criteria of scientific quality and filed relevance for this journal.

It is also suitably formatted for publication.

I recommend the manuscript for publication in MDPI Axioms, after taking care of a misprint on line 1: 'techniques' ---> 'technique'. Then, on line 3 and on the first line of page 8: 'Van der w(W)aals' ---> 'van Der Waals'.

Author Response

Answer: We have revised the written English and the Typos in the whole manuscript.

Author Response File: Author Response.pdf

Reviewer 4 Report

In this paper, iterative techniques are studied to solve nonlinear equation. Convergence order of the proposed iterative scheme is proved. Finally, four numerical examples arising from the Planck’s radiation, the Van der waals, the Beam designing and the isothermal continuous stirred tank reactor problems are used to verify the effectiveness of the proposed iterative scheme.

The title of this article is too broad. In fact, this paper only discusses the iterative method of finding the roots of nonlinear equations.

Finding single root and multi roots for nonlinear equations is a classical problem in the area of applied mathematics. A great deal of work has been published in recent years, including many third-order and the fourth-order iterative schemes. However, the authors did not mention them. In addition, only second-order iterative scheme is studied.

The four numerical examples studied in Section 4 are too simple and can not show the advantages of the proposed iterative scheme. Moreover, there are many grammar errors and typos.

In short, this paper lacks of novelty and is not well-written. I don’t recommend its publication.

Author Response

Please see the attachement.

Author Response File: Author Response.pdf

Reviewer 5 Report

Review of the article

Approximate solutions of applied mathematical problems by

iterative techniques

by

Ramandeep Behl, Himani Arora, Eulalia Mart?nez, Tajinder Singh

The article is devoted to the study of Newtonian iterative methods for solving nonlinear one-dimensional equations $f(x)=0$, in the construction of which the derivative $f'(x)$ of the function $f(x)$ is replaced by a difference operator. An extensive review of works in this direction is presented, but there are no references to works in which this problem is studied in Banach spaces. See, for example, the book Krasnoselsky M.A., Vainikko G.M., Zabreiko P.P. and others. Approximate solution of operator equations, paper V. A. Kurchatov, “On a method for solving nonlinear functional equations”, Dokl. AN USSR, 189:2 (1969), 247–249, which have current references.

In addition, the authors did not note the fundamental difference between their work and the known ones.

In addition, the stability of the algorithms constructed in this work has not been studied.

The article needs a major revision.

Comments for author File: Comments.pdf

Author Response

Answer: 1) We have included the references suggested mentioning that the theoretical study of local and semilocal convergence in Banach spaces plays an important role in the research area of iterative methods. 2) As we set in motivation section, the main difference between this new iterative methods and the known ones is that we have avoid the use of the derivative f'(x) in the iterative expression by introduce the approximation of the derivative by divided differences, with the aim of developing iterative methods for non differentiable operators in the case of approximating multiple roots.

3) We admit that we have not study the stability of the iterative methods by performing a dynamical study, we think that the the generalization, which is normally done by carrying out the dynamic study on simple polynomials of degree 2 or 3 and then inferring it to any non-linear  equation, is still too ambitious. But it can be checked in the Tables that we include the
numerical results the good behavior of the iterative methods.

We have emphasize in the Tables the better result in all problems and coincides in Tables 2 and 3 with the new method RM2, while in Table 5 we remark that new methods perform less iterations for reaching the same tolerance that the known ones.

Author Response File: Author Response.pdf

Reviewer 6 Report

Derivative free methods for multiple roots are very rare in literature. Because, it is not  an easy task to retain the same convergence order(as the simple roots) and the calculation  work is very hard and time consuming . Due to the rapid development of digital computers,  advanced computer languages and software, the production of derivative free methods  for obtaining the multiple roots of nonlinear equations have become the new area of research. In this paper the Authors succesfully face this interesting problem and get a very efficient numerical algorithm. For this reason, I recommend this paper for the publication

Author Response

We would like to thank these reviewers again for his contributions
and suggestions, which have allowed us to signi cantly improve our work.

Author Response File: Author Response.pdf

Round 2

Reviewer 4 Report

The title of this paper is “Approximating multiple roots of applied mathematical problems by iterative techniques”. Applied mathematical problems are a large category. In fact, this paper only discusses a simple iterative method for finding the roots of nonlinear equations, which is also a simple problem in applied mathematics. So, I still think that the title of this article is too broad.

The contribution of this paper is rather weak. Many efficient iterative methods with high convergence rate have been studied recently. However, the authors did not mention them. The four numerical examples studied in Section 4 are too simple and can not show the advantages of the proposed iterative scheme. Moreover, there are many grammar errors and typos.

In short, this paper lacks of novelty and is not well-written. I don’t recommend its publication.