Laguerre wavelet solution of Bratu and Duffing equations

The aim of this study is to solve the Bratu and Duffing equations by using the Laguerre wavelet method. The solution of these nonlinear equations is approximated by Laguerre wavelets which are defined by well known Laguerre polynomials. One of the advantages of the proposed method is that it does not require the approximation of the nonlinear term like other numerical methods. The application of the method converts the nonlinear differential equation to a system of algebraic equations. The method is tested on four examples and the solutions are compared with the analytical and other numerical solutions and it is observed that the proposed method has a better accuracy.Publisher's Versio

Nonstandard finite differences for a truncated Bratu–Picard model

In this paper, we consider theoretical and numerical properties of a nonlinear boundary-value problem which is strongly related to the well-known Gelfand–Bratu model with parameter λ. When approximating the nonlinear term in the model via a Taylor expansion, we are able to find new types of solutions and multiplicities, depending on the final index N in the expansion. The number of solutions may vary from 0, 1, 2 to ∞. In the latter case of infinitely many solutions, we find both periodic and semi-periodic solutions. Numerical experiments using a non-standard finite-difference (NSFD) approximation illustrate all these aspects. We also show the difference in accuracy for different denominator functions in NSFD when applied to this model. A full classification is given of all possible cases depending on the parameters N and λ

Nonstandard finite differences for a truncated Bratu–Picard model

In this paper, we consider theoretical and numerical properties of a nonlinear boundary-value problem which is strongly related to the well-known Gelfand–Bratu model with parameter λ. When approximating the nonlinear term in the model via a Taylor expansion, we are able to find new types of solutions and multiplicities, depending on the final index N in the expansion. The number of solutions may vary from 0, 1, 2 to ∞. In the latter case of infinitely many solutions, we find both periodic and semi-periodic solutions. Numerical experiments using a non-standard finite-difference (NSFD) approximation illustrate all these aspects. We also show the difference in accuracy for different denominator functions in NSFD when applied to this model. A full classification is given of all possible cases depending on the parameters N and λ

New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus

This reprint focuses on exploring new developments in both pure and applied mathematics as a result of fractional behaviour. It covers the range of ongoing activities in the context of fractional calculus by offering alternate viewpoints, workable solutions, new derivatives, and methods to solve real-world problems. It is impossible to deny that fractional behaviour exists in nature. Any phenomenon that has a pulse, rhythm, or pattern appears to be a fractal. The 17 papers that were published and are part of this volume provide credence to that claim. A variety of topics illustrate the use of fractional calculus in a range of disciplines and offer sufficient coverage to pique every reader's attention

Factors Influencing Customer Satisfaction towards E-shopping in Malaysia

Online shopping or e-shopping has changed the world of business and quite a few people have decided to work with these features. What their primary concerns precisely and the responses from the globalisation are the competency of incorporation while doing their businesses. E-shopping has also increased substantially in Malaysia in recent years. The rapid increase in the e-commerce industry in Malaysia has created the demand to emphasize on how to increase customer satisfaction while operating in the e-retailing environment. It is very important that customers are satisfied with the website, or else, they would not return. Therefore, a crucial fact to look into is that companies must ensure that their customers are satisfied with their purchases that are really essential from the ecommerce’s point of view. With is in mind, this study aimed at investigating customer satisfaction towards e-shopping in Malaysia. A total of 400 questionnaires were distributed among students randomly selected from various public and private universities located within Klang valley area. Total 369 questionnaires were returned, out of which 341 questionnaires were found usable for further analysis. Finally, SEM was employed to test the hypotheses. This study found that customer satisfaction towards e-shopping in Malaysia is to a great extent influenced by ease of use, trust, design of the website, online security and e-service quality. Finally, recommendations and future study direction is provided. Keywords: E-shopping, Customer satisfaction, Trust, Online security, E-service quality, Malaysia

Numerical Methods for Nonlinear Elliptic Boundary Value Problems with Parameter Dependence

This thesis discusses numerical methods for nonlinear elliptic partial differential equations with parameter dependence such as the Gelfand-Bratu model and singularly perturbed convection-diffusion-reaction equations. In numerical investigations, more accurate and efficient nonstandard finite difference and multigrid methods are adopted to solve parameter dependent elliptic boundary-value problems. Aim and objectives of the research are given in chapter 1. Further, it describes the background of Gelfand-Bratu and singularly perturbed problems. In addition, an overview of nonstandard finite difference and multigrid methods is given followed by an outline of the thesis. In chapter 2, standard and nonstandard finite difference approximations are employed to find the numerical solutions of the one-dimensional truncated Bratu-Picard (BP) model. Numerical results show the existence of infinitely many solutions, which are calculated numerically (for a large, but finite, set of solutions). These new types of solutions are either periodic or semi-periodic. We observe that the nonstandard finite difference schemes provide more accurate and efficient results than standard finite difference schemes. In chapter 3, we propose a higher order non-uniform finite difference grid, to solve singularly perturbed boundary value problems with steep boundary-layers. Theoretical properties concerning the extremum values and the asymptotic value at the right boundary point are presented. Several examples are provided, which demonstrate the effectiveness of the proposed numerical strategy. We establish numerically, not only 4th-order but also a 6th-order of accuracy by considering only three-point central non-uniform finite differences. Numerical results illustrate that to achieve the 6th-order of accuracy, the proposed method needs approximately a factor of 5-10 fewer grid points than the uniform case. In chapter 4, we propose three numerical methods, viz, a finite-difference approximation and two multigrid (MG) approaches: Newton-MG and Full Approximation Storage (FAS). A comparison, in terms of convergence, accuracy and efficiency among the three numerical methods demonstrate improvement for the whole parameter range λ ∈ (0, λc]. Further, we investigate the bifurcation behaviour of solutions and find new multiplicity of solutions in the case of a cubic approximation of the nonlinear exponential term. We demonstrate that the convergence of all solutions namely, unique, lower, upper, periodic and semi-periodic is obtained for small values of the parameter λ. Particularly, FAS-MG is found to be more efficient than the other two methods. In chapter 5, we present a numerical study of the Gelfand-Bratu model for higher dimensions. For three dimensions, we adopt an accurate and efficient nonlinear multigrid approach, namely, FAS-MG extended with a Krylov method as a smoother. New types of solutions are obtained or specific values of the bifurcation parameter. Further, the numerical bifurcation curve of the Gelfand-Bratu problem in three dimensions shows the existence of two new turning points. Numerical results confirm the convergence of all types of solutions and demonstrate the effectiveness of the proposed numerical strategy. For even higher dimensions, numerical experiments show the existence of several types of solutions. Bifurcation curves confirm the theoretical results of the higher-dimensional Gelfand-Bratu problem as presented in the literature

Numerical Methods for Nonlinear Elliptic Boundary Value Problems with Parameter Dependence

This thesis discusses numerical methods for nonlinear elliptic partial differential equations with parameter dependence such as the Gelfand-Bratu model and singularly perturbed convection-diffusion-reaction equations. In numerical investigations, more accurate and efficient nonstandard finite difference and multigrid methods are adopted to solve parameter dependent elliptic boundary-value problems. Aim and objectives of the research are given in chapter 1. Further, it describes the background of Gelfand-Bratu and singularly perturbed problems. In addition, an overview of nonstandard finite difference and multigrid methods is given followed by an outline of the thesis. In chapter 2, standard and nonstandard finite difference approximations are employed to find the numerical solutions of the one-dimensional truncated Bratu-Picard (BP) model. Numerical results show the existence of infinitely many solutions, which are calculated numerically (for a large, but finite, set of solutions). These new types of solutions are either periodic or semi-periodic. We observe that the nonstandard finite difference schemes provide more accurate and efficient results than standard finite difference schemes. In chapter 3, we propose a higher order non-uniform finite difference grid, to solve singularly perturbed boundary value problems with steep boundary-layers. Theoretical properties concerning the extremum values and the asymptotic value at the right boundary point are presented. Several examples are provided, which demonstrate the effectiveness of the proposed numerical strategy. We establish numerically, not only 4th-order but also a 6th-order of accuracy by considering only three-point central non-uniform finite differences. Numerical results illustrate that to achieve the 6th-order of accuracy, the proposed method needs approximately a factor of 5-10 fewer grid points than the uniform case. In chapter 4, we propose three numerical methods, viz, a finite-difference approximation and two multigrid (MG) approaches: Newton-MG and Full Approximation Storage (FAS). A comparison, in terms of convergence, accuracy and efficiency among the three numerical methods demonstrate improvement for the whole parameter range λ ∈ (0, λc]. Further, we investigate the bifurcation behaviour of solutions and find new multiplicity of solutions in the case of a cubic approximation of the nonlinear exponential term. We demonstrate that the convergence of all solutions namely, unique, lower, upper, periodic and semi-periodic is obtained for small values of the parameter λ. Particularly, FAS-MG is found to be more efficient than the other two methods. In chapter 5, we present a numerical study of the Gelfand-Bratu model for higher dimensions. For three dimensions, we adopt an accurate and efficient nonlinear multigrid approach, namely, FAS-MG extended with a Krylov method as a smoother. New types of solutions are obtained or specific values of the bifurcation parameter. Further, the numerical bifurcation curve of the Gelfand-Bratu problem in three dimensions shows the existence of two new turning points. Numerical results confirm the convergence of all types of solutions and demonstrate the effectiveness of the proposed numerical strategy. For even higher dimensions, numerical experiments show the existence of several types of solutions. Bifurcation curves confirm the theoretical results of the higher-dimensional Gelfand-Bratu problem as presented in the literature