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Paper's Title:
A New Relaxed Complex-valued b-metric Type and Fixed Point Results
Author(s):
P. Singh, V. Singh and T. C. M. Jele
Department of Mathematics, University of
KwaZulu-Natal,
Private Bag X54001, Durban,
South Africa.
E-mail: singhp@ukzn.ac.za
singhv@ukzn.ac.za
thokozani.jele@nwu.ac.za
Abstract:
In this paper, we study the existence and uniqueness of fixed point in complex valued b-metric spaces and introduce a new relaxed α, β Complex-valued b-metric type by relaxing the triangle inequality and determine whether the fixed point theorems are applicable in these spaces.
Paper's Title:
A Generalization of a Partial b-metric and Fixed Point Theorems
Author(s):
Pravin Singh and Virath Singh
Department of Mathematics, University of
KwaZulu-Natal,
Private Bag X54001, Durban,
South Africa.
E-mail: singhp@ukzn.ac.za
singhv@ukzn.ac.za
Abstract:
The purpose of this paper is to introduce the concept of a Partial α, β b-metric as a generalization of a partial b-metric and prove theorems for some contractive type mapping.
Paper's Title:
Oblique Projectors from the Simpson Discrete Fourier Transformation Matrix
Author(s):
P. Singh and V. Singh
School of Mathematics, Computer Science
and Statistics,
University of Kwazulu-Natal,
Private Bag X54001, Durban 4001,
South Africa.
E-mail: singhp@ukzn.ac.za,
singhv@ukzn.ac.za
Abstract:
In this paper we examine the projectors of the Simpson Discrete Fourier Transform matrix of dimension two modulus four and show how they decompose the complex vector space into a direct sum of oblique eigenspaces. These projection operators are used to define a Simpson Discrete Fractional Fourier Transform (SDFRFT).
Paper's Title:
A New Relaxed b-metric Type and Fixed Point Results
Author(s):
P. Singh, V. Singh and Thokozani Cyprian Martin Jele
Department of Mathematics,
University of KwaZulu-Natal,
Private Bag X54001, Durban,
South Africa.
E-mail: singhp@ukzn.ac.za,
singhv@ukzn.ac.za,
thokozani.jele@nwu.ac.za
Abstract:
The purpose of this paper is to introduce a new relaxed α, β b-metric type by relaxing the triangle inequality. We investigate the effect that this generalization has on fixed point theorems.
Paper's Title:
Orthogonal Collocation on Finite Elements Using Quintic Hermite Basis
Author(s):
P. Singh, N. Parumasur and C. Bansilal
University of KwaZulu-Natal,
School of Mathematics Statistics and Computer Sciences,
Private Bag X54001,
Durban, 4000,
South Africa.
E-mail: singhprook@gmail.com
parumasurn1@ukzn.ac.za
christelle18@gmail.com
Abstract:
In this paper we consider the orthogonal collocation on finite elements (OCFE) method using quintic Hermite (second degree smooth) basis functions and use it to solve partial differential equations (PDEs). The method is particularly tailored to solve third order BVPS and PDEs and to handle their special solutions such as travelling waves and solitons, which typically is the case in the KdV equation. The use of quintic polynomials and collocation using Gauss points yields a stable high order superconvergent method. OCFE using quintic Hermite basis is optimal since it is computationally more efficient than collocation methods using (first degree smooth) piecewise-polynomials and more accurate than the (third degree smooth) B-splines basis. Various computational simulations are presented to demonstrate the computational efficiency and versatility of the OCFE method.
Paper's Title:
Introducing the Picard-S3 Iteration for Fixed Points
Author(s):
Pravin Singh, Virath Singh and Shivani Singh
University of KwaZulu-Natal,
School of Mathematics Statistics and Computer Sciences,
Private Bag X54001,
Durban, 4000
South Africa.
Unisa,
Department of Decision Sciences,
PO Box 392,
Pretoria, 0003
South Africa.
E-mail: singhprook@gmail.com
singhv@ukzn.ac.za
shivstarsingh@gmail.com
Abstract:
In this paper we introduce a three step iteration method and show that it can be used to approximate the fixed point of a weak contraction mapping. Furthermore, we prove that this scheme is equivalent to the Mann iterative scheme. A comparison is made with other three step iterative methods by examining the speed of convergence. Results are presented in tables to support our conclusion.
Paper's Title:
Bounds for the Extremal Eigenvalues of Positive Definite Matrices
Author(s):
Shivani Singh and Pravin Singh
Unisa, Department of Decision Sciences,
PO Box 392,
Pretoria,
0003,
South Africa.
E-mail: singhs2@unisa.ac.za
University of KwaZulu-Natal,
School of Mathematics Statistics and Computer Sciences
Private Bag X54001,
Durban,
4000,
South Africa.
E-mail: singhprook@gmail.com
Abstract:
We use a projection to achieve bounds for a vector function of the eigenvalues of a positive definite matrix. For various choices of the monotonic function we are able to obtain bounds for the extremal eigenvalues in terms of the traces of the matrix and its powers. These bounds are relatively simple to compute.
Paper's Title:
High Order Collocation Method for the Generalized Kuramoto-Sivashinsky Equation
Author(s):
Zanele Mkhize, Nabendra Parumasur and Pravin Singh
School of Mathematics, Statistics and
Computer Sciences,
University of KwaZulu-Natal,
Private Bag X 54001,
Durban 4000.
E-mail: mkhizez2@ukzn.ac.za
parumasurn1@ukzn.ac.za
singhp@ukzn.ac.za
URL: https://www.ukzn.ac.za
Abstract:
In this paper, we derive the heptic Hermite basis functions and use them as basis functions in the orthogonal collocation on finite elements (OCFE) method. We apply the method to solve the generalized Kuramoto-Sivashinsky equation. Various numerical simulations are presented to justify the computational efficiency of the proposed method.
Paper's Title:
Results on Bounds of the Spectrum of Positive Definite Matrices by Projections
Author(s):
P. Singh, S. Singh, V. Singh
University of KwaZulu-Natal,
Private Bag X54001,
Durban, 4001,
South Africa.
E-mail: singhp@ukzn.ac.za
University of South Africa, Department of
Decision Sciences,
PO Box 392, Pretoria, 0003,
South Africa.
E-mail: singhs2@unisa.ac.za
University of KwaZulu-Natal,
Private Bag X54001,
Durban, 4001,
South Africa.
E-mail: singhv@ukzn.ac.za
Abstract:
In this paper, we develop further the theory of trace bounds and show that in some sense that the earlier bounds obtained by various authors on the spectrum of symmetric positive definite matrices are optimal. Our approach is by considering projection operators, from which several mathematical relationships may be derived. Also criteria for positive lower bounds are derived.
Paper's Title:
Eigenvalue Bounds based on Projections
Author(s):
Pravin Singh, Shivani Singh, Virath Singh
University of KwaZulu-Natal,
Private Bag X54001,
Durban, 4001,
South Africa.
University of South Africa,
Department of Decision Sciences, PO Box 392,
Pretoria,0003,
South Africa.
University of KwaZulu-Natal,
Private Bag X54001,
Durban, 4001,
South Africa.
E-mail:
singhp@ukzn.ac.za,
singhs2@unisa.ac.za,
singhv@ukzn.ac.za
Abstract:
In this paper, we derive expressions for the bounds of the extremal eigenvalues of positive definite matrices. Our approach is to use a symmetric projection operator onto an n-2 dimensional subspace of the real space of n tuples. These bounds are based on traces of the matrix and its powers. They are relatively easy and inexpensive to compute.
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