## 1. Introduction

In recent years, many kinds of wavelet bases have been utilized to solve functional equations; for example, Shannon wavelets [

1], Daubechies wavelets [

2] and Chebyshev wavelets [

3,

4]. In this paper, we utilize Legendre wavelets. Legendre wavelets are derived from Legendre polynomials [

5]. These wavelets have been used in solving different kinds of functional equations such as integral equations [

6,

7], fractional equations [

8,

9], ordinary differential equations [

5], partial differential equations [

10,

11], etc.

In solving time-dependent problems, Legendre wavelets are often used for spatial discretization. Different techniques are implemented for time discretization. In some articles, Legendre wavelets are also applied for time discretization. Therefore, the collocation points should be defined for both time and spatial variables. Also in this technique, multi-dimensional wavelets should be used to approximate required functions, which deal with large matrices and require large storage space. For example, readers can refer to [

9].

There are many contexts that use collocation methods in solving functional equations. For example, Luo et al. [

12] presented three collocation methods based on a family of barycentric rational interpolation functions for solving a class of nonlinear parabolic partial differential equations. Furthermore, for solving a class of fractional subdiffusion equation, Luo et al. in 2016 [

13] used the quadratic spline collocation method.

Another path for time discretization uses a finite difference method. Islam et al. [

10] used a fully implicit scheme, which is based on the first-order Taylor expansion. Yin et al. [

11] employed the

$\theta $-weighted scheme for nonlinear Klein–Sine–Gordon equations. Stability is the important point in using finite difference methods. Thus, methods that are first-order accurate in time might be inappropriate.

Here, we exploit the three-step finite element method for time discretization [

14,

15,

16]. For the suitable differentiable function

$F(t)$, these three steps are defined as follows:

It can be shown that the above equations are equivalent to the third-order Taylor expansion. Therefore, this method is third-order accurate in

t. The first idea of using these three steps has been demonstrated by Jiang and Kawahara [

14]. Equations (

1)–(

3) are usually accompanied by the Galerkin finite element method, which is known as the three-step Taylor–Galerkin method [

17]. Kumar and Mehra [

2] proposed a three-step wavelet Galerkin method based on the Daubechies wavelets for solving partial differential equations subject to periodic boundary conditions. In this paper, motivated and inspired by the ongoing research, we develop a new effective method, which combines the Legendre wavelets collocation method for spatial discretization and the mentioned three steps for time discretization in the numerical solution of a linear time-dependent partial differential equation subject to the Dirichlet boundary conditions. We call this method the three-step wavelet collocation method. Furthermore, we explain the asymptotic stability of the proposed method.

The organization of this paper is as follows. In

Section 2, fundamental properties of the Legendre wavelets are described. The three-step wavelet collocation method is presented in

Section 3. The analysis of asymptotic stability is performed in

Section 4. Some numerical examples are presented in

Section 5. Finally,

Section 6 provides the conclusions of the study.

## 4. Stability Analysis

For stability analysis, we use the asymptotic (or absolute) stability of a numerical method, which is defined in [

20]. In a numerical scheme, when we fix the final time

$t=n\Delta t$ and let

$n\to \infty $, we want the corresponding numerical solution to remain bounded; a scheme satisfying this property is called stable. Therefore, a stability analysis needs a restriction on the mesh size

$\Delta t$. In practice, we can only choose a finite and proper mesh size. It is then important to study the region of absolute stability in order to to choose the proper mesh size in practical computation.

Let us start from the typical evolution equation:

where the non-linear operator

f contains the spatial part of the partial differential equation. Let us abbreviate

$u({x}_{j},{t}_{n})$ by

${u}_{j}^{n}$. We shall approximate

${u}_{j}^{n}$ by

${U}_{j}^{n}$. Following the general formulation of the proposed method, the semi-discrete version is:

where

${u}_{j}$ is the spectral approximation to

u,

${f}_{j}$ denotes the spectral approximation to the operator

f and

${Q}_{j}$ is the projection operator, which characterizes the scheme. Let us set

$U(t)={Q}_{j}{u}_{j}(t)$. Then, the previous discrete problem can be written in the form:

As is often done, we confine our discussion of time-discretizations to the linearized version of (

28):

where

L is the diagonalizable matrix resulting from the implementation of spectral method on the spatial variable of the partial differential equation.

According to different contexts, the time discretization is said to be stable if

${U}^{n}$, the computed solution at the time

${t}_{n}=n\Delta t$, has been upper bounded, i.e., there exists a constant

M such that:

In many problems, the solution is bounded in some norm for all

$t>0$. In these cases, a method that produces the exponential growth allowed by Estimate (

30) is not practical for long-time integrations. For such problems, the notion of asymptotic (or absolute) stability is useful.

**Definition** **1.** The region of absolute stability of a numerical method is defined for the scalar model problem:to be the set of all $\lambda \Delta t$ such that $\parallel {U}^{n}\parallel $ is bounded as $t\to \infty $ [

20].

Finally, we say that a numerical method is asymptotically stable for a particular problem if, for sufficiently small

$\Delta t$, the product of

$\Delta t$ times every eigenvalue of

L lies within the region of absolute stability. In the following items, we summarize some remarkable characteristics of absolute stability [

21]:

- 1.
An absolutely stable method is one that generates a solution ${u}^{n}$ that tends to zero as ${t}_{n}$ tends to infinity,

- 2.
A method is said to be A-stable, if it is absolutely stable for any possible choice of the time-step, $\Delta t$, otherwise a method is called conditionally stable.

- 3.
Absolutely stable methods keep the perturbation controlled,

- 4.
The analysis of absolute stability for the linear model problem can be exploited to find stability conditions on the time step when considering some nonlinear problems.

Since the three-step Equations (

12)–(

14) are equivalent to the third-order Taylor expansion, to demonstrate the stability region and achieve the stability condition, we use Equation (

10). For simplicity, consider Equation (

6), where

$\mu =0$ and

$f(x,t)=0$. Then, successive differentiations of the obtained equation indicate that:

In Equation (

32), we use Euler’s formula to avoid the third-order space derivatives, as it is used in the finite element context [

22]. By rearranging Equation (

10) and substitution of Equations (

31) and (

32), we have the semi-discrete equation:

After applying the wavelet collection method, Equation (

33) transforms into the following equation:

where:

and

${\left\{{x}_{i}\right\}}_{i=1}^{{2}^{k}(M+1)}$ are the collocation and boundary points. Here, the matrix

L, which is introduced in Equation (

29), is defined as

$L={A}^{-1}B$.

There is a similar process to the one-step method. Lambert provided an explanation for how to draw the stability region. Readers can refer to [

23], Chapter 3. Briefly, we can plot the region of absolute stability,

${R}_{L}$, by the meaning of the first and second characteristic polynomials. If we set,

$\widehat{h}=\lambda \Delta t$, the region of absolute stability is a function of the method and the complex parameter

$\widehat{h}$ only, so that we are able to plot the region

${R}_{L}$ in the complex

$\widehat{h}$-plane.

First of all, we can write Equation (

34) as a usual linear multi-step method given by:

where

k is the number of steps required for the method, and

${\alpha}_{j}$ and

${\beta}_{j}$ are constants subject to the conditions:

According to Equations (

34) and (

35), we have:

Afterward, the first and second characteristic polynomials are defined as follows, respectively:

where

$\xi \in \mathbb{C}$ is a dummy variable. Using the values of

k and

${\{{\alpha}_{j},{\beta}_{j}\}}_{j=0}^{1}$ in (

36), for the proposed method, we have:

Then, we plot the boundary of

${R}_{L}$, which consists of the contour

$\partial {R}_{L}$. The contour

$\partial {R}_{L}$ in the complex

$\widehat{h}$-plane is defined by the requirement that for all

$\widehat{h}\in \partial {R}_{L}$, one of the roots of

$\pi (r,\widehat{h}):=\rho (r)-\widehat{h}\sigma (r)$ has modulus one, that is, it is of the form

$r=exp(i\theta )$. Thus, for all

$\widehat{h}$ in

$\partial {R}_{L}$, the identity:

must hold. This equation is readily solved for

$\widehat{h}$, and we have that the locus of

$\partial {R}_{L}$ is given by:

Finally, we use (

37) to plot

$\widehat{h}(\theta )$ for a range of

$\theta \in [0,2\pi ]$ and link consecutive plotted points by straight lines to get a representation of

$\partial {R}_{L}$.

Therefore, according to Lambert’s book and the above explanations, the stability region of the three-step and one-step wavelet collocation methods is the circle with center $(-1,0)$ and radius one. Therefore, these methods will be stable if the eigenvalues of the corresponding system and $\Delta t$ satisfy $Re({\lambda}_{j}\Delta t)\in [-2,0]$.