M. B. Almatrafi, Abdulghani Alharbi^*

CMES-Computer Modeling in Engineering & Sciences, Vol.137, No.1, pp. 827-841, 2023, DOI:10.32604/cmes.2023.027344

Abstract The extraction of traveling wave solutions for nonlinear evolution equations is a challenge in various mathematics, physics, and engineering disciplines. This article intends to analyze several traveling wave solutions for the modified regularized long-wave (MRLW) equation using several approaches, namely, the generalized algebraic method, the Jacobian elliptic functions technique, and the improved Q-expansion strategy. We successfully obtain analytical solutions consisting of rational, trigonometric, and hyperbolic structures. The adaptive moving mesh technique is applied to approximate the numerical solution of the proposed equation. The adaptive moving mesh method evenly distributes the points on the high error areas. This method perfectly and… More >

Mudassir Shams¹, Nasreen Kausar^2,*, Shams Forruque Ahmed³, Irfan Anjum Badruddin⁴, Syed Javed⁴

CMC-Computers, Materials & Continua, Vol.74, No.3, pp. 5331-5347, 2023, DOI:10.32604/cmc.2023.033528

Abstract A fifth-order family of an iterative method for solving systems of nonlinear equations and highly nonlinear boundary value problems has been developed in this paper. Convergence analysis demonstrates that the local order of convergence of the numerical method is five. The computer algebra system CAS-Maple, Mathematica, or MATLAB was the primary tool for dealing with difficult problems since it allows for the handling and manipulation of complex mathematical equations and other mathematical objects. Several numerical examples are provided to demonstrate the properties of the proposed rapidly convergent algorithms. A dynamic evaluation of the presented methods is also presented utilizing basins… More >

Saima Akram^1,*, Allah Nawaz¹, Muhammad Bilal Riaz², Mariam Rehman³

Intelligent Automation & Soft Computing, Vol.31, No.3, pp. 1467-1482, 2022, DOI:10.32604/iasc.2022.019767

Abstract In this article, the maximum possible numbers of periodic solutions for the quartic differential equation are calculated. In this regard, for the first time in the literature, we developed new formulae to determine the maximum number of periodic solutions greater than eight for the quartic equation. To obtain the maximum number of periodic solutions, we used a systematic procedure of bifurcation analysis. We used computer algebra Maple 18 to solve lengthy calculations that appeared in the formulae of focal values as integrations. The newly developed formulae were applied to a variety of polynomials with algebraic and homogeneous trigonometric coefficients of… More >

Mohammad Ghalandari¹, Ibrahim Mahariq², Majid Pourghasem³, Hasan Mulki², Fahd Jarad^4,5,*

CMC-Computers, Materials & Continua, Vol.70, No.2, pp. 3383-3397, 2022, DOI:10.32604/cmc.2022.020245

Abstract The middle layer model has been used in recent years to better describe the connection behavior in composite structures. The influencing parameters including low pre-screw and high preload have the main effects on nonlinear behavior of the connection as well as the amplitude of the excitation force applied to the structure. Therefore, in this study, the effects of connection behavior on the general structure in two sections of increasing damping and reducing the stiffness of the structures that lead to non-linear phenomena have been investigated. Due to the fact that in composite structure we are faced to the limitation of… More >

Ahad Alloqmani¹, Omimah Alsaedi¹, Nadia Bahatheg¹, Reem Alnanih^1,*, Lamiaa Elrefaei^1,2

Computer Systems Science and Engineering, Vol.40, No.3, pp. 1023-1042, 2022, DOI:10.32604/csse.2022.019704

Abstract Interactive learning tools can facilitate the learning process and increase student engagement, especially tools such as computer programs that are designed for human-computer interaction. Thus, this paper aims to help students learn five different methods for solving nonlinear equations using an interactive learning tool designed with common principles such as feedback, visibility, affordance, consistency, and constraints. It also compares these methods by the number of iterations and time required to display the result. This study helps students learn these methods using interactive learning tools instead of relying on traditional teaching methods. The tool is implemented using the MATLAB app and… More >

Mustafa Turkyilmazoglu^1,2,*

CMES-Computer Modeling in Engineering & Sciences, Vol.127, No.1, pp. 1-22, 2021, DOI:10.32604/cmes.2021.012595

Abstract The present paper is devoted to the convergence control and accelerating the traditional Decomposition Method of Adomian (ADM). By means of perturbing the initial or early terms of the Adomian iterates by adding a parameterized term, containing an embedded parameter, new modified ADM is constructed. The optimal value of this parameter is later determined via squared residual minimizing the error. The failure of the classical ADM is also prevented by a suitable value of the embedded parameter, particularly beneficial for the Duan–Rach modification of the ADM incorporating all the boundaries into the formulation. With the presented squared residual error analysis,… More >

Obadah Said Solaiman¹, Samsul Ariffin Abdul Karim², Ishak Hashim^1,*

CMC-Computers, Materials & Continua, Vol.67, No.2, pp. 1951-1962, 2021, DOI:10.32604/cmc.2021.015344

Abstract There are several ways that can be used to classify or compare iterative methods for nonlinear equations, for instance; order of convergence, informational efficiency, and efficiency index. In this work, we use another way, namely the basins of attraction of the method. The purpose of this study is to compare several iterative schemes for nonlinear equations. All the selected schemes are of the third-order of convergence and most of them have the same efficiency index. The comparison depends on the basins of attraction of the iterative techniques when applied on several polynomials of different degrees. As a comparison, we determine… More >

Obadah Said Solaiman, Ishak Hashim^*

Intelligent Automation & Soft Computing, Vol.27, No.2, pp. 379-390, 2021, DOI:10.32604/iasc.2021.015285

Abstract A new iterative technique for nonlinear equations is proposed in this work. The new scheme is of three steps, of which the first two steps are based on the sixth-order modified Halley’s method presented by the authors, and the last is a Newton step, with suitable approximations for the first derivatives appeared in the new scheme. The eighth-order of convergence of the new method is proved via Mathematica code. Every iteration of the presented scheme needs the evaluation of three functions and one first derivative. Therefore, the scheme is optimal in the sense of Kung-Traub conjecture. Several test nonlinear problems… More >

Obadah Said Solaiman, Ishak Hashim^*

CMC-Computers, Materials & Continua, Vol.66, No.2, pp. 1427-1444, 2021, DOI:10.32604/cmc.2020.012610

Abstract In this paper, we propose a fifth-order scheme for solving systems of nonlinear equations. The convergence analysis of the proposed technique is discussed. The proposed method is generalized and extended to be of any odd order of the form 2n − 1. The scheme is composed of three steps, of which the first two steps are based on the two-step Homeier’s method with cubic convergence, and the last is a Newton step with an appropriate approximation for the derivative. Every iteration of the presented method requires the evaluation of two functions, two Fréchet derivatives, and three matrix inversions. A comparison… More >

Syahmi Afandi Sariman¹, Ishak Hashim^{1, *}

CMC-Computers, Materials & Continua, Vol.65, No.1, pp. 69-85, 2020, DOI:10.32604/cmc.2020.010836

Abstract The classical iterative methods for finding roots of nonlinear equations, like the Newton method, Halley method, and Chebyshev method, have been modified previously to achieve optimal convergence order. However, the Householder method has so far not been modified to become optimal. In this study, we shall develop two new optimal Newton-Householder methods without memory. The key idea in the development of the new methods is the avoidance of the need to evaluate the second derivative. The methods fulfill the Kung-Traub conjecture by achieving optimal convergence order four with three functional evaluations and order eight with four functional evaluations. The efficiency… More >