Type Iterative Method Research Articles

Fractal as Julia sets of complex functions via a new generalized viscosity approximation type iterative method

In this article, we study and explore novel variants of Julia set patterns that are linked to the complex exponential function $W(z)=pe^{z^n}+qz+r$, and complex cosine function $T(z)=\cos({z^n})+dz+c$, where $n\geq 2$ and $c,d,p,q,r\in \mathbb{C}$ by employing a generalized viscosity approximation type iterative method introduced by Nandal et al. (Iteration process for fixed point problems and zero of maximal monotone operators, Symmetry, 2019) to visualize these sets. We utilize a generalized viscosity approximation type iterative method to derive an escape criterion for visualizing Julia sets. This is achieved by generalizing the existing algorithms, which led to visualization of beautiful fractals as Julia sets. Additionally, we present graphical illustrations of Julia sets to demonstrate their dependence on the iteration parameters. Our study concludes with an analysis of variations in the images and the influence of parameters on the color and appearance of the fractal patterns. Finally, we observe intriguing behaviors of Julia sets with fixed input parameters and varying values of $n$ via proposed algorithms.

Development and performance evaluation of the Block-product type iterative methods with the variable grouping strategy

Development and performance evaluation of the Block-product type iterative methods with the variable grouping strategy

Generation of Julia and Mandelbrot fractals for a generalized rational type mapping via viscosity approximation type iterative method extended with $ s $-convexity

<abstract><p>A dynamic visualization of Julia and Mandelbrot fractals involves creating animated representations of these fractals that change over time or in response to user interaction which allows users to gain deeper insights into the intricate structures and properties of these fractals. This paper explored the dynamic visualization of fractals within Julia and Mandelbrot sets, focusing on a generalized rational type complex polynomial of the form $ S_{c}(z) = a z^{n}+\frac{b}{z^{m}}+c $, where $ a, b, c \in \mathbb{C} $ with $ |a| > 1 $ and $ n, m \in \mathbb{N} $ with $ n > 1 $. By applying viscosity approximation-type iteration processes extended with $ s $-convexity, we unveiled the intricate dynamics inherent in these fractals. Novel escape criteria was derived to facilitate the generation of Julia and Mandelbrot sets via the proposed iteration process. We also presented graphical illustrations of Mandelbrot and Julia fractals, highlighting the change in the structure of the generated sets with respect to the variations in parameters.</p></abstract>

On the viscosity approximation type iterative method and its non-linear behaviour in the generation of Mandelbrot and Julia sets

In this paper, we visualise and analyse the dynamics of fractals (Julia and Mandelbrot sets) for complex polynomials of the form T(z)=zn+mz+r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T(z) = z^{n} + mz + r$$\\end{document}, where n≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 2$$\\end{document} and m,r∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m, r \\in \\mathbb {C}$$\\end{document}, by adopting the viscosity approximation type iteration process which is most widely used iterative method for finding fixed points of non-linear operators. We establish a convergence condition in the form of escape criterion which allows to adapt the escape-time algorithm to the considered iteration scheme. We also present some graphical examples of the Mandelbrot and Julia fractals showing the dependency of Julia and Mandelbrot sets on complex polynomials, contraction mappings, and iteration parameters. Moreover, we propose two numerical measures that allow the study of the dependency of the set shape change on the values of the iteration parameters. Using these two measures, we show that the dependency for the considered iteration method is non-linear.

A new family of Sakurai–Torii–Sugiura type iterative methods with high order of convergence

In this paper, we introduce a new family of iterative methods for finding simultaneously all zeros (multiple or simple) of a polynomial. The proposed family is constructed by combining the known Sakurai–Torii–Sugiura iteration function with an arbitrary iteration function. We provide a detailed convergence analysis in the following two directions: local convergence if the polynomial has multiple zeros with known multiplicity and semilocal convergence if the polynomial has only simple zeros. As an application, we study the convergence of several particular iterative methods with high order of convergence.

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

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.

Iterative learning algorithms for boundary tracing problems of nonlinear fractional diffusion equations

<abstract><p>In this paper, the iterative learning control technique is extended to distributed parameter systems governed by nonlinear fractional diffusion equations. Based on $ P $-type and $ PI^{\theta} $-type iterative learning control methods, sufficient conditions for the convergences of systems are given. Finally, numerical examples are presented to illustrate the efficiency of the proposed iterative schemes. The numerical results show that the closed-loop iterative learning control scheme converges faster than the open-loop iterative learning control scheme and the $ PI^{\theta} $-type iterative learning control scheme converges faster than the $ P $-type and the $ PI $-type iterative learning control scheme.</p></abstract>

Iterative schemes for numerical reckoning of fixed points of new nonexpansive mappings with an application

<abstract><p>The goal of this manuscript is to introduce a new class of generalized nonexpansive operators, called $ (\alpha, \beta, \gamma) $-nonexpansive mappings. Furthermore, some related properties of these mappings are investigated in a general Banach space. Moreover, the proposed operators utilized in the $ K $-iterative technique estimate the fixed point and examine its behavior. Also, two examples are provided to support our main results. The numerical results clearly show that the $ K $-iterative approach converges more quickly when used with this new class of operators. Ultimately, we used the $ K $-type iterative method to solve a variational inequality problem on a Hilbert space.</p></abstract>

Extending the Applicability of Cordero Type Iterative Method

Symmetries play a vital role in the study of physical systems. For example, microworld and quantum physics problems are modeled on the principles of symmetry. These problems are then formulated as equations defined on suitable abstract spaces. Most of these studies reduce to solving nonlinear equations in suitable abstract spaces iteratively. In particular, the convergence of a sixth-order Cordero type iterative method for solving nonlinear equations was studied using Taylor expansion and assumptions on the derivatives of order up to six. In this study, we obtained order of convergence six for Cordero type method using assumptions only on the first derivative. Moreover, we modified Cordero’s method and obtained an eighth-order iterative scheme. Further, we considered analogous iterative methods to solve an ill-posed problem in a Hilbert space setting.

An Application of Viscosity Approximation Type Iterative Method in the Generation of Mandelbrot and Julia Fractals

In this paper, we present an application of the viscosity approximation type iterative method introduced by Nandal et al. (Iteration Process for Fixed Point Problems and Zeros of Maximal Monotone Operators, Symmetry, 2019) to visualize and analyse the Julia and Mandelbrot sets for a complex polynomial of the type T(z) = z^{n} + p z + r, where p, rin {mathbb {C}}, and n ge 2. This iterative method has many applications in solving various fixed point problems. We derive an escape criterion to visualize Julia and Mandelbrot sets via the proposed viscosity approximation type method. Moreover, we present several graphical examples of the fractals generated with the proposed iteration method.

Sobolev gradient type iterative solution methods for a nonlinear 4th order elastic plate equation

This paper considers the numerical solution of an elastic bending model of a thin plate, based on material nonlinearity, which leads to a nonlinear elliptic 4th order boundary value problem. Our goal is to summarize possible efficient iterative solvers, adapted to this problem, based on a Sobolev space background. It is proved that the proposed methods exhibit robust behaviour, that is, convergence rates are bounded independently of the considered Galerkin discretization subspace.

A Convergence Theorem of an Implicit Type Iteration in a Geodesic Space

In this paper, we define a quadrilateral inequality, Browder type and implicit iterative method in a CAT(1) space. Furthermore, we prove a convergence theorem of the implicit iterative method in the same space used the quadrilateral inequality.

Convergence of Jungck-Kirk Type Iteration Method with Applications

The aim of this article is to define a new Jungck-Kirk type iteration method and to examine the convergence result under appropriate conditions together with other Jungck-Kirk type iteration methods in the literature. It is also to analyze whether the newly defined iteration method is stable. In addition, it has been shown through numerical examples that the new iteration method has a better convergence rate than the others. Finally, to show the validity of convergence and stability results, some examples are given. The results obtained in this paper may be interpreted as a refinement and improvement of the previously known results.

A robust and efficient fingerprint image restoration method based on a phase-field model

In this study, we present a robust and efficient fingerprint image restoration algorithm using the nonlocal Cahn–Hilliard (CH) equation, which was proposed for modeling the microphase separation of diblock copolymers. We take a small local region embedding the damaged domain and solve the nonlocal CH equation to restore the fingerprint image. A Gauss–Seidel type iterative method, which is efficient and simple to implement, is used. The proposed method has the advantage in that the pixel values in the damaged fingerprint domain can be obtained using the image information from the outside of the damaged fingerprint region. Fingerprint restoration based on adjacent pixel information can ensure the accuracy of the fingerprint information with a low computational cost. Computational experiments demonstrated the superior performance of the proposed fingerprint restoration algorithm.

Finite element approximation of steady flows of colloidal solutions

We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions–Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method.

New Modified Newton Type Iterative Methods

In this work, we present two Newton type iterative methods for finding the solution of nonlinear equations of single variable. One is obtained as variant of McDougall and Wotherspoon method, and another is obtained by amalgamation of Potra and Pta’k method and our newly introduced method. The order of convergence of these methods are 1 + √2 and 3.5615. Some numerical examples are given to compare the performance of these methods with some similar existing methods.

Schwarz domain decomposition methods for the fluid-fluid system with friction-type interface conditions

In this paper we consider a fluid-fluid system coupled with a friction-type interface condition which is described by a jump of the velocities along the tangential interface. Based on domain decomposition, we propose a Schwarz type iterative method to decouple the different physical process in each subdomain occupied by a single fluid, which allows solving in each subdomain only the physical process described by a Stokes problem. Using energy estimate, we prove that the algorithm converges for any traction coefficient. A more detailed analysis based on Fourier analysis shows that the convergence rate depends on both the fluids' properties, the traction coefficient and the time stepsize, but is independent of the spatial mesh. We finally use numerical examples to illustrate the theoretical results.

Modified Krasnoselski\u2013Mann iterative method for hierarchical fixed point problem and split mixed equilibrium problem

In this paper, we introduce a modified Krasnoselski–Mann type iterative method for capturing a common solution of a split mixed equilibrium problem and a hierarchical fixed point problem of a finite collection of k-strictly pseudocontractive nonself-mappings. Many of the algorithms for solving the split mixed equilibrium problem involve a step size which depends on the norm of a bounded linear operator. Since the computation of the operator norm is very difficult, we formulate our iterative algorithm in such a way that the implementation of the proposed algorithm does not require any prior knowledge of operator norm. Weak convergence results are established under mild conditions. We also establish strong convergence results for a certain class of hierarchical fixed point and split equilibrium problem. Our results generalize some important results in the recent literature.

A projective averaged Kaczmarz iteration for nonlinear ill-posed problems

The averaged Kaczmarz iteration is a hybrid of the Landweber method and Kaczmarz method with easy implementation and increased stability for solving problems with multi nonlinear equations. In this paper, we propose an accelerated averaged Kaczmarz type iterative method by introducing the search direction of homotopy perturbation Kaczmarz and a projective strategy. The new iterate is updated by using an average over the intermediate variables. These variables are obtained by the metric projection of previous iterates onto the stripes which are related to the property of forward operator and noise level. We present the convergence analysis of the proposed method under the similar assumptions of Landweber Kaczmarz method. The numerical experiments on parameter identification problem validate that the proposed method has evident acceleration effect and reconstruction stability.

Unification of some iterative and proximal like methods for asymptotically nonexpansive and quasi-nonexpansive mappings

In this paper, we introduce the concept of a strongly asymptotically quasi-nonexpansive sequence of mappings in the context of a Hilbert space. Firstly, we prove the weak convergence of a Picard type iterative method to a common fixed point for the sequence as well as the strong convergence when a Halpern type regularization scheme is considered. Among other features, our results are applied to get convergence to a fixed point of iterative procedure of Ishikawa and Halpern–Ishikawa for a lipschitzian asymptotically quasi-nonexpansive mappings (resp. as the Picard type iteration and the Halpern type iteration for the sequence of strongly asymptotically quasi-nonexpansive mappings) and, the convergence of proximal like methods for asymptotically nonexpansive mappings. Finally, we show an example of an asymptotically nonexpansive mappings and compute some of the methods studied in the paper.