Efficient adaptive step-size formulation of an optimized two-step hybrid block method for directly solving general second-order initial-value problems

Singla, Rajat; Singh, Gurjinder; Kanwar, V.; Ramos, Higinio

doi:10.1007/s40314-021-01599-z

Cited by 7 publications

(2 citation statements)

References 22 publications

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“…The methodology was employed to address practical issues, and a comparative analysis was conducted with existing approaches in the literature to ascertain the superiority of the novel approach. Additional studies that have focused on the optimization of hybrid points through the minimization of the LTE include the works of [21,22,23,24].…”

Section: Introductionmentioning

confidence: 99%

One-step three-parameter optimized hybrid block method for solving first order initial value problems of ordinary differential equations

E. A. Areo,

Gbenro,

Olabode

et al. 2024

J. Math. Anal. Model.

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A one-step three-parameter optimized hybrid block method and second derivative hybrid block method with optimized points were proposed to solve first-order ordinary differential equations. The techniques of interpolation and collocation were adopted for the derivation of the methods using a three-parameter approximation. The hybrid points were obtained by optimizing the local truncation error of the method. The schemes obtained were reformulated to reduce the number of occurrences of the source term. The hybrid points were used in the derivation of the second derivative hybrid block method. The discrete schemeswere produced as a by-product of the continuous scheme and used to simultaneously solve initial value problems (IVPs) in block mode. The resulting schemes are self-starting, do not require the creation of individual predictors, and are consistent, zero-stable, and convergent. The accuracy and efficiency of the methods were ascertained using several numerical test problems. The numerical results were favourably compared to some techniques from the cited literature.

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Section: Introductionmentioning

confidence: 99%

One-step three-parameter optimized hybrid block method for solving first order initial value problems of ordinary differential equations

E. A. Areo,

Gbenro,

Olabode

et al. 2024

J. Math. Anal. Model.

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show abstract

“…The methods with constant coefficients, like the majority of classical methods, are derived primarily using polynomials as basis functions, such as power series and orthogonal polynomials ( [6,16,17,26,28,31]). Many of these approaches do not perform well in the case of oscillating outcomes due to the nature of the solutions.…”

Section: Introductionmentioning

confidence: 99%

A Novel Optimized Exponentially Fitted Algorithm with Time-Efficiency

Abdulganiy

Qureshi

Soomro

et al. 2022

Preprint

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There are numerous models in the form of ordinary differential equations that require numerical treatment, for they have no closed-form solutions due to some reasons, including non-linearity, stiffness, singularity, and stability. Having in mind the increasing demand for efficient and cost-effective algorithms yet straightforward to code; an attempt is made in this research paper to introduce a new time-efficient optimal exponentially fitted numerical algorithm whose qualitative analysis unfolds its local truncation error, zero-stability, A-stability, consistency, convergence, and accuracy (at least third-order). The proposed algorithm is also implemented in an adaptive step-size mode, whereas both constant and adaptive approaches outperform several existing algorithms when tested upon differential models taken from applied sciences. The efficiency curves further confirm the time-effectiveness of the proposed exponentially fitted one-step optimal method.

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A novel two‐parameter class of optimized hybrid block methods for integrating differential systems numerically

Singh

Garg²,

Singla

et al. 2021

Comp and Math Methods

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In this article, a two-parameter class of hybrid block methods for integrating first-order initial value ordinary differential systems is proposed. The methods exhibit hybrid nature which helps in bypassing the first Dahlquist barrier existing for linear multistep methods. The approach used in the development of a class of methods is purely interpolation and collocation technique. The class of methods is based on four intra-step points from which two intra-step points have been optimized by using an optimization strategy. In this optimization strategy, the values of two intra-step points are obtained by minimizing the local truncation errors of the formulas at the points x n+1∕2 and x n+1 .The order of accuracy of the proposed methods is six. A method as a special case of this class of methods is considered and developed into a block form which produces approximate numerical solutions at several points simultaneously. Further, the method is formulated into an adaptive step-size algorithm using an embedded type procedure. This method which is a special case of this class of methods has been tested on six well-known first-order differential systems. K E Y W O R D Sblock methods, embedded-type procedure, hybrid methods, ordinary differential equations INTRODUCTIONNumerous numerical integrators are available for solving first-order initial value ordinary differential systems numerically. Runge-Kutta and linear multistep methods are the usual classes of methods that are being used for solving these systems numerically. These days many built-in computer codes based on these classes of methods are available in software like Matlab, Mathematica, and so forth. To develop more accurate and efficient codes for solving the differential systems is an ongoing process of research. This article is concerned with the development of a class of optimized codes for solving the following system:whereTo proceed further, we assume that the system (1) has a unique continuously differentiable solution that we denote by Y(x). As a first step, the interval of interest [x 0 , x N ] is discretized as follows:x n = x 0 + nh, n = 0, 1, 2, 3, … , N; h = x n+1 − x n and an approximate numerical solution of the system (1) at x n is denoted as Y n ≈ Y(x n ).

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Efficient adaptive step-size formulation of an optimized two-step hybrid block method for directly solving general second-order initial-value problems

Cited by 7 publications

References 22 publications

One-step three-parameter optimized hybrid block method for solving first order initial value problems of ordinary differential equations

One-step three-parameter optimized hybrid block method for solving first order initial value problems of ordinary differential equations

A Novel Optimized Exponentially Fitted Algorithm with Time-Efficiency

A novel two‐parameter class of optimized hybrid block methods for integrating differential systems numerically

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