Review of: "New Computational Methods Using Seventh Derivative Type for the Solution of First Order Initial Value Problems"

Review comments and questions I recommend publication of the research article in Qeios after minor revision and proper responses to the questions raised.

Review of: "New Computational Methods Using Seventh Derivative Type for the Solution of First Order Initial Value Problems"

This paper is interesting. I recommend it to be published in your journal after answering the following questions: 1. Why use 7th order instead of 3rd or 5th order when constructing algorithms? 2. How was it derived from (9) to derive ( 10) -( 20)?

The manuscript demonstrates a high standard of quality and is suitable for publication. However, it requires minor corrections in English. Please review the manuscript thoroughly to address these issues. Additionally, incorporate recent references by reviewing the latest literature, and ensure that the work is compared with recent studies in the field.

In this research, the authors have examined a class of implicit block methods of a seventh derivative type through interpolation and collocation techniques using finite power series as the basis functions.They have presented the discrete schemes, which are implicit two-point block methods, obtained by carefully and unevenly choosing collocation points that ensure better methods' stability via test.However, these schemes require seventh derivative functions, unlike other existing numerical formulae.The new methods are found, investigated, and proven to be convergent and A-stable.The implementation of the methods is achieved by using the Newton-Raphson method.Experiments show the efficiency and accuracy of the developed formulae on different classes of first-order initial value problems, including SIR, growth models, and the Prothero-Robinson oscillatory problem, and in comparison to such existing methods.In addition, it is observed that the uneven positioning of collocation points greatly influences the efficiency and accuracy of numerical methods.This paper can be considered for publication after the authors address the following comments and suggestions:1. What is the main advantage of the proposed method?Please provide more details in the revised version.2. Can the proposed method be developed for other classes of differential equations, such as integro-differential equations or partial differential equations?3.In conclusion, can you describe the direction of your further work?4. The paper also has some typos and language issues that need to be checked and corrected.

In this paper, linear multi-step hybrid block methods with three-, four- and five-step numbers are developed for approximating directly the solution of second order Initial and Boundary Value Problems (IBVPs). Multiple finite difference formulas are derived and combined in a block formulation to form a numerical integrator that provides direct solution to second order IBVPs over sub-intervals. A new class of orthogonal polynomials constructed as basis function to develop the hybrid block methods adopting collocation technique with a non-negative weight function. The scheme is applied as simultaneous integrator to second order initial value and boundary value problems of ODEs. The properties and convergence of the proposed method are discussed. The derived schemes were used to solve some problems and the numerical result shows the effectiveness, accuracy and superiority of the method over the existing methods found in the literature.

Second order initial value problems with non-absolute integrals in ordered Banach spaces

The paper studies the differential equation $$y'' + (\rho ^2 \varphi ^2 (x) - q(x))y = 0$$ (*) on the interval I = [0, 1], containing a finite number of zeros 0 < x1 < x2 < ... < xm < 1 of ϕ2, i.e. so-called turning points. Using asymptotic estimates from [6] for appropriate fundamental systems of solutions of (*) as |ρ| → ∞, it is proved that, if there is an asymptotic solution of the initial value problem generated by (*) in the interval [0, x1), then the asymptotic solutions in the remaining intervals can be obtained recursively. Furthermore, an infinite product representation of solutions of (*) is studied. The representations are useful in the study of inverse spectral problems for such equations.

Inverse linear multistep methods (ILMMs) for first and second order differential equations have been proved to be suitable numerical methods for the solution of inverse initial value problems (IVPs). This paper presents the hybrid version of the ILMMs for the numerical solution of second order inverse IVPs. The stability of the proposed methods is represented in the boundary locus graph. The applicability of the schemes is demonstrated herein for the solution of linear and nonlinear problems. Computational results on the problems are compared with those from the existing method and ode45 (the explicit Runge–Kutta method).

A computational method for the numerical solution of the singularly perturbed boundary value problem having a boundary layer at one end is presented in this paper. The original problem is reduced to a first order initial value problem using the asymptotic approximation. Then a fitting factor is introduced and the initial value problem is solved by using Box–Trapezoidal scheme. It is shown that the proposed technique provides first order accuracy independent of the perturbation parameter. Parameter-uniform error estimates are derived for the numerical solution. Numerical experiments are carried out to support the theoretical results.

Differential equations of the form y′ = f(t, y, y′), where f is not necessarily linear in its arguments, represent certain physical phenomena and solutions have been known for quite some time. The well known Clairut′s and Chrystal′s equations fall into this category. Earlier existence of solutions of first order initial value problems and stability of solutions of first order ordinary differential system of the above type were established. In this paper we study boundedness and asymptotic stability in the large of solutions of an ordinary differential system of the above type under certain natural hypotheses on f.

Order six generalised hybrid block method for the solution of third order initial value problems

We develop the lower and upper solutions method for first order initial value problems as well as for first order periodic problems in case the nonlinearity presents singularities. More precisely we prove that if we have a lower solution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and an upper solution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta"> <mml:semantics> <mml:mi> β </mml:mi> <mml:annotation encoding="application/x-tex">\beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of these problems, which are not necessarily continuous nor ordered, we have a solution wedged between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="min left-brace right-brace comma alpha comma beta"> <mml:semantics> <mml:mrow> <mml:mo movablelimits="true" form="prefix">min</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi> α </mml:mi> <mml:mo>,</mml:mo> <mml:mi> β </mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\min \{\alpha ,\beta \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="max left-brace right-brace comma alpha comma beta"> <mml:semantics> <mml:mrow> <mml:mo movablelimits="true" form="prefix">max</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi> α </mml:mi> <mml:mo>,</mml:mo> <mml:mi> β </mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\max \{\alpha ,\beta \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

This paper considered the development of numerical integrator for the solution of second order initial value problems. The method was derived through the interpolation and collocation of the basis polynomial which is com- bination of power series and exponential function to derive a continuous linear multistep method. The method was implemented in block method which gave solution at a non overlapping interval. The method was found to be conver- gence and A stable. The efficiency of the method was tested on some numerical examples.

This paper examines the development of one step, five hybrid point method for the solution of first order initial value problems. We adopted the method of collocation and interpolation of power series approximate solution to generate a continuous linear multistep method. The continuous linear multistep method was evaluated at selected grid points to give the discrete linear multistep method. The method was implemented using a constant order predictor of order seven over an overlapping interval. The basic properties of the derived corrector was investigated and found to be zero stable, consistent and convergent. The region of absolute stability was also investigated. The method was tested on some numerical experiments and found to compare favorably with the existing methods.