Search

In this article, families of non-linear subdivision schemes are presented that are based on univariate polynomials up to degree three. Theses families of schemes are constructed by using dynamic iterative re-weighed least squares method. These schemes are suitable for fitting scattered data with noise and outliers. Although these schemes are non-interpolatory, but have the ability to preserve the shape of the initial polygon in case of non-noisy initial data. The numerical examples illustrate that the schemes constructed by non-linear polynomials give better performance than the schemes that are constructed by linear polynomials (Computer-Aided Design, 58, 189-199). Moreover, the numerical examples show that these schemes have the ability to reproduce polynomials and do not cause over and under fitting of the data. Furthermore, families of non-linear bivariate subdivision schemes are also presented that are based on linear and non-linear bivariate polynomials., Comment: There are 98 figures and 41 pages in this paper

Toplitz matrix completion (TMC) is to fill a low-rank Toeplitz matrix from a small subset of its entries. Based on the augmented Lagrange multiplier (ALM) algorithm for matrix completion, in this paper, we propose a new algorithm for the TMC problem using the smoothing technique of the approximation matrices. The completion matrices generated by the new algorithm are of Toeplitz structure throughout iteration, which save computational cost of the singular value decomposition (SVD) and approximate well the solution. Convergence results of the new algorithm are proved. Finally, the numerical experiments show that the augmented Lagrange multiplier algorithm with smoothing is more effective than the original ALM and the accelerated proximal gradient (APG) algorithms.

This paper focuses on the delay-dependent stability problem of time-varying delay systems. A generalized free-matrix-based integral inequality (GFMBII) is presented. This inequality is able to deal with time-varying delay systems without using the reciprocal convexity lemma. It overcomes the drawback that the Bessel–Legendre inequality is inconvenient to cope with a time-varying delay system as the resultant bound contains a reciprocal convexity. Through the use of the derived inequality and by constructing a suitable Lyapunov–Krasovskii function (LKF), improved stability criteria are presented in the form of linear matrix inequalities (LMIs). Two numerical examples are carried out to demonstrate that the results outperform the state of the art in the literature.

Irrational transfer function has been widely used in modelling and identification. But time response analysis of systems with irrational transfer functions is hard to be achieved in comparison with the rational ones. One of the main reasons is that irrational transfer function generally has infinite poles or zero. In this paper, the closed-loop time response of fractional-system with irrational transfer function is analyzed based on numerical inverse Laplace transform. The numerical solutions and stability evaluation of irrational fractional-order systems are presented. Several examples of fractional-order systems with irrational transfer functions are shown to verify the effectiveness of the proposed algorithm in both time and frequency domain analysis. The MATLAB codes developed to solve the fractional differential equations using numerical Laplace transform are also provided. The results of this paper can be used on analysis and design of control system described by irrational fractional-order or integer-order transfer function.

In this paper, we present functional definitions of all types of generalized inverses related to the {1}-inverse, which is a continuation of the work of Campbell and Meyer (2009). According to these functional definitions, we further derive novel representations for all types of generalized inverses related to the {1}-inverse in terms of the bases for R(A*), N(A) and N(A*). Based on these representations, we present the corresponding algorithm for computing various generalized inverses related to the {1}-inverse of a matrix and analyze the computational complexity of our algorithm for a constant matrix. Finally, we implement our algorithm and several known algorithms for symbolic computation of the Moore-–Penrose inverse in the symbolic computational package MATHEMATICA and compare their running times. Numerical experiments show that our algorithm outperforms these known algorithms when applied to compute the Moore–Penrose inverse of one-variable rational matrices, but is not the best choice for two-variable rational matrices in practice.

This paper presents, a novel semi-analytical algorithm, based on the Chebyshev collocation method, for the solution of space-fractional diffusion equations. The original fractional equation is transformed into a system of ordinary differential equations (ODEs) by the Chebyshev collocation method. A new semi-analytical method is then used to approximate the solution of the resulting system. To emphasize the reliability of the new scheme, a convergence analysis is presented. It is shown that highly accurate solutions can be achieved with relatively few approximating terms and absolute errors are rapidly decrease as the number of approximating terms is increased. By presenting some numerical examples, we show that the proposed method is a powerful and reliable algorithm for solving space-fractional diffusion equations and can be extended to solve another space-fractional partial differential equations. Comparison with other methods in the literature, demonstrates that the proposed method is both efficient and accurate.

In this paper, we study the solution of coupled forward-backward stochastic differential equation driven by G-Brownian motion with monotone coefficients. Besides, we prove that the solution is the minimal one.

In this paper, Lie symmetry method is employed to solve the fractional neutral ordinary differential equations. The symmetry analysis was carried out and infinite dimension symmetry algebras are obtained. An application of Lie symmetry techniques is used to obtain the exact solutions.

A highly accurate meshfree approach based on the barycentric Lagrange basis functions is reported for solving the linear and nonlinear multi-dimensional systems of Fredholm integral equations (FIEs) of the second kind. The method is an improved Lagrange interpolation technique which possesses high precision and inexpensive procedure. The systems of FIEs are handled with the barycentric formula and transformed into the corresponding linear and nonlinear systems of algebraic equations. The stability of the solved integral equation systems is guaranteed by the numerical stability of the barycentric Lagrange formula, which is dominated by barycentric weights. The convergence analysis and error estimation are included. Some computational results indicate that the method is simple and efficient.

In this paper, we show that the univariate interpolatory subdivision schemes with the optimal approximation order, which are the D-D interpolatory ones, can be obtained from the cubic B-spline scheme. Such schemes are derived in this paper by suitably using the push-back operation, a connection between the approximating and interpolatory subdivision. Then, the process to obtain the D-D schemes is generalized to the bivariate case and bivariate interpolatory schemes with the optimal approximation order are generated based on the triangular meshes. Compared with the existing bivariate interpolatory schemes with the optimal approximation order, the newly constructed ones own some advantages, such as a better performance in terms of the number of nonzero coefficients in the mask. Several examples are given and explicitly computed.

This paper proposes a quadrature method based on multi-variate Bernstein polynomials. The method is used to solve multidimensional Volterra integral equations with weakly singular kernels. Firstly, we use multi-variate Bernstein polynomials to approximate the unknown function of an equation, then a discrete function equation can be obtained by substituting the approximate solution into the equation. Secondly, the discrete function system is transformed into an algebra equation system by using some discrete points. We can perform the integral operations without discrete kernel function, and the weakly singular integrals can be calculated directly by using quadrature method, so the method is easy to implement. Thirdly, we prove the existence and uniqueness of the solution of the approximate equation, as well as the error analysis of the proposed method. Six numerical examples are given to illustrate the efficiency of this method.

The objective of this study is to contribute a general family for solving Ito-type stochastic ordinary differential equations. The proposed scheme is implicit and comprises a free parameter. Theoretical aspects are provided to show its convergence. The extension of the new approach for the system of stochastic differential equations is also attained.

One important application of the Sherman–Morrison formula is the construction of an efficient method for computing the inverse of an arrowhead matrix. Our intention is to apply the Sherman–Morrison–Woodbury formula in finding new representations for numeric or symbolic computation of the inverse of block arrowhead matrices. In addition, a generalization of the well known notion of the Schur complement is defined and exploited. Finally, we present a new representation for numeric and symbolic computing of the Moore–Penrose inverse of special type of block arrowhead matrices. Estimated computational complexity as well as presented numerical and symbolic results show the effectiveness of the proposed methods.

In this research, we study the numerical solution of the singular Abel's equation of the second kind. Solving this equation is challengeable, because of the nonlinear and singularity. For this purpose, we present an efficient algorithm based on the Galerkin method using biorthogonal Hermite cubic spline multiwavelets (BHCSMWs). Because of the sparse multiscale representations of functions and operators by these wavelets, the CPU time and computer memory are reduced by the proposed algorithm. Also, the convergence analysis of the method is discussed.

In this paper, we consider a semi-explicit form of Volterra integro-differential-algebraic equations (IDAEs) and investigate the existence and uniqueness of solution of these systems by using differentiability index. Numerical method based on discontinuous piecewise polynomial approximation is proposed for the solution of the semi-explicit IDAEs and global convergence results are established. The performance of the numerical scheme is illustrated by means of some test problems.

This paper briefly surveys Markov’s theorem related to zeros of orthogonal polynomials. Monotonicity of zeros of some families of orthogonal polynomials are reviewed in detail.

In this work we derive new alternatives for efficient computation of the matrix exponential which is useful when solving Linear Initial Value Problems, vibratory systems or after semidiscretization of PDEs. We focus especially on the two classes of normal and nonnegative matrices and we present intervals of applications for rational L ∞ approximations of various degrees for these types of matrices in the lines of [7, 8]. Our method relies on Remez algorithm for rational approximation while the innovation here is the choice of the starting set of non-symmetrical Chebyshev points. Only one Remez iteration is then usually enough to quickly approach the actual L ∞ approximant.

Some new Frobenius norm bounds of the unique solution to certain structured Sylvester equation are derived. Based on the derived norm upper bounds, new multiplicative perturbation bounds are provided both for subunitary polar factors and positive semi-definite polar factors. Some previous results are then improved., Comment: 22 pages. Accepted for publication in Applied Mathematics and Computation

In this paper, inspired by the idea of invariant subspace method and combined with elementary integral method, we introduced a novel approach for investigating exact solutions of a time-fractional nonlinear partial differential equation (NPDE). Based on hypothetical structure of solution of separated variable, a time-fractional NPDE defined by time and space variables can be reduced to a nonlinear ordinary differential equation (NODE) or NODEs defined by space variable alone, and then using the elementary integral method to solve the NODE or NODEs, different kinds of exact solutions of a time-fractional NPDE are obtained finally. As examples, the time-fractional Hunter–Saxton equation and time-fractional Li–Olver equation were studied. Different kinds of exact solutions of these equations were obtained and their dynamical properties were illustrated.

In this paper, two iterative algorithms are proposed to solve stochastic algebraic Riccati matrix equations arising in the linear quadratic optimal control problem of linear stochastic systems with state-dependent noise. In the first algorithm, a standard Riccati matrix equation needs to be solved at each iteration step, and in the second algorithm a standard Lyapunov matrix equation needs to be solved at each iteration step. In the proposed algorithms, a weighted average of the estimates in the last and the previous steps is used to update the estimate of the unknown variable at each iteration step. Some properties of the sequences generated by these algorithms under appropriate initial conditions are presented, and the convergence properties of the proposed algorithms are analyzed. Finally, two numerical examples are employed to show the effectiveness of the proposed algorithms.

Variational integrators are particularly suitable for simulation of mechanical systems, where features such as symplecticity and momentum preservation are essential. They also exhibit excellent long-time energy behavior even if external forcing is involved. Motivated by this fact, we present a new approach, that is based on the local path fitting technique, to construct variational integrators for forced mechanical systems. The core technology exploited is to fit the local trajectory as the Lagrange interpolation polynomial by requiring that the forced Euler–Lagrange equations hold at the internal interpolation nodes. This operation also yields the essential terms of the discrete forced Euler–Lagrange equations and consequently formulates the final integrator. This new approach not only avoids numerical quadrature involved in the classical construction, but also significantly improves the precision of the resulting integrator, as illustrated by the given examples.

This paper proposes an algorithm based on the Shultz iterative method and divided differences to find the roots of nonlinear equations. Using the new method, we present a rapid and powerful algorithm to compute an approximate inverse of an invertible tensor. Analysis of the convergence error shows that the convergence order of the method is a linear combination of the Fibonacci sequence and also is rapid and powerful in finding and keeping sparsity of the obtained approximate inverse of the sparse tensors. The algorithm is extended for computing the Moore-Penrose inverse of a tensor. As an application, we use the iterates obtained by the algorithm as a preconditioner for the tensorized Krylov subspace method, e.g., LSQR based tensor form to solve the multilinear system A ★ N X = B . Several examples are also provided to show the efficiency of the proposed method. Finally, some concluding remarks are given.

In this paper, a novel online mode-free integral reinforcement learning algorithm is proposed to solve the multiplayer non-zero sum games. We first collect and learn the subsystems information of states and inputs; then we use the online learning to compute the corresponding N coupled algebraic Riccati equations. The policy iterative algorithm proposed in this paper can solve the coupled algebraic Riccati equations corresponding to the multiplayer non-zero sum games. Finally, the effectiveness and feasibility of the design method of this paper is proved by simulation example with three players.

Yaman, Olha Ivanyshyn/0000-0002-1727-9461, WOS: 000439036700003, We propose two methods based on boundary integral equations for the numerical solution of the planar exterior Robin boundary value problem for the Laplacian in a multiply connected domain. The methods do not require any a-priori information on the logarithmic capacity. Investigating the properties of the integral operators and employing the Riesz theory we prove that the obtained boundary integral equations for both methods are uniquely solvable. The feasibility of the numerical methods is illustrated by examples obtained via solving the integral equations by the Nystrom method based on weighted trigonometric quadratures on an equidistant mesh. (C) 2018 Elsevier Inc. All rights reserved.

We consider the numerical solution of a singular boundary value problem on the half line for a second order nonlinear ordinary differential equation. Due to the fact that the nonlinear differential equation has a singularity at the origin and the boundary value problem is posed on an unbounded domain, the proposed approaches are complex and require a considerable computational effort. In the present paper, we describe an alternative approach, based on the reduction of the original problem to an integro-differential equation. Though the problem is posed on the half-line, we just need to approximate the solution on a finite interval. By analyzing the behavior of the numerical approximation on this interval, we identify the solution that satisfies the prescribed boundary condition. Although the numerical algorithm is much simpler than the ones proposed before, it provides accurate approximations. We illustrate the proposed methods with some numerical examples.

In this paper, numerical techniques are presented for solving initial value problems of nonlinear fractional differential equations. The method is implemented by applying alternative Legendre polynomials. The operational matrix of fractional integration and the product for the alternative Legendre polynomials are derived in order to transform the nonlinear equations into a system of algebraic equations. The study of the error analysis of the obtained method is also considered. Furthermore, numerical examples demonstrate that this method is applicable and accurate.

A united form of the classical Hermite interpolation and shape-preserving interpolation is presented in this paper. The presented interpolation method provides higher order continuous shape-preserving interpolation splines. The given interpolants are explicit piecewise rational expressions without solving a linear or nonlinear system of consistency equations. By setting parameter values, the interpolation curve can be generated by choosing the classical piecewise Hermite interpolation polynomials or the presented piecewise rational expressions. For monotonicity-preserving and convexity-preserving interpolation, the appropriate values of a parameter are given on each subinterval. Numerical examples indicate that the given method produces visually pleasing curves.

In this paper, we consider a new fractional function based on Legendre and Laguerre polynomials for solving a class of linear and nonlinear time-space fractional partial differential equations with variable coefficients. The concept of the fractional derivative is utilized in the Caputo sense. The idea of solving these problems is based on operational and pseudo-operational matrices of integer and fractional order integration with collocation method. We convert the problem to a system of algebraic equations by applying the operational matrices, pseudo-operational matrices and collocation method. Also, we calculate the upper bound for the error of integral operational matrix of the fractional order. We illustrated the efficiency and the applicability of the approach by considering several numerical examples in the format of table and graph. We also describe the physical application of some examples.

We address the problem of the numerical bifurcation analysis of general nonlinear delay equations, including integral and integro-differential equations, for which no software is currently available. Pseudospectral discretization is applied to the abstract reformulation of equations with infinite delay to obtain a finite dimensional system of ordinary differential equations, whose properties can be numerically studied with well-developed software. We explore the applicability of the method on some test problems and provide some numerical evidence of the convergence of the approximations.

The characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. Finding efficient algorithms for computing characteristic polynomial of graphs is an active area of research and for some graph classes, like threshold graphs, there exist very fast algorithms which exploit combinatorial structure of the graphs. In this paper, we put forward some novel ideas based on divisor technique to obtain fast algorithms for computing the characteristic polynomial of threshold and chain graphs.

High index differential algebraic equations (DAEs) are ordinary differential equations (ODEs) with constraints and arise frequently from many mathematical models of physical phenomenons and engineering fields. In this paper, we generalize the idea of differential elimination with Dixon resultant to polynomially nonlinear DAEs. We propose a new algorithm for index reduction of DAEs and establish the notion of differential Dixon resultant, which can provide the differential resultant of the enlarged system of original equations. To make use of structure of DAEs, variable pencil technique is given to determine the termination of differentiation. Moreover, we also provide a heuristic method for removing the extraneous factors from differential resultant. The experimentation shows that the proposed algorithm outperforms existing ones for many examples taken from the literature.

In this paper, the problem of finding numerical solution for a system of Cauchy type singular integral equations of first kind with index zero is considered. The analytic solution of such system is known. But it is of limited use as it is a nontrivial task to use it practically due to the presence of singularity in the known solution itself. Therefore, a residual based Galerkin method is proposed with Legendre polynomials as basis functions to find its numerical solution. The proposed method converts the system of Cauchy type singular integral equations into a system of linear algebraic equations which can be solved easily. Further, Hadamard conditions of well-posedness are established for system of Cauchy singular integral equations as well as for system of linear algebraic equations which is obtained as a result of approximation of system of singular integral equations with Cauchy kernel. The theoretical error bound is derived which can be used to obtain any desired accuracy in the approximate solution of system of Cauchy singular integral equations. The derived theoretical error bound is also validated with the help of numerical examples.

In the present paper, a modification of hat functions (MHFs) has been considered for solving a class of nonlinear fractional integro-differential equations with weakly singular kernels, numerically. The fractional order operational matrix of integration is introduced. We provide an error estimation for the approximation of a function by a series of MHFs. To suggest a numerical method, the main problem is converted to an equivalent Volterra integral equation of the second kind and operational matrices of MHFs are used to reduce the problem to the solution of bivariate polynomial equations. Finally, illustrative examples are provided to confirm the accuracy and validity of the proposed method.

In this work, we will introduce generalized tribonacci and generalized tricobsthal polynomials. We introduce definitions, formulas for both families of polynomials and the Binet formulas, generating functions. We analyze special points for considered polynomials and present some of polynomials pictorially.

In this paper, a new method for solving coupled systems of nonlinear fractional order integro-differential equations is proposed. The idea is to use Bernoulli wavelets and operational matrix. The main purpose of the technique is to transform the studied systems of fractional order integro-differential equations into systems of algebraic equations which can be solved easily. Illustrative examples and comparisons with Haar wavelets and Legendre wavelets are included to reveal the effectiveness of the method and the accuracy of the convergence analysis.

In this paper, we perform a further investigation for the Bernoulli polynomials of the second kind. By making use of the generating function methods and summation transform techniques, we establish a higher-order convolution identity for the Bernoulli polynomials of the second kind. We also present some illustrative special cases as well as immediate consequences of the main result.

Let D be a quasi-residual design with an automorphism group G. In this paper, we classify flag-transitive, block-primitive or point-primitive quasi-residual designs with sporadic socle. Furthermore, we show that if G is flag-transitive block-primitive with sporadic socle, then G is point-primitive.

In this paper, we study a triangular spectral-element method based on a one-to-one mapping between the rectangle and the triangle. We construct a new approximation space where the integral singularity brought by the mapping can be removed in a naive and stable way. We build aquasi-interpolation triangular spectral-element approximation, and analyze its approximation error. Based on this quasi-interpolation spectral-element approximation, we put forward a new triangular spectral-element method for the elliptic problems. We present the approximation scheme, analyze the convergence, and do some experiments to test the effectiveness. At last, we implement this triangular spectral-element method to solve the steady Stokes problem.

A polyhedral graph is a three connected simple planar graph. An automorphism of a graph is a bijection on its vertices which preserves the edge set. In this paper, we compute the automorphism group of cubic polyhedral graphs whose faces are triangles, quadrangles, pentagons and hexagons. In continuing, we classify all cubic polyhedral graphs with Cayley graph structure.

This paper discusses the range of the eigenvalues of a class of matrices. By using the eigenvalues range of a class of matrices, an extension of the inner product type Cauchy–Schwarz inequality is obtained, the convergence proof of the least squares based iterative algorithm for solving the coupled Sylvester matrix equations is given and the best convergence factor is determined. Moreover, by using the eigenvalues range of this class of matrices, an iterative algorithm for solving linear matrix equation is established. Three numerical examples are offered to illustrate the effectiveness of the results suggested in this paper.

For the quadratic Lagrange interpolation function, an algorithm is proposed to provide explicit and verified bound for the interpolation error constant that appears in the interpolation error estimation. The upper bound for the interpolation constant is obtained by solving an eigenvalue problem along with explicit lower bound for its eigenvalues. The lower bound for interpolation constant can be easily obtained by applying the Rayleigh-Ritz method. Numerical computation is performed to demonstrate the sharpness of lower and upper bounds of the interpolation constants over triangles of different shapes. An online computing demo is available at http://www.xfliu.org/onlinelab/., 16 pages, 5 figures, 1 table

Birkhoff (or lacunary) interpolation is an extension of polynomial interpolation that appears when observation gives irregular information about some function and its derivatives. A Birkhoff interpolation problem is not always solvable even in the appropriate polynomial or rational space. In this paper we split up the initial problem in subproblems having a unique polynomial solution and use multinode rational basis functions in order to obtain a global interpolant.

The Weierstrass and Riemann walks are non trivial discrete random processes to model and characterize the underlying “noise” in the dynamics of fluctuations for out of equilibrium systems, and, in more general contexts, to simulate complex dynamics like order-disorder phase transitions and anomalous diffusion properties in physical, biological and financial systems. In this work simple algorithms, implemented in GNU-R, for both Riemann and Weierstrass discrete processes are presented. Explicit formulas for the probability distributions of n steps are obtained. Finally a way to connect both random processes is commented.

Singular perturbation problems have been discussed using collocation technique with quintic Hermite interpolating polynomials as base functions. These polynomials have the property to interpolate the function as well as its tangent at node points. To discretize the problem in temporal direction forward difference operator has been applied. The given technique is a combination of collocation and difference scheme. Parameter uniform convergence has been studied using the method given by Farrell and Hegarty (1991). Rate of convergence of quintic Hermite difference scheme has been found to depend upon node points. Applicability and computational effect of the scheme has been examined through numerical examples. Results have been presented graphically through surface plots as well as in tabular form.

In this paper, a numerical method is implemented to solve the stochastic Ito-Volterra integral equations. In this approach, operational matrices have been applied to reduce the stochastic Ito-Volterra integral equations to linear algebraic equations. Then collocation method is applied to solve the algebraic equations. The error, convergence, and stability analysis of the proposed method are discussed. Also, the steps of the proposed method have been presented in the form of an algorithm. Numerical examples are introduced to confirm the efficiency and reliability of the proposed scheme.

In this work, we present an iterative algorithm to solve a class of generalized coupled Sylvester-conjugate matrix equations over the generalized Hamiltonian matrices. We show that if the equations are consistent, a generalized Hamiltonian solution can be obtained within finite iteration steps in the absence of round-off errors for any initial generalized Hamiltonian matrix by the proposed iterative algorithm. Furthermore, we can obtain the minimum-norm generalized Hamiltonian solution by choosing the special initial matrices. Finally, numerical examples show that the iterative algorithm is effective.

Nonhomogeneous linear systems of second order differential equations with pure delay and multiple delays are considered. Representations of their solutions without a commutativity assumption on the matrix coefficients are obtained by using delayed matrix functions and Laplace transform. Our results improve and extend previous results.

The main intention of the current paper is to describe a scheme for the numerical solution of boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations result from boundary value problems of Laplace’s equations with linear Robin boundary conditions. The method approximates the solution using the radial basis function (RBF) expansion with polynomial precision in the discrete collocation method. The collocation method for solving logarithmic boundary integral equations encounters more difficulties for computing the singular integrals which cannot be approximated by the classical quadrature formulae. To overcome this problem, we utilize the non-uniform composite Gauss–Legendre integration rule and employ it to estimate the singular logarithm integrals appeared in the method. Since the scheme is based on the use of scattered points spread on the analyzed domain and does not need any domain elements, we can call it as the meshless discrete collocation method. The new algorithm is successful and easy to solve various types of boundary integral equations with singular kernels. We also provide the error estimate of the proposed method. The efficiency and accuracy of the new approach are illustrated by some numerical examples.

We extend some Nielson and Marshall type interpolation operators to the case of a triangle with one curved side. We study the interpolation properties of the obtained operators and of their product and Boolean sum operators, the orders of accuracy and the remainders of the corresponding interpolation formulas. Finally, we give some numerical examples.

Under the assumption of known root multiplicity m ∈ N , a triparametric family of two-point optimal quartic-order methods locating multiple zeros are investigated in this paper by introducing a weight function dependent on a function-to-function ratio. Special cases of weight functions with selected free parameters are considered and studied through various test equations and numerical experiments to support the theory developed in this paper. In addition, we explore the relevant dynamics of proposed methods via Mobius conjugacy map when applied to a prototype polynomial ( z − a ) m ( z − b ) m . The results of such dynamics are visually illustrated through a variety of parameter spaces as well as dynamical planes.