Research

14 pages, 1502 KiB

Open AccessArticle

Extension of the Optimal Auxiliary Function Method to Solve the System of a Fractional-Order Whitham–Broer–Kaup Equation

by Zainab Alsheekhhussain, Khaled Moaddy, Rasool Shah, Saleh Alshammari, Mohammad Alshammari, M. Mossa Al-Sawalha and Aisha Abdullah Alderremy

Fractal Fract. 2024, 8(1), 1; https://doi.org/10.3390/fractalfract8010001 - 19 Dec 2023

Cited by 4 | Viewed by 1006

Abstract

In this paper, we introduce and implement the optimal auxiliary function method to solve a system of fractional-order Whitham–Broer–Kaup equations, a class of nonlinear partial differential equations with broad applications in mathematical physics. This method provides a systematic and efficient approach to finding [...] Read more.

In this paper, we introduce and implement the optimal auxiliary function method to solve a system of fractional-order Whitham–Broer–Kaup equations, a class of nonlinear partial differential equations with broad applications in mathematical physics. This method provides a systematic and efficient approach to finding accurate solutions for complex systems of fractional-order equations. We give a full analysis using tables and figures to demonstrate the reliability and accuracy of our approach. We confirm the effectiveness of our suggested method in solving the considered equations using numerical simulations and comparisons, emphasizing its potential for applications in a variety of scientific and engineering areas. Full article

(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)

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19 pages, 2269 KiB

Open AccessArticle

A Comparative Analysis of Laplace Residual Power Series and a New Iteration Method for Fitzhugh-Nagumo Equation in the Caputo Operator Framework

by Azzh Saad Alshehry

Fractal Fract. 2023, 7(12), 867; https://doi.org/10.3390/fractalfract7120867 - 06 Dec 2023

Viewed by 853

Abstract

In this paper, I utilize the Laplace residual power series method (LRPSM) along with a novel iteration technique to investigate the Fitzhugh-Nagumo equation within the framework of the Caputo operator. The Fitzhugh-Nagumo equation is a fundamental model for describing excitable systems, playing a [...] Read more.

In this paper, I utilize the Laplace residual power series method (LRPSM) along with a novel iteration technique to investigate the Fitzhugh-Nagumo equation within the framework of the Caputo operator. The Fitzhugh-Nagumo equation is a fundamental model for describing excitable systems, playing a crucial role in understanding various physiological and biological phenomena. The Caputo operator extends the conventional derivative to handle non-local and non-integer-order differential equations, making it a potent tool for modeling complex processes. Our study involves transforming the Fitzhugh-Nagumo equation into its Laplace domain representation, applying the LRPSM to derive a series solution. We then introduce a novel iteration technique to enhance the solution’s convergence properties, enabling more accurate and efficient computations. This approach offers a systematic methodology for solving the Fitzhugh-Nagumo equation with the Caputo operator, providing deeper insights into excitable system dynamics. Numerical examples and comparisons with existing methods demonstrate the accuracy and efficiency of the LRPSM with the new iteration technique, showcasing its potential for solving diverse differential equations involving the Caputo operator and advancing mathematical modeling in various scientific and engineering domains. Full article

(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)

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17 pages, 2267 KiB

Open AccessArticle

Fractal Complexity of a New Biparametric Family of Fourth Optimal Order Based on the Ermakov–Kalitkin Scheme

by Alicia Cordero, Renso V. Rojas-Hiciano, Juan R. Torregrosa and Maria P. Vassileva

Fractal Fract. 2023, 7(6), 459; https://doi.org/10.3390/fractalfract7060459 - 03 Jun 2023

Viewed by 834

Abstract

In this paper, we generalize the scheme proposed by Ermakov and Kalitkin and present a class of two-parameter fourth-order optimal methods, which we call Ermakov’s Hyperfamily. It is a substantial improvement of the classical Newton’s method because it optimizes one that extends the [...] Read more.

In this paper, we generalize the scheme proposed by Ermakov and Kalitkin and present a class of two-parameter fourth-order optimal methods, which we call Ermakov’s Hyperfamily. It is a substantial improvement of the classical Newton’s method because it optimizes one that extends the regions of convergence and is very stable. Another novelty is that it is a class containing as particular cases some classical methods, such as King’s family. From this class, we generate a new uniparametric family, which we call the KLAM, containing the classical Ostrowski and Chun, whose efficiency, stability, and optimality has been proven but also new methods that in many cases outperform these mentioned, as we prove. We demonstrate that it is of a fourth order of convergence, as well as being computationally efficienct. A dynamical study is performed allowing us to choose methods with good stability properties and to avoid chaotic behavior, implicit in the fractal structure defined by the Julia set in the related dynamic planes. Some numerical tests are presented to confirm the theoretical results and to compare the proposed methods with other known methods. Full article

(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)

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16 pages, 777 KiB

Open AccessArticle

A Seventh Order Family of Jarratt Type Iterative Method for Electrical Power Systems

by Saima Yaseen, Fiza Zafar and Francisco I. Chicharro

Fractal Fract. 2023, 7(4), 317; https://doi.org/10.3390/fractalfract7040317 - 06 Apr 2023

Cited by 3 | Viewed by 1386

Abstract

A load flow study referred to as a power flow study is a numerical analysis of the electricity that flows through any electrical power system. For instance, if a transmission line needs to be taken out of service for maintenance, load flow studies [...] Read more.

A load flow study referred to as a power flow study is a numerical analysis of the electricity that flows through any electrical power system. For instance, if a transmission line needs to be taken out of service for maintenance, load flow studies allow us to assess whether the remaining line can carry the load without exceeding its rated capacity. So, we need to understand about the voltage level and voltage phase angle on each bus under steady-state conditions to keep the bus voltage within a specific range. In this paper, our goal is to present a higher order efficient iterative method to carry out a power flow study to determine the voltages (magnitude and angle) for a specific load, generation and network conditions. We introduce a new seventh-order three-step iterative scheme for obtaining approximate solution of nonlinear systems of equations. We attain the seventh-order convergence by using four function evaluations which makes it worthy of interest. Moreover, we show its applicability to the electrical power system for calculating voltages and phase angles. By calculating the bus angle and voltage level, we conclude that the performance of the power system is assessed in a more efficient manner using the new scheme. In addition, dynamical planes of the methods applied on nonlinear systems of equations show global convergence. Full article

(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)

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21 pages, 877 KiB

Open AccessArticle

Convergence Analysis of a New Implicit Iterative Scheme and Its Application to Delay Caputo Fractional Differential Equations

by Austine Efut Ofem, Mfon Okon Udo, Oboyi Joseph, Reny George and Chukwuka Fernando Chikwe

Fractal Fract. 2023, 7(3), 212; https://doi.org/10.3390/fractalfract7030212 - 24 Feb 2023

Cited by 4 | Viewed by 1010

Abstract

This article presents a new three-step implicit iterative method. The proposed method is used to approximate the fixed points of a certain class of pseudocontractive-type operators. Additionally, the strong convergence results of the new iterative procedure are derived. Some examples are constructed to [...] Read more.

This article presents a new three-step implicit iterative method. The proposed method is used to approximate the fixed points of a certain class of pseudocontractive-type operators. Additionally, the strong convergence results of the new iterative procedure are derived. Some examples are constructed to authenticate the assumptions in our main result. At the end, we use our new method to solve a fractional delay differential equation in the sense of Caputo. Our main results improve and generalize the results of many prominent authors in the existing literature. Full article

(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)

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12 pages, 516 KiB

Open AccessArticle

Structured Doubling Algorithm for a Class of Large-Scale Discrete-Time Algebraic Riccati Equations with High-Ranked Constant Term

by Bo Yu, Chengxu Jiang and Ning Dong

Fractal Fract. 2023, 7(2), 193; https://doi.org/10.3390/fractalfract7020193 - 14 Feb 2023

Viewed by 1017

Abstract

Consider the computation of the solution for a class of discrete-time algebraic Riccati equations (DAREs) with the low-ranked coefficient matrix G and the high-ranked constant matrix H. A structured doubling algorithm is proposed for large-scale problems when A is of lowrank. Compared [...] Read more.

Consider the computation of the solution for a class of discrete-time algebraic Riccati equations (DAREs) with the low-ranked coefficient matrix G and the high-ranked constant matrix H. A structured doubling algorithm is proposed for large-scale problems when A is of lowrank. Compared to the existing doubling algorithm of

O (2^{k} n)

flops at the k-th iteration, the newly developed version merely needs

O (n)

flops for preprocessing and

O ({(k + 1)}^{3} m^{3})

flopsfor iterations and is more proper for large-scale computations when

m ≪ n

. The convergence and complexity of the algorithm are subsequently analyzed. Illustrative numerical experiments indicate that the presented algorithm, which consists of a dominant time-consuming preprocessing step and a trivially iterative step, is capable of computing the solution efficiently for large-scale DAREs. Full article

(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)

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22 pages, 685 KiB

Open AccessArticle

Order of Convergence, Extensions of Newton–Simpson Method for Solving Nonlinear Equations and Their Dynamics

by Santhosh George, Ajil Kunnarath, Ramya Sadananda, Jidesh Padikkal and Ioannis K. Argyros

Fractal Fract. 2023, 7(2), 163; https://doi.org/10.3390/fractalfract7020163 - 06 Feb 2023

Cited by 1 | Viewed by 1036

Abstract

Local convergence of order three has been established for the Newton–Simpson method (NS), provided that derivatives up to order four exist. However, these derivatives may not exist and the NS can converge. For this reason, we recover the convergence order based only on [...] Read more.

Local convergence of order three has been established for the Newton–Simpson method (NS), provided that derivatives up to order four exist. However, these derivatives may not exist and the NS can converge. For this reason, we recover the convergence order based only on the first two derivatives. Moreover, the semilocal convergence of NS and some of its extensions not given before is developed. Furthermore, the dynamics are explored for these methods with many illustrations. The study contains examples verifying the theoretical conditions. Full article

(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)

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15 pages, 452 KiB

Open AccessArticle

Spectral Collocation Approach via Normalized Shifted Jacobi Polynomials for the Nonlinear Lane-Emden Equation with Fractal-Fractional Derivative

by Youssri Hassan Youssri and Ahmed Gamal Atta

Fractal Fract. 2023, 7(2), 133; https://doi.org/10.3390/fractalfract7020133 - 31 Jan 2023

Cited by 29 | Viewed by 1796

Abstract

Herein, we adduce, analyze, and come up with spectral collocation procedures to iron out a specific class of nonlinear singular Lane–Emden (LE) equations with generalized Caputo derivatives that appear in the study of astronomical objects. The offered solution is approximated as a truncated [...] Read more.

Herein, we adduce, analyze, and come up with spectral collocation procedures to iron out a specific class of nonlinear singular Lane–Emden (LE) equations with generalized Caputo derivatives that appear in the study of astronomical objects. The offered solution is approximated as a truncated series of the normalized shifted Jacobi polynomials under the assumption that the exact solution is an element in

L_{2}

. The spectral collocation method is used as a solver to obtain the unknown expansion coefficients. The Jacobi roots are used as collocation nodes. Our solutions can easily be a generalization of the solutions of the classical LE equation, by obtaining a numerical solution based on new parameters, by fixing these parameters to the classical case, we obtain the solution of the classical equation. We provide a meticulous convergence analysis and demonstrate rapid convergence of the truncation error concerning the number of retained modes. Numerical examples show the effectiveness and applicability of the method. The primary benefits of the suggested approach are that we significantly reduce the complexity of the underlying differential equation by solving a nonlinear system of algebraic equations that can be done quickly and accurately using Newton’s method and vanishing initial guesses. Full article

(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)

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20 pages, 394 KiB

Open AccessArticle

Stability and Bifurcation Analysis of Fifth-Order Nonlinear Fractional Difference Equation

by Abdul Khaliq, Irfan Mustafa, Tarek F. Ibrahim, Waleed M. Osman, Bushra R. Al-Sinan, Arafa Abdalrhim Dawood and Manal Yagoub Juma

Fractal Fract. 2023, 7(2), 113; https://doi.org/10.3390/fractalfract7020113 - 23 Jan 2023

Cited by 6 | Viewed by 1783

Abstract

In this paper, a rational difference equation with positive parameters and non-negative conditions is used to determine the presence and direction of the Neimark–Sacker bifurcation. The neimark–Sacker bifurcation of the system is first studied using the characteristic equation. In addition, we study bifurcation [...] Read more.

In this paper, a rational difference equation with positive parameters and non-negative conditions is used to determine the presence and direction of the Neimark–Sacker bifurcation. The neimark–Sacker bifurcation of the system is first studied using the characteristic equation. In addition, we study bifurcation invariant curves from the perspective of normal form theory. A computer simulation is used to illustrate the analytical results. Full article

(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)

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12 pages, 494 KiB

Open AccessArticle

Necessary and Sufficient Conditions for Existence and Uniqueness of Solutions to Nabla Fractional Systems

by Jikai Yang, Hongli Li and Long Zhang

Fractal Fract. 2022, 6(12), 723; https://doi.org/10.3390/fractalfract6120723 - 07 Dec 2022

Viewed by 967

Abstract

In this paper, we study the existence and uniqueness of solutions for nabla fractional systems. By using the properties of bijective functions, we obtain a necessary and sufficient condition ensuring the existence and uniqueness of solutions for a class of fractional discrete systems. [...] Read more.

In this paper, we study the existence and uniqueness of solutions for nabla fractional systems. By using the properties of bijective functions, we obtain a necessary and sufficient condition ensuring the existence and uniqueness of solutions for a class of fractional discrete systems. Furthermore, we derive two sufficient conditions guaranteeing the existence of solutions by means of a nonlinear functional analysis method. In addition, the above conclusions are extended to high-dimensional delayed systems. Finally, two examples are given to illustrate the validity of our results. Full article

(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)

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21 pages, 959 KiB

Open AccessArticle

Extended Comparison between Two Derivative-Free Methods of Order Six for Equations under the Same Conditions

by Samundra Regmi, Ioannis K. Argyros, Christopher I. Argyros and Debasis Sharma

Fractal Fract. 2022, 6(11), 634; https://doi.org/10.3390/fractalfract6110634 - 30 Oct 2022

Viewed by 923

Abstract

Under the same conditions, we propose the extended comparison between two derivative free schemes of order six for addressing equations. The existing convergence technique used the standard Taylor series approach, which requires derivatives up to order seven. In contrast to previous researchers, our [...] Read more.

Under the same conditions, we propose the extended comparison between two derivative free schemes of order six for addressing equations. The existing convergence technique used the standard Taylor series approach, which requires derivatives up to order seven. In contrast to previous researchers, our convergence theorems only demand the first derivative. In addition, formulas for determining the region of uniqueness for the solution, convergence radii, and error estimations are suggested. As a consequence, we broaden the utility of these productive schemes. Moreover, we present a comparison of attraction basins for these schemes to obtain roots of complex polynomial equations. The confirmation of our convergence findings on application problems brings this research to a close. Full article

(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)

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15 pages, 1460 KiB

Open AccessArticle

Parametric Family of Root-Finding Iterative Methods: Fractals of the Basins of Attraction

by José J. Padilla, Francisco I. Chicharro, Alicia Cordero and Juan R. Torregrosa

Fractal Fract. 2022, 6(10), 572; https://doi.org/10.3390/fractalfract6100572 - 08 Oct 2022

Cited by 4 | Viewed by 1650

Abstract

Research interest in iterative multipoint schemes to solve nonlinear problems has increased recently because of the drawbacks of point-to-point methods, which need high-order derivatives to increase the order of convergence. However, this order is not the only key element to classify the iterative [...] Read more.

Research interest in iterative multipoint schemes to solve nonlinear problems has increased recently because of the drawbacks of point-to-point methods, which need high-order derivatives to increase the order of convergence. However, this order is not the only key element to classify the iterative schemes. We aim to design new multipoint fixed point classes without memory, that improve or bring together the existing ones in different areas such as computational efficiency, stability and also convergence order. In this manuscript, we present a family of parametric iterative methods, whose order of convergence is four, that has been designed by using composition and weight function techniques. A qualitative analysis is made, based on complex discrete dynamics, to select those elements of the class with best stability properties on low-degree polynomials. This stable behavior is directly related with the simplicity of the fractals defined by the basins of attraction. In the opposite, particular methods with unstable performance present high-complexity in the fractals of their basins. The stable members are demonstrated also be the best ones in terms of numerical performance of non-polynomial functions, with special emphasis on Colebrook-White equation, with wide applications in Engineering. Full article

(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)

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