Browsing by Author "Singh, Pravin."

Now showing 1 - 9 of 9

A comparative study of collocation methods for the numerical solution of differential equations.
(2008) Kajotoni, Margaret Modupe.; Parumasur, Nabendra.; Singh, Pravin.
The collocation method for solving ordinary differential equations is examined. A detailed comparison with other weighted residual methods is made. The orthogonal collocation method is compared to the collocation method and the advantage of the former is illustrated. The sensitivity of the orthogonal collocation method to different parameters is studied. Orthogonal collocation on finite elements is used to solve an ordinary differential equation and its superiority over the orthogonal collocation method is shown. The orthogonal collocation on finite elements is also used to solve a partial differential equation from chemical kinetics. The results agree remarkably with those from the literature.
Application of the wavelet transform for sparse matrix systems and PDEs.
(2009) Karambal, Issa.; Paramasur, Nabendra.; Singh, Pravin.
We consider the application of the wavelet transform for solving sparse matrix systems and partial differential equations. The first part is devoted to the theory and algorithms of wavelets. The second part is concerned with the sparse representation of matrices and well-known operators. The third part is directed to the application of wavelets to partial differential equations, and to sparse linear systems resulting from differential equations. We present several numerical examples and simulations for the above cases.
Bounds on the extremal eigenvalues of positive definite matrices.
(2018) Jele, Thokozani Cyprian Martin.; Singh, Virath Sewnath.; Singh, Pravin.
The minimum and maximum eigenvalues of a positive de nite matrix are crucial to determining the condition number of linear systems. These can be bounded below and above respectively using the Gershgorin circle theorem. Here we seek upper bounds for the minimum eigenvalue and lower bounds for the maximum eigenvalue. Intervals containing the extremal eigenvalues are obtained for the special case of Toeplitz matrices. The theory of quadratic forms is discussed in detail as it is fundamental in obtaining these bounds.
A discrete Fourier transform based on Simpson's rule.
(2013) Sibandze, Dan Behlule.; Singh, Pravin.; Singh, Virath Sewnath.
Fourier transforms are mathematical operations which play a vital role in the analysis of mathematical models for problems originating from a broad spectrum of elds. In this thesis, we formulate a discrete transform based on Simpson's quadrature for 4m + 2 quadrature nodes and analyse its various properties and provide detailed proofs thereof. In addition, we make applications to encryption and watermarking in the frequency domain.
A discrete Hartley transform based on Simpson's rule.
(2015) Ramsunder, Ashai.; Singh, Pravin.; Singh, Virath Sewnath.
The Discrete Hartley Transform and Discrete Fourier Transform are classical transfor- mations designed for e cient computations in the frequency domain. We introduce a relatively new transformation based on the existing Discrete Hartley Transform by applying Simpson's quadrature for N = 4m + 2 quadrature nodes. The majority of our investigation involves exploring the mathematical properties satis ed by our newly derived transformation. We formulate the convolution and cross correlation properties both in the real and frequency domain. An intensive spectral analysis is performed to ascertain the multiplicities of the eigenvalues corresponding to the transformation matrix.
Eigenvalue bounds for matrices.
(2018) Khambule, Pretty Nombuyiselo.; Singh, Pravin.; Singh, Virath Sewnath.
Eigenvalues are characteristic of linear operators. Once the spectrum of a matrix is known then its Jordan Canonical form can be determined which simplifies the un- derstanding of the matrix. For large matrices and spectral analysis sometimes it is only necessary to know the eigenvalues of smallest and largest absolute values. Hence we consider various strategies of bounding the spectrum in the complex plane. Such bounds may be numerically improved by various algorithms. The minimal and maximal eigenvalues are crucial to determine the condition number of linear systems.
Fixed point theory in various generalized metric-type spaces.
(2022) Jele, Thokozani Cyprian Martin.; Singh, Virath Sewnath.; Singh, Pravin.
In the theory of fixed points, there are numerous articles dealing with generalization of the basic Banach contraction mapping principle. There has been two lines of approach. The first one is concerned with generalizations of the contractive conditions on the mapping space. The other line of investigation deals with various generalizations of the metric spaces and the results that can be obtained in these new frameworks, referred to as metric-type spaces. In this thesis, we elected for the latter approach by providing a more general framework for a b-metric space , G-metric space and S-metric space. In this thesis, we proved that these new metric-type spaces equipped with various contractions type mappings have unique fixed points and provide numerous examples of each metric-type spaced mentioned.
Polynomial approximations to functions of operators.
(1994) Singh, Pravin.; Mika, Janusz R.
To solve the linear equation Ax = f, where f is an element of Hilbert space H and A is a positive definite operator such that the spectrum (T (A) ( [m,M] , we approximate -1 the inverse operator A by an operator V which is a polynomial in A. Using the spectral theory of bounded normal operators the problem is reduced to that of approximating a function of the real variable by polynomials of best uniform approximation. We apply two different techniques of evaluating A-1 so that the operator V is chosen either as a polynomial P (A) when P (A) approximates the n n function 1/A on the interval [m,M] or a polynomial Qn (A) when 1 - A Qn (A) approximates the function zero on [m,M]. The polynomials Pn (A) and Qn (A) satisfy three point recurrence relations, thus the approximate solution vectors P (A)f n and Q (A)f can be evaluated iteratively. We compare the procedures involving n Pn (A)f and Qn (A)f by solving matrix vector systems where A is positive definite. We also show that the technique can be applied to an operator which is not selfadjoint, but close, in the sense of operator norm, to a selfadjoint operator. The iterative techniques we develop are used to solve linear systems arising from the discretization of Freedholm integral equations of the second kind. Both smooth and weakly singular kernels are considered. We show that earlier work done on the approximation of linear functionals < x,g > , where 9 EH, involve a zero order approximation to the inverse operator and are thus special cases of a general result involving an approximation of arbitrary degree to A -1 .
A study of the interaction of strong electromagnetic waves and anisotropic ion beams with a background plasma.
(1989) Singh, Pravin.; Bharuthram, Ramesh.
The interaction of an anisotropic (in velocity space) ion beam with an isotropic background hydrogen plasma is theoretically investigated. The length and time scales are such that both the ions and electrons are magnetized. Using linear theory, the electrostatic dispersion relation is derived, and solved fully, using no approximations. It is shown that the anisotropy can significantly enhance the instability growth rates as compared to the isotropic case. The importance of ion magnetization is illustrated. Comparisons are made with unmagnetized plasma results. The modulational instability of an arbitrarily-large-amplitude electron cyclotron wave along the external magnetic field is investigated, taking into account the relativistic electron quiver velocity and the relativistic ponderomotive force. Three types of plasma slow responses, the forced-Raman, quasistatic and forced-quasistatic, are considered and a parameter study of the instability growth rates is carried out.