Fourth-order Method Research Articles

On obtaining convergence order of a fourth and sixth order method of Hueso et al. without using Taylor series expansion

Hueso et al. (2015) studied the fourth and sixth order methods to approximate a solution of a nonlinear equation in Rn, where the convergence analysis needs the involved operator to be five times differentiable and seven times differentiable for fourth-order and sixth-order methods, respectively. Also, they found no error estimate for those methods, as the Taylor series expansion played a leading role in proving the convergence. In this paper, we extended the method in the Banach space settings and relaxed the higher order derivative of the involved operator so that the methods can be used in a bigger class of problems which were not covered by the analysis in Hueso et al. (2015). Also, we obtained an error estimate without Taylor series expansion. This error estimate helps to get the number of iterations to achieve a given accuracy. Moreover, new sixth-order method is introduced by small modification and numerical examples were discussed for all these methods to validate our theoretical results and to study the dynamics.

Thermal analysis on Ferro Casson nanofluid flow over a Riga plate with thermal radiation and non-uniform heat source/sink

Ferro hybrid nanofluids can be used in electronics and microelectronics cooling applications to reduce heat accumulation and efficiently remove surplus heat. These nanofluids aid to maintain optimum operating temperatures and reduce device overheating by enhancing the heat transfer rate. With this motivation, the aim of the present numerical analysis is to study the three-dimensional incompressible hybrid nanofluid flow over a slippery Riga surface by combining the Casson fluid model. Mathematical modeling is constructed with nanoparticles as Fe3O4 and CoFe2O4 with base fluid as water. The non-uniform heat source/sink and thermal linear radiation effects are taken into account with the Hamilton–Crosser thermal conductivity model. A system of nonlinear PDEs is produced by the proposed problem and then the relevant similarity variables are implemented to transform the set of partial differential equations and their accompanying boundary conditions into the coupled nonlinear differential equations with one independent variable. These modified ordinary differential equations (ODEs) were then successfully solved with the Runge–Kutta fourth-order method by combining via the shooting technique. With the aid of graphical representations, the effects of various influencing parameters were presented and analyzed comprehensively. Furthermore, the impacts of relevant parameters on heat transfer rates and shear stress are concisely discussed and illustrated in tabular forms. The significant findings include the enhancement of the radiation parameter increases the thermal boundary layer thickness and the thickness increases whenever surface experiences slippery conditions. The axial and transverse momentum of the fluid are controlled with the Casson parameter. An effective connection is noticed once the current numerical solutions are verified under particular conditions that were previously described.

Stability and numerical solutions for second-order ordinary differential equations with application in mechanical systems

This study undertakes a comprehensive analysis of second-order Ordinary Differential Equations (ODEs) to examine animal avoidance behaviors, specifically emphasizing analytical and computational aspects. By using the Picard–Lindelöf and fixed-point theorems, we prove the existence of unique solutions and examine their stability according to the Ulam-Hyers criterion. We also investigate the effect of external forces and the system’s sensitivity to initial conditions. This investigation applies Euler and Runge–Kutta fourth-order (RK4) methods to a mass-spring-damper system for numerical approximation. A detailed analysis of the numerical approaches, including a rigorous evaluation of both absolute and relative errors, demonstrates the efficacy of these techniques compared to the exact solutions. This robust examination enhances the theoretical foundations and practical use of such ODEs in understanding complex behavioral patterns, showcasing the connection between theoretical understanding and real-world applications.

A fourth-order accurate extrapolation nonlinear difference method for fourth-order nonlinear PIDEs with a weakly singular kernel

A fourth-order accurate extrapolation nonlinear difference method for fourth-order nonlinear PIDEs with a weakly singular kernel

Determination of levofloxacin, norfloxacin, and moxifloxacin in pharmaceutical dosage form or individually using derivative UV spectrophotometry

PurposeIn this study, first, second, third, and fourth-order derivative spectrophotometric methods utilizing the peak—zero (P—O) and peak-peak (P—P) techniques of measurement were developed for the determination of levofloxacin, norfloxacin, and moxifloxacin. These methods were applied to their combined pharmaceutical dosage form or individually for levofloxacin, norfloxacin, and moxifloxacin.MethodsLinearity was established in the concentration range of 2–20 µg/mL. The procedures are simple, quick, and precise. The developed methods are sensitive, accurate, and cost-effective, demonstrating excellent correlation coefficients (R2 = 0.9998) and mean recovery values ranging from 99.20% to 100.08%, indicating a high level of precision.ResultsThe developed approach was effectively employed to determine the levofloxacin, norfloxacin, and moxifloxacin content in commercially available pharmaceutical dosages.ConclusionsStatistical analysis and recovery tests confirmed the method's linearity and accuracy. The results suggest that this method can be utilized for routine analysis in both bulk and commercial formulations. The simplicity, accuracy, and cost-effectiveness of the developed methods make them valuable for pharmaceutical analysis.

Novel high-order explicit energy-preserving schemes for NLS-type equations based on the Lie-group method

In this paper, by taking the NLS-type equations as examples, we propose a novel high-order explicit energy-preserving Lie-group method for Hamiltonian PDEs. The main idea is to combine recently developed generalized SAV (G-SAV) approach (Cheng et al., 2021) and explicit Lie group method, in which the fourth-order or higher-order Lie group method only composes two exponentials in each stage, and this is the key point that the proposed methods can preserve energy of Hamiltonian PDEs. This is our first attempt to construct high-order explicit energy-preserving methods without using projection technique (Hairer et al., 2006 [14]), and some stability conclusions of our proposed methods for the NLS-type equations are also presented. Finally, we present 2D and 3D numerical simulations to demonstrate the stability and accuracy.

Numerical solutions of the EW and MEW equations using a fourth-order improvised B-spline collocation method

Numerical solutions of the EW and MEW equations using a fourth-order improvised B-spline collocation method

Cattaneo-Christov heat flux effect on Darcy-Forchheimer dual-phase dusty shear-thickening carreau hybrid nanofluid flow along a stretched vertical cylinder

This research ventures into the convoluted behavior of dusty Copper - Titanium Dioxide and Water shear-thickening Carreau hybrid nanofluid flow within a porous vertical cylinder, incorporating the effects of viscous dissipation and the Cattaneo-Christov heat flux model. By applying the Runge-Kutta-Fehlberg fourth-order method alongside the shooting technique, we address the nonlinear coupled ordinary differential equations governing this flow. The study systematically investigates the impact of various dimensionless parameters on critical flow characteristics such as velocity and thermal profiles, skin friction coefficient, and thermal transfer rate. Our findings indicate that increasing local Weissenberg numbers and power law indices for the Carreau fluid model enhances velocity distribution while reducing the temperature profile. Higher Weissenberg numbers correlate with a decreased skin friction coefficient and an elevated Nusselt number. Initially, the curvature parameter shows a gradual increase in the fluid phase of the velocity profile, but it experiences a significant surge near the boundary. The heat transfer rate diminishes with rising values of the porosity and Forchheimer parameters. Notably, the study reveals the remarkable benefits of the dusty Carreau hybrid nanofluid, demonstrating a 39% increase in absolute shear stress compared to the dusty Carreau nanofluid, highlighting its potential for improved momentum transfer. Additionally, a 20% enhancement in heat transfer rate underscores its suitability for high-performance engineering and heat transfer systems. These results offer promising advancements for industries requiring efficient heat and momentum transfer, contributing to superior performance and energy efficiency.

Combined methods for solving degenerate unconstrained optimization problems

UDC 519.853.6 : 519.613.2 We present constructive second- and fourth-order methods for solving degenerate unconstrained optimization problems. The fourth-order method applied in the present work is a combination of the Newton method and the method that uses fourth-order derivatives. Our approach is based on the decomposition of ℝ n into the direct sum of the kernel of a Hessian matrix and its orthogonal complement. The fourth-order method is applied to the kernel of the Hessian matrix, whereas the Newton method is applied to its orthogonal complement. This method proves to be efficient in the case of a one-dimensional kernel of the Hessian matrix. In order to get the second-order method, Newton's method is combined with the steepest-descent method. We study the efficiency of these methods and analyze their convergence rates. We also propose a new adaptive combined quasi-Newton-type method (ACQNM) based on the use of the second- and fourth-order methods in the degenerate case. The efficiency of ACQNM is demonstrated by analyzing an example of some most common test functions.

Magneto-Micropolar Nanofluid Flow Over a Convectively Heated Sheet with Nonlinear Radiation, Chemical Reaction, and Viscous Dissipation

A mathematical model is presented for analyzing the flow of magneto-micropolar nanofluid above a convectively heated permeable stretching sheet with nonlinear radiation, chemical reaction, and viscous dissipation. Similarity transformation is utilized to change the governing partial differential equations to the system of nonlinear ordinary differential equations (ODEs). The resulting system of ODEs is sorted out mathematically by utilizing the shooting method with the Adams-Moulton method of fourth order, and the obtained mathematical outcomes are compared with published results. The obtained numerical results reflect very good agreement with those from open literature. Numerical values of physical quantities like skin friction coefficient, Sherwood number, and Nusselt number for emerging parameters such as Pr , Nb , Nt , Le , etc. are also computed and discussed in this work. The impact of pertinent flow parameters on nondimensional velocity, temperature, and concentration profiles is illustrated via tables and graphs.

Numerical integration of third-order BVPs using a fourth-order hybrid block method

This research paper introduces a new hybrid block method to solve the third-order boundary value problems (BVPs). The method combines interpolation and collocation and uses a power series polynomial to find an approximate solution to the considered third-order BVPs. Some third-order BVP models are numerically solved to verify the performance and efficiency of the proposed method, and the approximate solution from the proposed method is more efficient when compared to some existing numerical methods. In summary, the proposed method provides reliable and efficient accuracy for solving third-order BVPs, making it a valuable contribution to the fields of numerical analysis and computational mathematics. The advantages of the proposed method include improved computational time efficiency and accuracy in terms of maximum absolute errors for solving third-order BVPs.

Study of the Six-Compartment Nonlinear COVID-19 Model with the Homotopy Perturbation Method

The current study aims to utilize the homotopy perturbation method (HPM) to solve nonlinear dynamical models, with a particular focus on models related to predicting and controlling pandemics, such as the SIR model. Specifically, we apply this method to solve a six-compartment model for the novel coronavirus (COVID-19), which includes susceptible, exposed, asymptomatic infected, symptomatic infected, and recovered individuals, and the concentration of COVID-19 in the environment is indicated by S(t), E(t), A(t), I(t), R(t), and B(t), respectively. We present the series solution of this model by varying the controlling parameters and representing them graphically. Additionally, we verify the accuracy of the series solution (up to the (n−1)th-degree polynomial) that satisfies both the initial conditions and the model, with all coefficients correct at 18 decimal places. Furthermore, we have compared our results with the Runge–Kutta fourth-order method. Based on our findings, we conclude that the homotopy perturbation method is a promising approach to solve nonlinear dynamical models, particularly those associated with pandemics. This method provides valuable insight into how the control of various parameters can affect the model. We suggest that future studies can expand on our work by exploring additional models and assessing the applicability of other analytical methods.

Fuzzy Predator-Prey Systems by Extended Runge-Kutta Method with Polynomial Interpolation Technique

The growing significance of uncertainty quantification in mathematical modeling of physical phenomena is evident, with fuzzy sets emerging as an alternative approach. This paper focuses on exploring a numerical method for addressing fuzzy differential equations in two-dimensional problems like fuzzy predator-prey systems. The numerical solution employed the extended Runge-Kutta fourth-order method. To overcome challenges related to polynomial fuzzy differential equations, a combination of the polynomial interpolation technique and the extended Runge-Kutta fourth-order method was applied, mitigating the issues associated with high-degree polynomials. The proposed approach for fuzzy predator-prey systems is briefly described, accompanied by a numerical example. The obtained results underscore the efficacy of the extended Runge-Kutta fourth-order method with polynomial interpolation, showcasing its remarkable accuracy and potential as an alternative method for addressing various uncertainty-related problems.

Analysis of the hate speech and racism co-existence dissemination model with optimal control strategies

Hate speech, racism, and their co-existence are the human mind infections that are major factors that affect the living conditions of people, societal well-being, and political stability, economic and social disturbance throughout the world especially in underdeveloped nations of the world. The main objective of this study is to formulate and analyze the hate-speech and racism co-existence model with optimal control strategies and investigate the optimal effects of protection and improvement (rehabilitation) strategies to minimize and tackle the hate speech and racism co-existence dissemination in the community. In this study, we have computed the hate speech sub-model, the racism sub-model, and the hate speech and racism co-existence model equilibrium points and analyzed their stabilities, using the next generation operator approach all the models' effective reproduction numbers are computed, examined the phenomenon of backward bifurcation whenever the model effective reproduction number is less than one where at which both the positive co-existence free equilibrium point and the positive co-existence dominance equilibrium point exist simultaneously. To achieve the objective minimizing the hate speech and racism co-existence dissemination dynamics by implementing the efforts towards the protection and improvement (rehabilitation) strategies, the optimal control problem is re-formulated, and optimal control analysis is performed for the co-existence model using Pontryagin's maximum principle. Numerical simulations using MATLAB ode45 solver with fourth-order Runge-Kutta numerical methods to investigate the behavior of the co-existence model solutions and to explore the optimal effects of the protections and improvements (rehabilitation) strategies for the hate speech and racism co-existence dynamics are then performed. From the results, we observed that implementing the protection and improvement (rehabilitation) strategies simultaneously is the most effective approach to minimize and tackle the speech and racism co-existence propagation in the community.

Fourth-order compact block-centered splitting domain decomposition method for parabolic equations

It is of importance to develop efficient high-order solution-flux domain decomposition methods for solving the large scale time-dependent partial differential equations. In this paper, a fourth-order compact block-centered splitting domain decomposition method is developed for solving parabolic partial differential equations in large domains. Combining the time second-order splitting technique with non-overlapping domain decompositions, we propose the compact block-centered splitting domain decomposition algorithm to the solution-flux structured system of parabolic partial differential equations. The domain is divided into non-overlapping multi-block sub-domains of block-centered meshes. On the interfaces of sub-domains, the interface fluxes are predicted by the semi-implicit interface flux scheme while the solution and flux in the interiors of sub-domains are computed by the compact block-centered implicit solution-flux scheme. The feature of the method is that over block-centered grids, constructing high-order compact difference scheme to the structured system of solution and flux equations leads to efficient schemes for solving the problems over domain decompositions. Numerical experiments show that the method is of fourth-order convergence in space and of second order convergence in time and it is much efficient in parallel computing.

Double fast algorithm for solving time-space fractional diffusion problems with spectral fractional Laplacian

This paper presents an efficient and concise double fast algorithm to solve high dimensional time-space fractional diffusion problems with spectral fractional Laplacian. We first establish semi-discrete scheme of time-space fractional diffusion equation, which uses linear finite element or fourth-order compact difference method combining with matrix transfer technique to approximate spectral fractional Laplacian. Then we introduce a fast time-stepping L1 scheme for time discretization. The proposed scheme can exactly evaluate fractional power of matrix and perform matrix-vector multiplication at per time level by using discrete sine transform, which doesn't need to resort to any iteration method and can significantly reduce computation cost and memory requirement. Further, we address stability and convergence analyses of full discrete scheme based on fast time-stepping L1 scheme on graded time mesh. Our error analysis shows that the choice of graded mesh factor ω=(2−α)/α shall give an optimal temporal convergence O(N−(2−α)) with N denoting the number of time mesh. Finally, ample numerical examples are delivered to illustrate our theoretical analysis and the efficiency of the suggested scheme.

Bounds for Weierstrass Elliptic Function and Jacobi Elliptic Integrals of First and Second Kinds

The Weierstrass elliptic function is presented in connection with the Jacobi elliptic integrals of first and second kinds leading to comparing coefficients appearing in the Laurent series expansion with those of Eisenstein series for the cubic polynomial in the meromorphic Weierstrass function. It is unified in the formulation the Weierstrass elliptic function with Jacobi elliptic integral by considering motion of a unit mass particle in a cubic potential in terms of bounded and unbounded velocities and the time of flight with imaginary part in the complex function playing a major role. Numerical tools box used are the Konrad-Gauss quadrature and Runge-Kutta fourth order method.

Vibration Suppression of Piecewise-Linear Stiffness Nes System Under Random Excitation

Nonlinear energy sinks (NESs) are critically important for structural vibration suppression. They can absorb vibrational energy across a broad frequency spectrum, possess strong robustness, and have a relatively small mass. This study addresses the vibration suppression in a piecewise linear stiffness NES system under random excitation. Initially, a theoretical model of the piecewise linear stiffness NES system is developed. The piecewise linear stiffness function is approximated using Legendre polynomial approximation. Following this, the steady-state Fokker–Planck–Kolmogorov (FPK) equation of the system is formulated via the Generalized Harmonic Function Method. The FPK equation is solved using the fourth-order central difference method, and the effectiveness of this approach is validated by comparing the FDM results with numerical simulations. Lastly, the influence of varying system parameters on the stability of the piecewise linear stiffness NES system is analyzed.

Riemannian quantum circuit optimization for Hamiltonian simulation

Hamiltonian simulation, i.e. simulating the real time evolution of a target quantum system, is a natural application of quantum computing. Trotter-Suzuki splitting methods can generate corresponding quantum circuits; however, a faithful approximation can lead to relatively deep circuits. Here we start from the insight that for translation invariant systems, the gates in such circuit topologies can be further optimized on classical computers to decrease the circuit depth and/or increase the accuracy. We employ tensor network techniques and devise a method based on the Riemannian trust-region algorithm on the unitary matrix manifold for this purpose. For the Ising and Heisenberg models on a one-dimensional lattice, we achieve orders of magnitude accuracy improvements compared to fourth-order splitting methods. The optimized circuits could also be of practical use for the time-evolving block decimation algorithm.

A regularized fourth-order adaptive method for seamless integration of charging station to the grid for EV and critical loads

ABSTRACT This paper proposes a real-time implementation and regularised fourth-order adaptive (RFOA) method of three-phase grid connectedsolar photo voltaic (SPV) array and battery-based charging station. The proposed system mainly uses a solar photovoltaic array and battery system to reduce the burden on the grid. The system feeds the surplus energy to the grid as well as operates during the outage and support the EV and home loads/critical loads. To minimise the error at various operating scenarios, the RFOA method is utilised for voltage source converter (VSC). The battery is integrated with DC link with the help of bi-directional converter. It works in boost mode when it is discharging and works in buck mode when it is charging. The perturbation & observation (P&O) maximum power point tracking (MPPT) is used to generate the reference DC link voltage. In order to improve the power quality of the charging station, the proposed system is designed to act as an active filter, load leveller, power factor corrector and reduce the harmonics to 5% as per IEEE 519 standards. The proposed charging is examined for different balanced and unbalanced load conditions and at the different environmental situations in MATLAB/Simulink based platform as well as in experimental test bed based on OPAL-RT platform.