In this paper, general Runge–Kutta–Nyström (GRKN) methods are developed and analyzed, tailored for second-order initial value problems of the form y″=Ly′+My+g(t), where L,M∈Rn×n are constant matrices with n≥1. The construction of embedded pairs of orders 6(4) and 7(5), suitable for adaptive integration strategies, is emphasized. By utilizing rooted tree theory and recent simplifications for linear inhomogeneous systems, symbolic order conditions are derived, and efficient schemes are designed through algebraic and evolutionary techniques. Numerical tests verify the superiority of our new derived pairs. In particular, this work introduces novel embedded GRKN pairs with reduced-order conditions that exploit the linearity and structure of the underlying system, enabling the construction of low-stage, high-accuracy integrators. The methods incorporate FSAL (First Same As Last) formulations, making them computationally efficient. They are tested on representative physical systems in one, two, and three dimensions, demonstrating notable improvements in efficiency and accuracy over existing high-order RKN methods.

Solving Lur'e equations plays a critical role in addressing linear-quadratic optimal control (LQOC) problems, especially in cases where the control cost matrices are singular. This paper introduces, for the first time, two novel zeroing neural network (ZNN) models—ZNNLE and ZNNLE-LQOC—specifically designed to solve the Lur'e equation system and the LQOC problem, respectively. The proposed models extend the applicability of the ZNN methodology to these challenging scenarios by offering robust and efficient solutions to time-varying matrix equations. Theoretical analyses confirm the validity of both models, while numerical simulations and practical applications demonstrate their effectiveness. Moreover, a comparative study with an enhanced alternating-direction implicit (ADI) method highlights the superior performance of the ZNNLE-LQOC model in solving LQOC problems.

This study presents a novel low-order Adams–Bashforth–Moulton predictor–corrector algorithm. This new approach was able to incorporate any linear combination of the functions [ 1, x, x^2, e^I v x] . Stability zones are established for the novel approach. Using cases from chemistry and other fields, we test how well the recently suggested technique works.

The theory for computing the phase-lag and amplification-factor for both explicit and implicit multistep approaches for first-order differential equations was recently established by one of the authors. The goal of this work is to design high order Adams–Bashforth algorithms that eliminate the phase-lag, amplification-factor, and their respective derivatives. The stability zones of the recently developed approaches will also be presented. Additionally, we will discuss our findings from numerical experiments using the recently created techniques.

This study investigates a widely recognized ninth-order numerical technique within the explicit two-step family of methods (a.k.a. hybrid Numerov-type methods). To boost its performance, we incorporate an economical step-size control algorithm that, after each iteration, either preserves the current step length, reduces it by half, or doubles it. Any additional off-grid points needed by this strategy are computed using a local interpolation routine. Indicative numerical experiments confirm the substantial efficiency gains realized by this method. It is particularly adept at resolving differential equations with oscillatory dynamics, delivering high precision with fewer function evaluations. Furthermore, a detailed Mathematica implementation is supplied, enhancing usability and fostering further research in the field. By simultaneously improving computational efficiency and accuracy, this work offers a significant contribution to the numerical analysis community.

This study introduces a family of implicit four-point block methods for solving first-order initial value problems (IVPs) with oscillatory solutions. In addition to an eighth-order block method, amplification-fitted and phase-fitted implicit block methods are also derived. The methods are implemented in a predictor–corrector framework, where the predictor is a four-point explicit block method constructed with the corresponding properties. A comprehensive stability analysis is carried out to assess the robustness of the proposed approaches. Comparative evaluations with existing methods demonstrate the superior efficiency of the new algorithms. Numerical experiments further confirm that the proposed techniques provide significant improvements over traditional methods, particularly for oscillatory IVPs.

This paper introduces a novel Runge–Kutta (RK) pair of orders 8(6) designed specifically for solving linear inhomogeneous initial value problems (IVPs) with constant coefficients. The proposed method requires only 11 stages per iteration, a significant improvement over conventional RK pairs of orders 8(7), which typically demand 13 stages. The reduction in stages is achieved by leveraging a smaller set of order conditions tailored to linear inhomogeneous problems, where traditional simplification techniques are not applicable. To address the complexity of deriving such methods, the authors employ the Differential Evolution algorithm, a global optimization technique, to solve the resulting system of equations. The new RK pair, named NEW8(6)Lin, is tested on several benchmark problems, including scalar, linear inhomogeneous, and larger systems, demonstrating a superior performance in terms of accuracy and computational efficiency. The method’s high phase-lag accuracy and efficiency make it particularly suitable for problems requiring high precision over extended intervals. The coefficients of the method are provided with high precision, enabling direct implementation in computational environments like Mathematica. The results highlight the method’s potential as a robust tool for solving linear inhomogeneous IVPs, offering a balance between computational cost and accuracy. This work contributes to the ongoing development of specialized numerical methods for differential equations, particularly in scenarios where traditional approaches struggle with efficiency or stability.

The methodology for calculating the phase-lag and amplification-factor for both explicit and implicit multistep methods for first-order differential equations was recently developed by one of the authors. The objective of this study is to develop low-order Adams–Bashforth–Moulton predictor–corrector algorithms that eradicate phase-lag, amplification-factor. The stability regions of the newly established methodologies will also be highlighted. Furthermore, we will examine our results from numerical experiments employing the newly developed approaches.

In this paper, we introduce a new family of tenth-order hybrid Numerov-type methods designed for solving second-order initial value problems (IVPs) with enhanced accuracy and efficiency. Our proposed methods build on the foundation of existing high-order Numerov-type techniques and incorporate advanced strategies for step size adaptation and error control. We provide a comprehensive analysis of the theoretical underpinnings of these methods and validate their performance through extensive numerical experiments.

By using a strategy that accounts for fading phase-lag, phase-lag and all of its derivatives up to order six can be eliminated. The cost-efficient approach is a new strategy whose aims are to boost algebraic order (AOR) and reduce function evaluations (FEVs). The symbolic representation of the one-of-a-kind approach is PF6DPHFITN142SPS. This method is infinitely periodic since it is P-Stable. The proposed method is general enough to address a large class of periodic and oscillatory problems. This new method was used to solve the difficult problem of Schrödinger-type coupled differential equations in quantum chemistry. Given that each stage only requires 5 FEVs , the new method could be seen as a cost-effective strategy. With a AOR of 14, we can greatly enhance our current situation.

In the present note we present some comments on the papers: (1) The use of a multistep, cost-efficient fourteenth-order phase-fitting method to chemistry problems by Rong Xu, Bin Sun, Chia-Liang Lin, T. E. Simos, Journal of Mathematical Chemistry (2024) 62:1781–1807 https://doi.org/10.1007/s10910-024-01623-7 and (2) An effective multistep fourteenth-order phase-fitting approach to solving chemistry problems by Hui Huang, Cheng Liu, Chia-Liang Lin, T. E. Simos, Journal of Mathematical Chemistry (2024) 62:1860–1889, https://doi.org/10.1007/s10910-024-01628-2

Runge–Kutta (RK) pairs are widely used for numerically solving initial value problems (IVPs), but dealing with step rejections during integration is a common occurrence. Conventionally, when a step is rejected, all calculations made during that step are discarded, and a completely new set of computations is initiated. In our research, we propose a method to address this inefficiency by repurposing the previously computed RK stages from rejected steps. Our primary focus is on the renowned RKF45 pair, consisting of fifth‐ and fourth‐order methods. When a step rejection occurs, we leverage the stages computed in prior steps and introduce just three additional stages. These stages are then used to evaluate the results with a smaller step size. This approach effectively reduces computational costs in various challenging IVPs where RK algorithms with different step sizes encounter difficulties

It is possible to eliminate phase-lag and all of its derivatives up to order five by employing a method that takes fading phase-lag into consideration. Improving algebraic order (AOR) and decreasing function evaluations (FEvs) are the goals of the new method called the cost-efficient approach. The unique method is illustrated by the symbol PF5DPHFITN142SPS. This approach is P-Stable, which means it is infinitely periodic. A wide variety of periodic and oscillatory issues can be solved using the suggested approach. The challenging problem of Schrödinger-type coupled differential equations in quantum chemistry was tackled using this novel approach. With only 5 FEvs needed to complete each step, the new method could be considered as a cost-effective approach. An AOR of 14 allows us to significantly improve our present condition.

Applying a method with vanished phase–lag might potentially eliminate the phase–lag and its first, second, and third derivatives. Improving algebraic order (AOR) and decreasing function evaluations (FEvs) are the goals of the new strategy called the cost–efficient approach. Equation PF3DPHFITN142SPS demonstrates the unique method. The suggested approach is P–Stable, meaning it is indefinitely periodic. The suggested approach is applicable to a wide variety of periodic and/or oscillatory issues. The challenging problem of Schrödinger-type coupled differential equations was solved in quantum chemistry by using this novel approach. Since the new method only needs 5FEvs to run each stage, it may be considered a cost–efficient approach. With an AOR of 14, we can significantly improve our present predicament.

The stability of nonlinear systems in the control domain has been extensively studied using different versions of the algebraic Riccati equation (ARE). This leads to the focus of this work: the search for the time-varying quaternion ARE (TQARE) Hermitian solution. The zeroing neural network (ZNN) method, which has shown significant success at solving time-varying problems, is used to do this. We present a novel ZNN model called ’ZQ-ARE’ that effectively solves the TQARE by finding only Hermitian solutions. The model works quite effectively, as demonstrated by one application to quadrotor control and three simulation tests. Specifically, in three simulation tests, the ZQ-ARE model finds the TQARE Hermitian solution under various initial conditions, and we also demonstrate that the convergence rate of the solution can be adjusted. Furthermore, we show that adapting the ZQ-ARE solution to the state-dependent Riccati equation (SDRE) technique stabilizes a quadrotor’s flight control system faster than the traditional differential-algebraic Riccati equation solution.

Applying a phase-fitting method might potentially vanish the phase-lag and its first derivative. Improving algebraic order (AOR) and decreasing function evaluations (FEvs) are the goals of the new strategy called the cost-efficient approach. Equation PF1DPHFITN142SPS demonstrates the unique method. The suggested approach is P-Stable, meaning it is indefinitely periodic. The proposed method is applicable to a wide variety of periodic and/or oscillatory issues. The challenging problem of Schrödinger-type coupled differential equations was solved in quantum chemistry by using this novel approach. Since the new method only needs 5FEvs to run each stage, it may be considered a cost-efficient approach. With an AOR of 14, we can significantly improve our present predicament.

We present a new family of ninth‐order hybrid explicit Numerov‐type methods, effectively utilizing only eight stages, for solving the special second‐order initial value problem. After applying a number of simplifying assumptions, we arrive to a reduced set of order conditions. Then, we derive an optimal method with constant coefficients that requires one less stage than standard methods found in the literature that use nine stages at this moment. Numerical tests are conducted using quadruple precision arithmetic on several well‐known problems and the superiority of the new method is clear. Finally, in Section 6, a Mathematica package is presented that implements the corresponding algorithm.

Many variations of the algebraic Riccati equation (ARE) have been used to study nonlinear system stability in the control domain in great detail. Taking the quaternion nonsymmetric ARE (QNARE) as a generalized version of ARE, the time-varying QNARE (TQNARE) is introduced. This brings us to the main objective of this work: finding the TQNARE solution. The zeroing neural network (ZNN) technique, which has demonstrated a high degree of effectiveness in handling time-varying problems, is used to do this. Specifically, the TQNARE can be solved using the high order ZNN (HZNN) design, which is a member of the family of ZNN models that correlate to hyperpower iterative techniques. As a result, a novel HZNN model, called HZ-QNARE, is presented for solving the TQNARE. The model functions fairly well, as demonstrated by two simulation tests. Additionally, the results demonstrated that, while both approaches function remarkably well, the HZNN architecture works better than the ZNN architecture.

A theory for the calculation of the phase–lag and amplification–factor for explicit and implicit multistep techniques for first–order differential equations was recently established by the author. His presentation also covered how the approaches’ efficacy is affected by the elimination of the phase–lag and amplification–factor derivatives. This paper will apply the theory for computing the phase–lag and amplification–factor, originally developed for implicit multistep methods, to a subset of implicit methods, called backward differentiation formulae (BDF), and will examine the impact of the phase–lag and amplification–factor derivatives on the efficiency of these strategies. Next, we will show you the stability zones of these brand-new approaches. Lastly, we will discuss the results of numerical experiments and draw some conclusions about the established approaches.

Undoubtedly, one of the most common machine learning challenges is multiclass classification. In light of this, a novel bio-inspired neural network (NN) has been developed to address multiclass classification-related issues. Given that weights and structure determination (WASD) NNs have been acknowledged to alleviate the disadvantages of conventional back-propagation NNs, such as slow training pace and trapping in a local minimum, we developed a bio-inspired WASD algorithm for multiclass classification problems (BWASDC) by using the metaheuristic beetle antennae search (BAS) algorithm to enhance the WASD algorithm's learning process. The BWASDC's effectiveness is then evaluated through applications in occupational classification systems. It is important to mention that systems of occupational classification serve as a fundamental indicator of occupational exposure. For this reason, they are highly significant in social science research. According to the findings of four occupational classification experiments, the BWASDC model outperformed some of the most modern classification models obtainable through MATLAB's classification learner app on all fronts.