Topic Editors

School of Mathematical Sciences, Bohai University, Jinzhou 121013, China
Institute for Advanced Studies in Basic Sciences, Zanjan, Iran

Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems

Abstract submission deadline
30 April 2024
Manuscript submission deadline
30 June 2024
Viewed by
12846

Topic Information

Dear Colleagues,

Solving nonlinear equations is an important problem in the areas of science and technology. The multipoint iterative method is a class of efficiency methods for solving nonlinear equations that can reach high convergence orders with less computational cost. In recent years, with the rapid development of computer technology, research on the multipoint iterative method has also developed rapidly. The main aim of this Special Issue is to show some research results of the multipoint iterative method. We welcome manuscripts that consider, but are not limited to, the following areas: Study of the stability of the multipoint iterative method by using complex and real dynamics; Fractional order multipoint iterative method for solving nonlinear equations; Design of Newton-type iterative method with and without memory; Design of Steffensen-type iterative method with and without memory; Application of the multipoint iterative method for solving matrix sign function; Application in ODEs/PDEs/SDEs.

Dr. Xiaofeng Wang
Dr. Fazlollah Soleymani
Topic Editors

Keywords

  • multipoint iterative method
  • nonlinear equations
  • dynamical analysis
  • basin of attractor
  • convergence order
  • iterative method for matrix sign function
  • with memory
  • Steffensen-type iterative method
  • Newton-type iterative method
  • computational efficiency

Participating Journals

Journal Name Impact Factor CiteScore Launched Year First Decision (median) APC
Algorithms
algorithms
2.3 3.7 2008 15 Days CHF 1600 Submit
Axioms
axioms
2.0 - 2012 21.8 Days CHF 2400 Submit
Fractal and Fractional
fractalfract
5.4 3.6 2017 18.9 Days CHF 2700 Submit
Mathematics
mathematics
2.4 3.5 2013 16.9 Days CHF 2600 Submit
Symmetry
symmetry
2.7 4.9 2009 16.2 Days CHF 2400 Submit

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Published Papers (11 papers)

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18 pages, 6865 KiB  
Article
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 - 06 Mar 2024
Viewed by 856
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=12. Full article
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16 pages, 1616 KiB  
Article
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
Viewed by 975
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
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14 pages, 2444 KiB  
Article
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
Viewed by 931
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
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14 pages, 350 KiB  
Article
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 - 05 Nov 2023
Viewed by 736
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
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14 pages, 324 KiB  
Article
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 551
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
12 pages, 794 KiB  
Article
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 1202
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
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17 pages, 2768 KiB  
Article
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 5 | Viewed by 1032
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
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11 pages, 655 KiB  
Article
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 1 | Viewed by 1024
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
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18 pages, 4765 KiB  
Article
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 4 | Viewed by 1223
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
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15 pages, 683 KiB  
Article
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 1072
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
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12 pages, 894 KiB  
Article
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 1358
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 2α 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
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