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

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.

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.

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.

Potential competing interests: No potential competing

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.

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).

The direct integration of second order initial and boundary value problems is considered in this paper. We employ a new class of orthogonal polynomials constructed as basis function to develop a two-step hybrid block method (2SHBM) adopting collocation technique. The recursive formula of the class of polynomials have been constructed, and then we give analysis of the basic properties of 2SHBM as findings show that the method is accurate and convergent. The boundary locus of the proposed 2SHBM shows that the new scheme is A-stable.

Purpose The purpose of this paper is to develop a block method of order five for the general solution of the first-order initial value problems for Volterra integro-differential equations (VIDEs). Design/methodology/approach A collocation approximation method is adopted using the shifted Legendre polynomial as the basis function, and the developed method is applied as simultaneous integrators on the first-order VIDEs. Findings The new block method possessed the desirable feature of the Runge–Kutta method of being self-starting, hence eliminating the use of predictors. Originality/value In this paper, some information about solving VIDEs is provided. The authors have presented and illustrated the collocation approximation method using the shifted Legendre polynomial as the basis function to investigate solving an initial value problem in the class of VIDEs, which are very difficult, if not impossible, to solve analytically. With the block approach, the non-self-starting nature associated with the predictor corrector method has been eliminated. Unlike the approach in the predictor corrector method where additional equations are supplied from a different formulation, all the additional equations are from the same continuous formulation which shows the beauty of the method. However, the absolute stability region showed that the method is A-stable, and the application of this method to practical problems revealed that the method is more accurate than earlier methods.

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

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.

A continuous two-step method of order 8 with a block extension for [formula omitted