scholarly journals Problems and solutions by the application of Julia set theory to one-dot and multi-dots numerical methods

2001 ◽  
Vol 28 (9) ◽  
pp. 545-548
Author(s):  
Anna Tomova
Keyword(s):  
Numerical Methods ◽  
Numerical Method ◽  
Nonlinear Equations ◽  
Set Theory ◽  
Julia Set ◽  
Time Algorithm ◽  
Escape Time ◽  
Special Cases ◽  
Problems And Solutions ◽  
The One

In 1977 Hubbard developed the ideas of Cayley (1879) and solved in particular the Newton-Fourier imaginary problem. We solve the Newton-Fourier and the Chebyshev-Fourier imaginary problems completely. It is known that the application of Julia set theory is possible to the one-dot numerical method like the Newton's method for computing solution of the nonlinear equations. The secants method is the two-dots numerical method and the application of Julia set theory to it is not demonstrated. Previously we have defined two one-dot combinations: the Newton's-secants and the Chebyshev's-secants methods and have used the escape time algorithm to analyse the application of Julia set theory to these two combinations in some special cases. We consider and solve the Newton's-secants and Tchebicheff's-secants imaginary problems completely.

scholarly journals Fractals Parrondo’s Paradox in Alternated Superior Complex System

2021 ◽  
Vol 5 (2) ◽  
pp. 39
Author(s):  
Yi Zhang ◽  
Da Wang
Keyword(s):  
Complex System ◽  
Julia Set ◽  
Julia Sets ◽  
Time Algorithm ◽  
Mandelbrot Set ◽  
Escape Time ◽  
Parrondo’S Paradox ◽  
The One ◽  
Totally Disconnected ◽  
Superior Mandelbrot Set

This work focuses on a kind of fractals Parrondo’s paradoxial phenomenon “deiconnected+diconnected=connected” in an alternated superior complex system zn+1=β(zn2+ci)+(1−β)zn,i=1,2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria.

scholarly journals Shamanskii Method for Solving Parameterized Fuzzy Nonlinear Equations

2020 ◽  
Vol 11 (1) ◽  
pp. 24-29
Author(s):  
Sulaiman Mohammed Ibrahim ◽  
Mustafa Mamat ◽  
Puspa Liza Ghazali
Keyword(s):  
Numerical Methods ◽  
Nonlinear Equations ◽  
Set Theory ◽  
Fuzzy Set Theory ◽  
Iteration Process ◽  
Benchmark Problems ◽  
Fuzzy Variables ◽  
Computationally Expensive ◽  
And Storage ◽  
Approximate Jacobian

One of the most significant problems in fuzzy set theory is solving fuzzy nonlinear equations. Numerous researches have been done on numerical methods for solving these problems, but numerical investigation indicates that most of the methods are computationally expensive due to computing and storage of Jacobian or approximate Jacobian at every iteration. This paper presents the Shamanskii algorithm, a variant of Newton method for solving nonlinear equation with fuzzy variables. The algorithm begins with Newton’s step at first iteration, followed by several Chord steps thereby reducing the high cost of Jacobian or approximate Jacobian evaluation during the iteration process. The fuzzy coe?cients of the nonlinear systems are parameterized before applying the proposed algorithm to obtain their solutions. Preliminary results of some benchmark problems and comparisons with existing methods show that the proposed method is promising.

scholarly journals Solving Dual Fuzzy Nonlinear Equations via Shamanskii Method

2018 ◽  
Vol 7 (3.28) ◽  
pp. 89 ◽  
Author(s):  
Ibrahim Mohammed Sulaiman ◽  
Mustafa Mamat ◽  
Nurnadiah Zamri ◽  
Puspa Liza Ghazali
Keyword(s):  
Numerical Methods ◽  
Numerical Method ◽  
Nonlinear Equations ◽  
Computational Cost ◽  
Numerical Results ◽  
Parametric Form ◽  
Solution Point ◽  
New Ideas

New ideas on numerical methods for solving fuzzy nonlinear equations have spread quickly across the globe. However, most of the methods available are based on Newton’s approach whose performance is impaired by either discontinuity or singularity of the Jacobian at the solution point. Also, the study of dual fuzzy nonlinear equations is yet to be explored by many researchers. Thus, in this paper, a numerical method to investigate the solution of dual fuzzy nonlinear equations is proposed. This method reduces the computational cost of Jacobian evaluation at every iteration. The fuzzy coefficients are presented in its parametric form. Numerical results obtained have shown that the proposed method is efficient. 

scholarly journals Numerical Solution of Nonlinear Equations in Maple

2021 ◽  
Vol 8 (4) ◽  
Author(s):  
Qani Yalda
Keyword(s):  
Numerical Methods ◽  
Numerical Solution ◽  
Numerical Method ◽  
Nonlinear Equations ◽  
Secant Method ◽  
Bisection Method ◽  
Real Roots ◽  
Newton Raphson ◽  
Root Analysis ◽  
Raphson Method

The main purpose of this paper is to obtain the real roots of an expression using the Numerical method, bisection method, Newton's method and secant method. Root analysis is calculated using specific, precise starting points and numerical methods and is represented by Maple. In this research, we used Maple software to analyze the roots of nonlinear equations by special methods, and by showing geometric diagrams, we examined the relevant examples. In this process, the Newton-Raphson method, the algorithm for root access, is fully illustrated by Maple. Also, the secant method and the bisection method were demonstrated by Maple by solving examples and drawing graphs related to each method.

scholarly journals On the Connectivity Measurement of the Fractal Julia Sets Generated from Polynomial Maps: A Novel Escape-Time Algorithm

2021 ◽  
Vol 5 (2) ◽  
pp. 55
Author(s):  
Yang Zhao ◽  
Shicun Zhao ◽  
Yi Zhang ◽  
Da Wang
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Initial Point ◽  
Visual Display ◽  
Time Algorithm ◽  
Escape Time ◽  
Polynomial Maps ◽  
Distribution Rule ◽  
Connected Area ◽  
Criterion Method

In this paper, a novel escape-time algorithm is proposed to calculate the connectivity’s degree of Julia sets generated from polynomial maps. The proposed algorithm contains both quantitative analysis and visual display to measure the connectivity of Julia sets. For the quantitative part, a connectivity criterion method is designed by exploring the distribution rule of the connected regions, with an output value Co in the range of [0,1]. The smaller the Co value outputs, the better the connectivity is. For the visual part, we modify the classical escape-time algorithm by highlighting and separating the initial point of each connected area. Finally, the Julia set is drawn into different brightnesses according to different Co values. The darker the color, the better the connectivity of the Julia set. Numerical results are included to assess the efficiency of the algorithm.

Electromagnetic effects in the theory of rods

1985 ◽  
Vol 314 (1530) ◽  
pp. 311-352 ◽  
Keyword(s):  
Nonlinear Equations ◽  
Constitutive Equations ◽  
Free Space ◽  
Thermal Effects ◽  
Direct Approach ◽  
One Dimensional ◽  
Special Cases ◽  
Electromagnetic Effects ◽  
The One

This paper is concerned with the nonlinear and linear thermomechanical theories of deformable rod-like bodies in which account is taken of electromagnetic effects. The development is made by a direct approach with the use of the one-dimensional formulation of a theory of directed media called a Cosserat curve . The first part of the paper deals with the formulation of appropriate nonlinear equations governing the motion of a rod in the presence of electromagnetic and thermal effects. In the second part of the paper, emphasis is placed on the linearized version of the theory, a general discussion of the linear constitutive equations and determination of the constitutive coefficients, along with applications in a number of special cases including a magnetic thermoelastic rod and a non-conducting rod in free space.

Pattern Generation Based on Julia Set and its Application in the Design of Seamless Underwear Products

2011 ◽  
Vol 332-334 ◽  
pp. 702-705
Author(s):  
Li Chen ◽  
Rui Zhang
Keyword(s):  
Julia Set ◽  
Time Algorithm ◽  
Pattern Generation ◽  
Escape Time ◽  
Knitted Fabric ◽  
Image Generation ◽  
Pattern Design ◽  
Design Capacity ◽  
Fractal Image

In order to improve the design capacity of knitted fabric pattern and explore new design thought, this article studies the fractal image generation of Julia set and applies it into the pattern design of seamless underwear products. It adopts VB programming to generate Julia set pattern based on escape time algorithm, and then uses part imaging and amplification or changes the function orders to get a lot of colorful patterns. When these generated patterns are imported under the help of Photon software, color part in the patterns will be replaced with pre-designed fabric patterns and finally comes with patterns identifiable to seamless underwear knitting machine. Such method greatly expands the design thoughts of seamless underwear products.

On soft set-valued maps and integral inclusions

2021 ◽  
pp. 1-15
Author(s):  
Monairah Alansari ◽  
Shehu Shagari Mohammed ◽  
Akbar Azam
Keyword(s):  
Fixed Point ◽  
Set Theory ◽  
Fuzzy Set Theory ◽  
Fixed Point Theorems ◽  
Soft Set ◽  
Mathematical Tool ◽  
Multivalued Maps ◽  
Soft Sets ◽  
Special Cases ◽  
New Development

As an improvement of fuzzy set theory, the notion of soft set was initiated as a general mathematical tool for handling phenomena with nonstatistical uncertainties. Recently, a novel idea of set-valued maps whose range set lies in a family of soft sets was inaugurated as a significant refinement of fuzzy mappings and classical multifunctions as well as their corresponding fixed point theorems. Following this new development, in this paper, the concepts of e-continuity and E-continuity of soft set-valued maps and αe-admissibility for a pair of such maps are introduced. Thereafter, we present some generalized quasi-contractions and prove the existence of e-soft fixed points of a pair of the newly defined non-crisp multivalued maps. The hypotheses and usability of these results are supported by nontrivial examples and applications to a system of integral inclusions. The established concepts herein complement several fixed point theorems in the framework of point-to-set-valued maps in the comparable literature. A few of these special cases of our results are highlighted and discussed.

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