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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.

Numerical approximation of various ordinary and partial differential equations is of ongoing interest [

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 [

In recent time, WFFs have been popularly used for solving partial differential equations (PDE) [

The author develops a general recipe for higher order BVPs in [

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 [

This paper is organized as follows.

We start by discussing WFFs with some properties, followed by an approximation of a function using WFFs in Section

We discuss representation of various operators using WFFs in Section

In Section

We finish with a conclusion in Section

In this section we review in short frames, windowed Fourier frame functions, and the windowed Fourier frame transformation (WFFT) (to approximate any function

A frame is a family of vectors

The sequence

It is well established that it is possible to reconstruct a signal

Let

We discuss windowed Fourier frame and its transformation next. For

The figure shows window function

Let

Thus construction of a window function is important. So we aim to present some ideas to design window functions. Let us first give a brief explanation of forming a window function. Consider an interval

Property 1

Property 2

Property 3

Property 4

Now we can define

the number of windows

the number of points on each window

the window function

This figure shows frame coefficients

This figure shows the original function

Here we intend to define and compute the solution

multiplication by a function,

differentiation operator (derivative as an operator),

We define

When

When

When

Let us start with computing

Now

Next we consider

We show

There are some cases when the second derivative is replaced by its variational form. Now with appropriate boundary conditions the variational form can be written as

In the previous section we discuss the frame representation of various functions and operators. Here we aim to use the representations to approximate two-point boundary value problems. We consider

Now we aim to display some computational results obtained with the scheme discussed in this paper. Here we solve some one-dimensional BVPs of the form (

Let

Solution of the BVP: using windowed Fourier frames.

Consider the BVP

Solution of the BVP: using windowed Fourier frames.

Consider the BVP

Solution of the BVP: using windowed Fourier frames and using a standard finite difference scheme.

From Figures

In the paper windowed Fourier frames have been used to approximate two-point BVPs. From the approximation of functions we notice that the results agree with the exact solutions. We also note that a small amount of WFF coefficients is needed to reconstruct a function, and huge storage costs can be minimized. The scheme also does not require a lot of knowledge concerning the behavior of the solutions. The illustrative examples have been included to demonstrate the validity and applicability of the technique. These examples also exhibit the efficiency of the present method. There are some drawbacks: design of an efficient window function is very important. For one-dimensional problems the advantage of using the method is not very highly visible compared to the other existing numerical schemes, but we have a conjecture (our computational experiences) that this scheme can be used for higher dimensional problem where storage is a real problem for numerical computations. A multidimensional approximation using WFF would be of interest which is left as an open problem.

In this study we apply WFFs to some second-order linear BVPs. WFFs are not limited to these problems only. There are many linear higher order BVPs and nonlinear BVPs modeling scientific and engineering problems where WFFs can be applied to approximate the solutions which are left as open research problems.

Samir Kumar Bhowmik is on leave from the Department of Mathematics, University of Dhaka, Dhaka, Bangladesh.

The authors declare that there is no conflict of interests regarding the publication of this paper.

Samir Kumar Bhowmik would like to thank Chris C. Stolk of University of Amsterdam for his kind and cordial help.