A Two-Dimensional Chebyshev Wavelet Method for Solving Partial Di erential Equations

In this paper, we introduce a two-dimensional Chebyshev wavelet method (TCWM) for solving partial di erential equations (PDEs) in L2(R) space. In this method, the spatial variables appearing in the PDE each has its own kernel, as well as wavelet coecient for approxi- mating the unknown solution of the equation. The approximated solu- tion of the equation is fast and has higher number of vanishing moments as compared to the Chebyshev wavelet method with only one wavelet coecient for two or more separated kernels for the variables appearing in the PDE

Numerical Solution for Linear State Space Systems using Haar Wavelets Method

في هذا البحث، تم استخدام طريقة الموبجات الشعرية لإيجاد حل تقريبي لأنظمة فضاء الحالة الخطية. وان تقنية الحل هي تحويل أنظمة فضاء الحالة الخطية إلى نظام من المعادلات الخطية للفاصل الزمني من 0 إلى . كما يمكن تعزيز دقة متغيرات الحالة عن طريق زيادة دقة موجات هار ويفلت. تم تطبيق الطريقة المقترحة لأمثلة مختلفة وتم توضيح نتائج المحاكاة بالرسوم البيانية ومقارنتها بالحل الدقيق.In this research, Haar wavelets method has been utilized to approximate a numerical solution for Linear state space systems. The solution technique is used Haar wavelet functions and Haar wavelet operational matrix with the operation to transform the state space system into a system of linear algebraic equations which can be resolved by MATLAB over an interval from 0 to . The exactness of the state variables can be enhanced by increasing the Haar wavelet resolution. The method has been applied for different examples and the simulation results have been illustrated in graphics and compared with the exact solution

Linear reconstructions and the analysis of the stable sampling rate

The theory of sampling and the reconstruction of data has a wide range of applications and a rich collection of techniques. For many methods a core problem is the estimation of the number of samples needed in order to secure a stable and accurate reconstruction. This can often be controlled by the Stable Sampling Rate (SSR). In this paper we discuss the SSR and how it is crucial for two key linear methods in sampling theory: generalized sampling and the recently developed Parametrized Background Data Weak (PBDW) method. Both of these approaches rely on estimates of the SSR in order to be accurate. In many areas of signal and image processing binary samples are crucial and such samples, which can be modelled by Walsh functions, are the core of our analysis. As we show, the SSR is linear when considering binary sampling with Walsh functions and wavelet reconstruction. Moreover, for certain wavelets it is possible to determine the SSR exactly, allowing sharp estimates for the performance of the methods

Wavelets for Differential Equations and Numerical Operator Calculus

Differential equations are commonplace in engineering, and lots of research have been carried out in developing methods, both efficient and precise, for their numerical solution. Nowadays the numerical practitioner can rely on a wide range of tools for solving differential equations: finite difference methods, finite element methods, meshless, and so on. Wavelets, since their appearance in the early 1990s, have attracted attention for their multiresolution nature that allows them to act as a “mathematical zoom,” a characteristic that promises to describe efficiently the functions involved in the differential equation, especially in the presence of singularities. The objective of this chapter is to introduce the main concepts of wavelets and differential equation, allowing the reader to apply wavelets to the solution of differential equations and in numerical operator calculus

Introduction to wavelets in engineering

International audienceThe aim of this paper is to provide an introduction to the subject of wavelet analysis for engineering applications. The paper selects from the recent mathematical literature on wavelets the results necessary to develop wavelet-based numerical algorithms. In particular, we provide extensive details of the derivation of Mallat's transform and Daubechies' wavelet coefficients, since these are fundamental to gaining an insight into the properties of wavelets. The potential benefits of using wavelets are highlighted by presenting results of our research in one-and two-dimensional data analysis and in wavelet solutions of partial differential equations

A numerical solution for nonlinear heat transfer of fin problems using the Haar wavelet quasilinearization method

The aim of this paper is to study the new application of Haar wavelet quasilinearization method (HWQM) to solve one-dimensional nonlinear heat transfer of fin problems. Three different types of nonlinear problems are numerically treated and the HWQM solutions are compared with those of the other method. The effects of temperature distribution of a straight fin with temperature-dependent thermal conductivity in the presence of various parameters related to nonlinear boundary value problems are analyzed and discussed. Numerical results of HWQM gives excellent numerical results in terms of competitiveness and accuracy compared to other numerical methods. This method was proven to be stable, convergent and, easily coded