Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems

Journal Name	Impact Factor	CiteScore	Launched Year	First Decision (median)	APC
Algorithms algorithms	1.8	4.1	2008	18.9 Days	CHF 1600	Submit
Axioms axioms	1.9	-	2012	22.8 Days	CHF 2400	Submit
Fractal and Fractional fractalfract	3.6	4.6	2017	23.7 Days	CHF 2700	Submit
Mathematics mathematics	2.3	4.0	2013	18.3 Days	CHF 2600	Submit
Symmetry symmetry	2.2	5.4	2009	17.3 Days	CHF 2400	Submit

22 pages, 5414 KiB

Open AccessArticle

Generation of Julia Sets for a Novel Class of Generalized Rational Functions via Generalized Viscosity Iterative Method

by Iqbal Ahmad and Ahmad Almutlg

Axioms 2025, 14(4), 322; https://doi.org/10.3390/axioms14040322 - 21 Apr 2025

Viewed by 173

This article investigates and analyzes the diverse patterns of Julia sets generated by new classes of generalized exponential and sine rational functions. Using a generalized viscosity approximation-type iterative method, we derive escape criteria to visualize the Julia sets of these functions. This approach [...] Read more.

This article investigates and analyzes the diverse patterns of Julia sets generated by new classes of generalized exponential and sine rational functions. Using a generalized viscosity approximation-type iterative method, we derive escape criteria to visualize the Julia sets of these functions. This approach enhances existing algorithms, enabling the visualization of intricate fractal patterns as Julia sets. We graphically illustrate the variations in size and shape of the images as the iteration parameters change. The new fractals obtained are visually appealing and attractive. Moreover, we observe fascinating behavior in Julia sets when certain input parameters are fixed, while the values of n and m vary. We believe the conclusions of this study will inspire and motivate researchers and enthusiasts with a strong interest in fractal geometry. Full article

(This article belongs to the Topic Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems)

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18 pages, 6865 KiB

Open AccessArticle

A Class of Fifth-Order Chebyshev–Halley-Type Iterative Methods and Its Stability Analysis

by Xiaofeng Wang and Shaonan Guo

Fractal Fract. 2024, 8(3), 150; https://doi.org/10.3390/fractalfract8030150 - 6 Mar 2024

Viewed by 1503

Abstract

In this paper, a family of fifth-order Chebyshev–Halley-type iterative methods with one parameter is presented. The convergence order of the new iterative method is analyzed. By obtaining rational operators associated with iterative methods, the stability of the iterative method is studied by using [...] Read more.

In this paper, a family of fifth-order Chebyshev–Halley-type iterative methods with one parameter is presented. The convergence order of the new iterative method is analyzed. By obtaining rational operators associated with iterative methods, the stability of the iterative method is studied by using fractal theory. In addition, some strange fixed points and critical points are obtained. By using the parameter space related to the critical points, some parameters with good stability are obtained. The dynamic plane corresponding to these parameters is plotted, visualizing the stability characteristics. Finally, the fractal diagrams of several iterative methods on different polynomials are compared. Both numerical results and fractal graphs show that the new iterative method has good convergence and stability when

α = \frac{1}{2}

. Full article

(This article belongs to the Topic Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems)

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16 pages, 1616 KiB

Open AccessArticle

A Class of Sixth-Order Iterative Methods for Solving Nonlinear Systems: The Convergence and Fractals of Attractive Basins

by Xiaofeng Wang and Wenshuo Li

Fractal Fract. 2024, 8(3), 133; https://doi.org/10.3390/fractalfract8030133 - 26 Feb 2024

Cited by 3 | Viewed by 1648

Abstract

In this paper, a Newton-type iterative scheme for solving nonlinear systems is designed. In the process of proving the convergence order, we use the higher derivatives of the function and show that the convergence order of this iterative method is six. In order [...] Read more.

In this paper, a Newton-type iterative scheme for solving nonlinear systems is designed. In the process of proving the convergence order, we use the higher derivatives of the function and show that the convergence order of this iterative method is six. In order to avoid the influence of the existence of higher derivatives on the proof of convergence, we mainly discuss the convergence of this iterative method under weak conditions. In Banach space, the local convergence of the iterative scheme is established by using the

ω

-continuity condition of the first-order Fréchet derivative, and the application range of the iterative method is extended. In addition, we also give the radius of a convergence sphere and the uniqueness of its solution. Finally, the superiority of the new iterative method is illustrated by drawing attractive basins and comparing them with the average iterative times of other same-order iterative methods. Additionally, we utilize this iterative method to solve both nonlinear systems and nonlinear matrix sign functions. The applicability of this study is demonstrated by solving practical chemical problems. Full article

(This article belongs to the Topic Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems)

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14 pages, 2444 KiB

Open AccessArticle

On Constructing a Family of Sixth-Order Methods for Multiple Roots

by Young Hee Geum

Fractal Fract. 2023, 7(12), 878; https://doi.org/10.3390/fractalfract7120878 - 11 Dec 2023

Cited by 1 | Viewed by 1476

Abstract

A family of three-point, sixth-order, multiple-zero solvers is developed, and special cases of weight functions are investigated based on polynomials and low-order rational functions. The chosen cases of the proposed iterative method are compared with existing methods. The experiments show the superiority of [...] Read more.

A family of three-point, sixth-order, multiple-zero solvers is developed, and special cases of weight functions are investigated based on polynomials and low-order rational functions. The chosen cases of the proposed iterative method are compared with existing methods. The experiments show the superiority of the proposed schemes in terms of the number of divergent points and the average number of function evaluations per point. The dynamical characteristics of the developed methods, along with their illustrations, are represented with detailed analyses, comparisons, and comments. Full article

(This article belongs to the Topic Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems)

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14 pages, 350 KiB

Open AccessArticle

Bifurcating Limit Cycles with a Perturbation of Systems Composed of Piecewise Smooth Differential Equations Consisting of Four Regions

by Erli Zhang, Jihua Yang and Stanford Shateyi

Mathematics 2023, 11(21), 4555; https://doi.org/10.3390/math11214555 - 5 Nov 2023

Viewed by 1317

Abstract

Systems composed of piecewise smooth differential (PSD) mappings have quantitatively been searched for answers to a substantial issue of limit cycle (LC) bifurcations. In this paper, LC numbers (LCNs) of a PSD system (PSDS) consisting of four regions are dealt with. A Melnikov [...] Read more.

Systems composed of piecewise smooth differential (PSD) mappings have quantitatively been searched for answers to a substantial issue of limit cycle (LC) bifurcations. In this paper, LC numbers (LCNs) of a PSD system (PSDS) consisting of four regions are dealt with. A Melnikov mapping whose order is one is implicitly obtained by finding its originators when the system is perturbed under any nth degree of real polynomials. Then, the approach employing the Picard–Fuchs mapping is utilized to attain a higher boundary of bifurcation LCNs of systems composed of PSD functions with a global center. The method we used could be implemented to examine the problems related to the LC of other PSDS. Full article

(This article belongs to the Topic Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems)

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14 pages, 324 KiB

Open AccessArticle

Global Classical Solutions of the 1.5D Relativistic Vlasov–Maxwell–Chern–Simons System

by Jing Chen, Omar Bazighifan, Chengjun Luo and Yanlai Song

Axioms 2023, 12(7), 627; https://doi.org/10.3390/axioms12070627 - 25 Jun 2023

Viewed by 978

Abstract

We investigate the kinetic model of the relativistic Vlasov–Maxwell–Chern–Simons system, which originates from gauge theory. This system can be seen as an electromagnetic fields (i.e., Maxwell–Chern–Simons fields) perturbation for the classical Vlasov equation. By virtue of a nondecreasing function and an iteration method, [...] Read more.

We investigate the kinetic model of the relativistic Vlasov–Maxwell–Chern–Simons system, which originates from gauge theory. This system can be seen as an electromagnetic fields (i.e., Maxwell–Chern–Simons fields) perturbation for the classical Vlasov equation. By virtue of a nondecreasing function and an iteration method, the uniqueness and existence of the global solutions for the 1.5D case are obtained. Full article

(This article belongs to the Topic Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems)

12 pages, 794 KiB

Open AccessArticle

A Fast Computational Scheme for Solving the Temporal-Fractional Black–Scholes Partial Differential Equation

by Rouhollah Ghabaei, Taher Lotfi, Malik Zaka Ullah and Stanford Shateyi

Fractal Fract. 2023, 7(4), 323; https://doi.org/10.3390/fractalfract7040323 - 12 Apr 2023

Cited by 1 | Viewed by 1856

Abstract

In this work, we propose a fast scheme based on higher order discretizations on graded meshes for resolving the temporal-fractional partial differential equation (PDE), which benefits the memory feature of fractional calculus. To avoid excessively increasing the number of discretization points, such as [...] Read more.

In this work, we propose a fast scheme based on higher order discretizations on graded meshes for resolving the temporal-fractional partial differential equation (PDE), which benefits the memory feature of fractional calculus. To avoid excessively increasing the number of discretization points, such as the standard finite difference or meshfree methods, and, at the same time, to increase the efficiency of the solver, we employ discretizations on spatially non-uniform meshes with an attention on the non-smoothness area of the underlying asset. Therefore, the PDE problem is transformed to a linear system of algebraic equations. We perform numerical simulations to observe and check the behavior of the presented scheme in contrast to the existing methods. Full article

(This article belongs to the Topic Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems)

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17 pages, 2768 KiB

Open AccessArticle

Stability Analysis of Simple Root Seeker for Nonlinear Equation

by Xiaofeng Wang and Wenshuo Li

Axioms 2023, 12(2), 215; https://doi.org/10.3390/axioms12020215 - 18 Feb 2023

Cited by 6 | Viewed by 1671

Abstract

In this paper, the stability of a class of Liu–Wang’s optimal eighth-order single-parameter iterative methods for solving simple roots of nonlinear equations was studied by applying them to arbitrary quadratic polynomials. Under the Riemann sphere and scaling theorem, the complex dynamic behavior of [...] Read more.

In this paper, the stability of a class of Liu–Wang’s optimal eighth-order single-parameter iterative methods for solving simple roots of nonlinear equations was studied by applying them to arbitrary quadratic polynomials. Under the Riemann sphere and scaling theorem, the complex dynamic behavior of the iterative method was analyzed by fractals. We discuss the stability of all fixed points and the parameter spaces starting from the critical points with the Mathematica software. The dynamical planes of the elements with good and bad dynamical behavior are given, and the optimal parameter element with stable behavior was obtained. Finally, a numerical experiment and practical application were carried out to prove the conclusion. Full article

(This article belongs to the Topic Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems)

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11 pages, 655 KiB

Open AccessArticle

From Fractal Behavior of Iteration Methods to an Efficient Solver for the Sign of a Matrix

by Tao Liu, Malik Zaka Ullah, Khalid Mohammed Ali Alshahrani and Stanford Shateyi

Fractal Fract. 2023, 7(1), 32; https://doi.org/10.3390/fractalfract7010032 - 28 Dec 2022

Cited by 8 | Viewed by 1659

Abstract

Investigating the fractal behavior of iteration methods on special polynomials can help to find iterative methods with global convergence for finding special matrix functions. By employing such a methodology, we propose a new solver for the sign of an invertible square matrix. The [...] Read more.

Investigating the fractal behavior of iteration methods on special polynomials can help to find iterative methods with global convergence for finding special matrix functions. By employing such a methodology, we propose a new solver for the sign of an invertible square matrix. The presented method achieves the fourth rate of convergence by using as few matrix products as possible. Its attraction basin shows larger convergence radii, in contrast to its Padé-type methods of the same order. Computational tests are performed to check the efficacy of the proposed solver. Full article

(This article belongs to the Topic Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems)

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18 pages, 4765 KiB

Open AccessArticle

Choosing the Best Members of the Optimal Eighth-Order Petković’s Family by Its Fractal Behavior

by Xiaofeng Wang and Wenshuo Li

Fractal Fract. 2022, 6(12), 749; https://doi.org/10.3390/fractalfract6120749 - 19 Dec 2022

Cited by 6 | Viewed by 1804

Abstract

In this paper, by applying Petković’s iterative method to the Möbius conjugate mapping of a quadratic polynomial function, we attain an optimal eighth-order rational operator with a single parameter r and research the stability of this method by using complex dynamics tools on [...] Read more.

In this paper, by applying Petković’s iterative method to the Möbius conjugate mapping of a quadratic polynomial function, we attain an optimal eighth-order rational operator with a single parameter r and research the stability of this method by using complex dynamics tools on the basis of fractal theory. Through analyzing the stability of the fixed point and drawing the parameter space related to the critical point, the parameter family which can make the behavior of the corresponding iterative method stable or unstable is obtained. Lastly, the consequence is verified by showing their corresponding dynamical planes. Full article

(This article belongs to the Topic Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems)

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15 pages, 683 KiB

Open AccessArticle

Interactions of Logistic Distribution to Credit Valuation Adjustment: A Study on the Associated Expected Exposure and the Conditional Value at Risk

by Yanlai Song, Stanford Shateyi, Jianying He and Xueqing Cui

Mathematics 2022, 10(20), 3828; https://doi.org/10.3390/math10203828 - 17 Oct 2022

Cited by 2 | Viewed by 1711

Abstract

In Basel III, the credit valuation adjustment (CVA) was given, and it was discussed that a bank covers mark-to-market losses for expected counterparty risk with a CVA capital charge. The purpose of this study is threefold. Using the logistic distribution, it is shown [...] Read more.

In Basel III, the credit valuation adjustment (CVA) was given, and it was discussed that a bank covers mark-to-market losses for expected counterparty risk with a CVA capital charge. The purpose of this study is threefold. Using the logistic distribution, it is shown how the expected exposure can be derived for an interest rate swap. Secondly, the risk measure of VaR is contributed for the CVA under this distribution. Thirdly, generalizations for the CVA VaR and CVA CVaR are given by considering both the credit spread and the expected positive exposure to follow the logistic distributions with different parameters. Finally, several simulations are provided to uphold the theoretical discussions. Full article

(This article belongs to the Topic Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems)

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12 pages, 894 KiB

Open AccessArticle

Inverse Multiquadric Function to Price Financial Options under the Fractional Black–Scholes Model

by Yanlai Song and Stanford Shateyi

Fractal Fract. 2022, 6(10), 599; https://doi.org/10.3390/fractalfract6100599 - 15 Oct 2022

Cited by 7 | Viewed by 2034

Abstract

The inverse multiquadric radial basis function (RBF), which is one of the most important functions in the theory of RBFs, is employed on an adaptive mesh of points for pricing a fractional Black–Scholes partial differential equation (PDE) based on the modified RL derivative. [...] Read more.

The inverse multiquadric radial basis function (RBF), which is one of the most important functions in the theory of RBFs, is employed on an adaptive mesh of points for pricing a fractional Black–Scholes partial differential equation (PDE) based on the modified RL derivative. To solve this problem, discretization along space is carried out on a non-uniform grid in order to focus on the hot area, at which the initial condition of the pricing model, i.e., the payoff, has discontinuity. The L1 scheme having the convergence order

2 - α

is used along the time fractional variable. Then, our proposed numerical method is built by matrices of differentiations to be as efficient as possible. Computational pieces of evidence are brought forward to uphold the theoretical discussions and show how the presented method is efficient in contrast to the exiting solvers. Full article

(This article belongs to the Topic Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems)

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