Modern Scientific Press

ABSTRACT: The numerical range has been a subject of interest to many researchers and scholars in the recent past. Based on the research outputs, many results have been obtained. Besides, several generalizations of the classical numerical range have also been made. The recent developments have focused on the theory of operators on Hilbert spaces. The determination of the numerical ranges of linear and nonlinear operators have been given in both the Hilbert and Banach spaces. In addition, results of these numerical ranges have been extended to the case of two operators in both spaces. It is important to note that more generalizations have been made in Hilbert spaces as compared to those that have been made in the Banach spaces. The Banach space has two major numerical ranges which are: the spatial and algebraic numerical ranges. This research focuses on determining the numerical range for a finite number of linear operators in the Banach space based on the classical definition. Properties which hold for the classical numerical range have been shown to hold for the Banach space numerical range. The property of convexity has been established using the Toeplitz-Hausdorff theorem under the condition that the Banach space is smooth. Furthermore, the numerical radius and the spectrum of these operators have also been determined.

ABSTRACT: Nonlinear partial differential equations appear in a wide variety of scientific applications such as plasma physics, solid state physics, optical fibers, biology, fluid dynamics and chemical kinetics. In this paper, we proposed Lifting scheme for the numerical solution of Fisher’s equations by different wavelet filter coefficients. The obtained numerical results using this scheme are compared with the exact solution to reveal the accuracy and also speed up convergence in lesser computational time as compared with existing scheme. Some test problems are presented for the applicability and validity of the schemes.

ABSTRACT: In this paper, we consider a new approach to solve type of partial differential equation by using coupled Laplace transformation with decomposition method to find the exact solution for non–linear non–homogenous equation with initial conditions. The reliability for suggested approach illustrated by solving model equations such as second order linear and nonlinear Klein–Gordon equation. The application results show the efficiency and ability for suggested approach.

ABSTRACT: The use of skewed error distributions in volatility modelling has not been much given research attention in Nigeria. This study models the volatility in Nigerian stock market using skewed error distribution with the objective of determining the combination of volatility model and the skewed error distribution that best capture the dynamics in the volatility of Nigerian stock market. Using data on daily All Share Index(ASI) for Nigeria between 02/10/2001 and 29/03/2018, the study estimate the parameters of GARCH(1,1), APARCH(1,1), GJR-GRACH(1,1), IGARCH(1,1) and EGARCH(1,1) using skewed normal, skewed Student-t and skewed generalized error distributions. The parameters of these volatility models at each of the error distribution were estimated using rugarch package in R software. Result reveals that among the competing error distributions, in most of the models, the skewed normal distribution was found to be the best error distribution both in terms of fitness. Result of the 50 days trading out of sample forecast also favours the skewed normal distribution. Based on the least RMSE, the best forecasting model was recommended as APARCH (1,1)-skewed normal distribution. Result shows evidence of volatility clustering and high persistence of volatility indicating high level of risk and uncertainties in the Nigerian stock market. This signals possibilities of excessive gain or losses by investors but nevertheless the use of APARCH (1, 1)-skewed normal distribution in volatility prediction is hoped to help reduce the chances of excessive losses by investors and traders in the stock market. There is a need for investors and traders in Nigerian stock market to exercise caution while trading and the Nigerian Stock Exchange must come out with policies that will make Nigerian stocks less volatile.

ABSTRACT: In this paper, we study semi-analytical methods entitled Homotopy Analysis Method (HAM) to solve fuzzy impulsive fractional differential equations based on the concept of generalized Hukuhara differentiability. At the end first of Homotopy Analysis Method is defined and its properties are considered completely. Then econvergence theorem for the solution are proved and we will show that the approximate solution convergent to the exact solution. Numerical example indicate that this method can be easily applied to many linear and nonlinear problems.

ABSTRACT: This paper presents a modified wavelet multigrid technique for solving elliptic type partial differential equations namely Helmholtz equation. The solution is first obtained on the coarser grid points, and then it is refined by obtaining higher accuracy by increasing the level of resolution. The implementation of the classical numerical methods has been found to involve some difficulty to observe fast convergence in low computational time. To overcome this, we have proposed modified wavelet multigrid method using wavelet intergrid operators similar to classical intergrid operators. Some of the numerical test problems are presented to demonstrate the applicability and attractiveness of the present implementation.

ABSTRACT: The purpose of this study is to explore the feasibility of integrating random ksatisfiability in Hopfield neural network. Random kSatisfiability (RAN-kSAT) logical representation is a decision problem that decides whether there exists a suitable assignment of truth values to a given Boolean variables that makes a given random Boolean formula in Conjunctive Normal Form (CNF) satisfiable. RAN-kSAT problem is formulated and then incorporated into Hopfield neural networks (HNN) to form a single network model. The hybrid computational model RAN-kSAT-HNN has been developed based on the network energy minimization scheme of HNN. The feasibility of the proposed hybrid model was evaluated based on computer simulation developed based on DEV C++. The result generated and plotted in form of graphs reveals the effectiveness of Random kSatisfiability logical (RAN-kSAT) to be represented optimally in the Hopfield neural network (HNN).

ABSTRACT: The wavelet based approximation of differential equations has been attracting the attention of so many researchers since from three decades. In this paper, we have investigated the numerical solution of Harmonic wave equation via modified wavelet-Galerkin. The proposed scheme is rather simple than the existing ones, the supremacy of this technique is illustrated by comparing the numerical results with exact ones. The solutions presented here are comparably good and give higher accuracy than the Haar wavelet collocation method (HWCM) with exact solution by increasing the resolution level.