Abstract and Applied Analysis

Volume 2015, Article ID 153010, 7 pages

http://dx.doi.org/10.1155/2015/153010

## Windowed Fourier Frames to Approximate Two-Point Boundary Value Problems

Department of Mathematics, College of Science, Al Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia

Received 8 September 2014; Revised 6 January 2015; Accepted 6 January 2015

Academic Editor: Maria A. Ragusa

Copyright © 2015 Abdullah Aljouiee and Samir Kumar Bhowmik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Boundary value problems arise while modeling various physical and engineering reality. In this communication we investigate windowed Fourier frames focusing two-point BVPs. We approximate BVPs using windowed Fourier frames. We present some numerical results to demonstrate the efficiency of such frame functions to approximate BVPs.

#### 1. Introduction

Numerical approximation of various ordinary and partial differential equations is of ongoing interest [1–5]. There are several popular schemes to approximate such models. Schemes based on special functions are increasingly popular [6].

The windowed Fourier transform (Gabor transform) has been a widely used tool in signal processing. This technique uses a single window function to Fourier-transform a signal locally. This process is repeated while shifting the window through the real line. This single window shifting and modulation mechanism of the Gabor transform produces some undesirable effects [5]. A set of frame functions, the windowed Fourier frames (WFFs), have been used to serve such purpose as well [5].

In recent time, WFFs have been popularly used for solving partial differential equations (PDE) [7]. In [7], the authors consider an elliptic PDE and develop an efficient solver using a combination of the symbol of the operator and WFFs. They discuss window functions, discretisation, and implementations in detail. They also study the efficiency of using such functions.

The author develops a general recipe for higher order BVPs in [2]. He considers Tchebychev polynomials to approximate BVPs by reducing the order. In fact, the higher order BVPs problems have been converted to first-order BVPs to approximate the problem using global polynomials efficiently. The author exhibits some numerical results to demonstrate the efficiency of the proposed scheme.

Here, in this paper, we focus on approximating the solutions of two-point boundary value problems using windowed Fourier frames. We motivate ourselves to develop a scheme based on windowed frame functions to approximate various operators in a spare way for one-dimensional academic problems (with an aim to approximate higher dimensional operators using WFFs in the near future). One needs a single window function to generate a family of windowed Fourier frame functions. Thus presentation of the operator becomes neat and simple. The advantage of using windowed frame functions is that they have a flexibility to use for various purpose; the windowed Fourier transformation operator generates a spare differential operator which is easy to store; as a result computations become simple (compared to the spectral collocation/global polynomial approximations for the differential operator). The superiority of the technique has been well discussed in [5, 8]. In this paper we use tight frames to approximate a function, its derivatives, as well as various inner products. Then we apply the frame representations to approximate the solutions of the BVPs.

This paper is organized as follows.(i)We start by discussing WFFs with some properties, followed by an approximation of a function using WFFs in Section 2.(ii)We discuss representation of various operators using WFFs in Section 3.(iii)In Section 4 we approximate some two-point BVPs using WFFs.(iv)We finish with a conclusion in Section 5.

#### 2. A Short Review of Frames

In this section we review in short frames, windowed Fourier frame functions, and the windowed Fourier frame transformation (WFFT) (to approximate any function ). We start by discussing frames and windowed Fourier frames. Then we discuss construction of an efficient window function briefly and use this function to construct windowed Fourier frames.

A frame is a family of vectors that characterizes any function from its inner product . It is possible to recover a vector in a Hilbert space from its inner products with a family of vectors . The index set might be finite or infinite and one can define a frame operator so that

Theorem 1. *The sequence is a frame of if there exist two constants , such that for any **
If this condition is satisfied then is called a frame operator. When the frame is said to be tight [5, 9].*

It is well established that it is possible to reconstruct a signal from its frame transformation using the concept of pseudo inverse which is a bounded operator expressed with a dual frame [5]. Note that a pseudo inverse is denoted by and satisfies where is the adjoint of . If is a frame operator with frame bounds and then . The pseudo inverse of a frame operator is related to a dual fame family, which is expressed by the following result.

Theorem 2 (see [5]). *Let be a frame with bounds and . The dual frame defined by
**
satisfies
**
Then the frame is tight (i.e., ). Here is adjoint of and is the pseudo inverse of .*

We discuss windowed Fourier frame and its transformation next. For , the translation can be defined by and modulation operator can be defined as . Operators of the form or are called time-frequency shifts. Given a nonzero window function and lattice parameters , the set of time-frequency shifts is called a Gabor system. If is a frame for , it is called Gabor frame, Weyl-Heisenberg frame, or windowed Fourier frame [8]. From now we will be denoting such frames by the windowed Fourier frames. Figure 1 shows a sample window function and its Fourier transform. Gabor frame can be constructed using A detailed construction process of window functions can be found in [7].