Exact and asymptotic solutions of Burgers’ equation on half-line

Summary

1. Introduction

In this paper, we study the large time asymptotic behavior of the solution of the initial boundary value problem (IBVP):

where |$ \tilde{u}_{0} \gt 0 $|⁠, |$ \hat{u}_{0} $| are real numbers, and |$ \tilde{u}_{0} \gt\hat{u}_{0} $|⁠. Further, |$ u_{0}(x) $| is an integrable function on |$ [0,~{}\infty] $|⁠.

This type of IBVP appears in the study of vertical infiltration of water into a non-swelling soil with prescribed concentration conditions at the boundary (see (1)). The nonlinear partial differential equation (1.1) is the well-known Burgers’ equation. Burgers’ equation was first derived by Bateman (2) and then by Burgers (3). One may refer to (4) and (5) for a detailed discussion on the Burgers’ equation.

It is well-known that Cole–Hopf transformation (see (6) and (7)) transforms Burgers’ equation (1.1) to heat equation. Therefore, one can obtain the exact solution of IBVP (1.1)–(1.3) by solving the relevant IBVP for the heat equation. Our study here is based on the analysis of the exact solution of (1.1) obtained in different regions of xt-plane for |$ x~{}\geqslant~{}0,\, t~{}\geqslant~{}0. $| One may note that the construction of solutions for reaction–diffusion equations using the method of matched asymptotic expansions on the quarter plane takes the following steps:

• (i)

  Construction of the asymptotic solutions for |$ t=o(1). $| Here, for |$ x\rightarrow 0 $|⁠, |$ x=O(1) $| and |$ x\rightarrow\infty $|⁠, the small time asymptotic solutions are obtained.

• (ii)

  Construction of the asymptotic solution for |$ t=O(1). $| Here, for |$ x\rightarrow\infty $|⁠, the asymptotic solutions are obtained. This step makes use of the asymptotic behavior of the initial condition as |$ x\to\infty $|⁠.

• (iii)

  Construction of the asymptotic representations as |$ t\rightarrow\infty. $| In this step, one discusses the asymptotics for |$ x\rightarrow\infty, $|  |$ x=O(1), $| and |$ x\rightarrow 0. $|

See (8) for a good discussion of the method of matched asymptotic expansions to construct the large time asymptotic solutions of reaction–diffusion equations. Inspired by this approach, we obtained the asymptotics to the solution of the IBVP (1.1)–(1.3) in the regions specified in steps (i)–(iii) in the quarter plane |$ x\,\gt\,0,\, t\,\gt\,0. $| Our analysis here takes the following steps:

• (a)

  The construction of the exact solution of the IBVP (1.1)–(1.3) using Cole–Hopf transformation. The solution of (1.1)–(1.3) involves undetermined integrals.

• (b)

  Description of asymptotic profiles of the exact solution of (1.1)–(1.3) in different regions of the quarter plane |$ x \gt 0, t \gt 0. $|

• (c)

  We identify the large time asymptotic behavior of the solution of (1.1)–(1.3) by identifying the most dominant terms as |$ t\rightarrow\infty. $| Depending on the dominance of different terms of the solution of (1.1)–(1.3) in the limit |$ t\rightarrow\infty $|⁠, we arrive at the traveling wave solution, stationary solution or slow traveling wave solution as large time asymptotics.

• (d)

  Keeping track of the most dominant terms in the left out terms of the solution give us the large time asymptotic behavior of the error committed.

We also tabulate the results for the convenience of the researchers who want to use them without worrying about the analysis or calculations. These tables give the most dominant terms as |$ t\rightarrow\infty $|⁠, the large time asymptotic behavior of the error and the large time asymptotic behavior of the solution of the IBVP (1.1)–(1.3).

We are of the view that a good understanding of the structure of the solution of the Burgers’ equation may help the construction of the large time asymptotics for generalized Burgers’ equations.

Hanaç (9) studied the IBVP (1.1)–(1.3) with |$ u_{0}(x)=0 $| using the method of matched asymptotic expansions when |$ \tilde{u}_{0}+\hat{u}_{0}\neq 0 $|⁠.

Joseph and Sachdev (10) studied the following IBVP for Burgers’ equation:

Here, |$ \nu \gt 0. $| Using Cole–Hopf transformation, they obtained the solution |$ u^{\nu} $| of the IBVP (1.4)–(1.6):

when |$ \hat{u}_{0}+\tilde{u}_{0}=0 $| and

when |$ \hat{u}_{0}+\tilde{u}_{0}\neq 0 $|⁠. Here |$ A_{-}^{\nu},A_{+}^{\nu}, $| and |$ B^{\nu} $| are given by

They obtained the following asymptotic behaviors of |$ u^{\nu}(x, t) $| as |$ t\rightarrow\infty $|⁠.

• (i)
  If |$ \hat{u}_{0} \lt 0,\, \tilde{u}_{0} \lt 0 $| and |$ \hat{u}_{0} \gt\tilde{u}_{0}, $| then

  $$\begin{align*}\lim_{t\rightarrow\infty}u(x, t)=-\hat{u}_{0}\ \coth\left(\frac{\hat{u}_{0}x}{\nu}+c\right),\quad c=\frac{1}{2}\ln\left(\frac{\tilde{u}_{0}-\hat{u}_{0}}{\tilde {u}_{0}+\hat{u}_{0}}\right).\end{align*}$$
• (ii)
  If |$ \hat{u}_{0} \lt 0,\, \hat{u}_{0} \lt\tilde{u}_{0} $| and |$ \hat{u}_{0}+\tilde{u}_{0} \lt 0, $| then

  $$\begin{align*}\lim_{t\rightarrow\infty}u(x, t)=-\hat{u}_{0}\ \tanh\left(\frac{\hat{u}_{0}x}{\nu}+c\right),\quad c=\frac{1}{2}\ln\left(\frac{\hat{u}_{0}-\tilde{u}_{0}}{\tilde {u}_{0}+\hat{u}_{0}}\right).\end{align*}$$
• (iii)
  If |$ \hat{u}_{0}=0 $| and |$ \tilde{u}_{0} \lt 0, $| then

  $$\begin{align*}\lim_{t\rightarrow\infty}u(x, t)=\frac{\tilde{u}_{0}}{1-\tilde{u}_{0}\frac{x}{\nu}}.\end{align*}$$
• (iv)
  For all other cases,

  $$\begin{align*}\lim_{t\rightarrow\infty}u(x, t)=\tilde{u}_{0}.\end{align*}$$

Liu and Nishihara (11) studied the generalized Burgers’ equation:

on the domain |$ -\infty \lt x\,\lt\,0 $| and |$ t\,\gt\,0 $| with the boundary condition |$ u(0, t)=\tilde{u}_{0} $| and initial function

They discussed the asymptotic stability of the traveling wave with negative wave speed on |$ (-\infty, 0) $| using the weighted energy method.

We present briefly the results of (12) relevant to the IBVP (1.1)–(1.3). They discussed five cases: (i) |$ \tilde{u}_{0} \lt\hat{u}_{0} \lt 0 $|⁠; (ii) |$ \tilde{u}_{0} \lt\hat{u}_{0}=0 $|⁠; (iii) |$ \tilde{u}_{0} \lt 0 \lt\hat{u}_{0} $|⁠; (iv) |$ 0=\tilde{u}_{0} \lt\hat{u}_{0} $|⁠; and (v) |$ 0 \lt\tilde{u}_{0} \lt\hat{u}_{0} $|⁠. For cases (i)–(ii), they showed that the large time asymptotic behavior of the solution of the IBVP (1.1)–(1.3) is given by the stationary solution. For cases (iv) and (v), the large time behavior of solutions of the IBVP (1.1)–(1.3) was shown to be given by the rarefaction wave solution. For case (iii), the large time behavior of the solutions of the IBVP was shown to be described by the combination of the stationary solution and the rarefaction wave solution of the IBVP (1.1)–(1.3). It may be noted that their work dealt with more general partial differential equation.

Shagi-Di (13) discussed a slow moving viscous shock wave for the IBVP (1.4)–(1.6) with |$ \nu\,=\,2\epsilon $|⁠, where |$ \epsilon $| is small.

One may refer to Liu and Yu (14) and Joseph and Sachdev (15) for the study of Burgers’ equation on semi-infinite line.

A result of Nishihara (16) is that if |$ v(x, t) $| is the solution of Burgers’ equation (1.4) with |$ \nu\,=\,2\epsilon $| on the whole real line with initial condition:

for some positive constants |$ K_{\pm} $|⁠, then

where

Here |$ x_{0} $| is chosen in such a way that |$ \int_{-\infty}^{\infty}(v_{0}(x)-V(x))dx\,=\,0 $| and |$ v_{-} \gt v_{+}. $|

Hattori and Nishihara (17) discussed the stability of the rarefaction wave solution of Burgers’ equation (1.4) on the whole real line.

A discussion on the asymptotic stability of traveling wave solutions of (1.7) with convex f and non-convex f can be seen in (18) and (19), respectively.

The scheme of the paper is as follows. In section 2, we derive relevant special solutions of Burgers’ equation (1.1). In section 3, we obtain the exact solution of the IBVP (1.1)–(1.3) using Cole–Hopf transformation. In section 4, we discuss the asymptotic representations of the solution u of the IBVP (1.1)–(1.3) for |$ u_{0}(x)=u_{\infty}\mathrm{e}^{-\sigma x} $| in different regions of xt-plane. Section 5 puts forward the conclusions.

2. Some special solutions of Burgers’ equation (1.1)

In this section, we derive some relevant special solutions of Burgers’ equation (1.1). All these solutions of the Burgers’ equation are well-known. However, we derive these solutions here to make the paper self-contained.

2.1 A traveling wave solution

In this subsection, we derive a traveling wave solution of the Burgers’ equation (1.1). The traveling wave solution |$ u(x, t)=U_{{\rm tw}}(z),z\,=\,x-c^{*}t-c_{0} $| of the Burgers’ equation (1.1) satisfying the equations

is given by

where |$ k^{*}=(\tilde{u}_{0}-\hat{u}_{0})/2 $| and |$ A $| is an arbitrary constant. Note that the wave position of |$ u_{\rm tw}(x, t) $| is |$ P(t)=c^{*}t\,+\,c_{0} $|⁠. We look for the convergence rate of the solution of the IBVP (1.1)–(1.3) to the traveling wave |$ U_{{\rm tw}} $| presented in (2.2).

2.2 A slow traveling wave solution

Assume that |$ \hat{u}_{0}=-\tilde{u}_{0} $| and |$ u(x, t)=U_{{\rm stw}}(z) $|⁠, where |$ z\,=\,x-\Omega(x, t) $|⁠, |$ \Omega\to\infty $|⁠, |$ \Omega_{t}\to 0 $|⁠, |$ \Omega_{x}\to 0 $| and |$ \Omega_{xx}\to 0 $| as |$ t\to\infty $|⁠. Now (1.1) becomes

Keeping the leading order terms in (2.3) for large t, we arrive at

The relevant initial and boundary conditions are

The exact solution of the boundary value problem (2.4)–(2.6) is given by

With |$ A\,=\,1 $|⁠, we get an exact solution of (2.4)–(2.6):

It may be observed that |$ \Omega(x, t) $| turns out to be

as |$ t\to\infty $| and satisfies all the conditions stated above. Note that, in this case, wave position is given by |$ P(t)=\ln(\tilde{u}_{0}^{2}t)/\tilde{u}_{0} $| and the wave (2.7) is moving slowly compared to the traveling wave (2.2) moving with speed |$ c^{*} $| and hence we are referring to this traveling wave solution as a slow traveling wave solution. The slowly moving wave is helpful to construct the solution of Burgers’ equation (1.1) when |$ \tilde{u}_{0}+\hat{u}_{0}=0 $| which is unsolved in (9).

2.3 Stationary solution

Assume that the solution u is independent of t, that is, |$ u(x, t)=U_{{\rm stn}}(x) $|⁠. Then the IBVP (1.1)–(1.3) becomes

An integration of (2.8) and the use of the boundary condition (2.9) lead to

Again, integrating (2.11) and the using the condition (2.10), we arrive at

when |$ \tilde{u}_{0} \lt\hat{u}_{0} \lt 0 $| and

when |$ \hat{u}_{0} \lt 0 \lt\tilde{u}_{0} $| and |$ \hat{u}_{0}+\tilde{u}_{0} \lt 0 $|⁠. One may note that the stationary solution (2.12) satisfies (2.9) only when |$ \hat{u}_{0} \lt 0. $| For |$ \hat{u}_{0}+\tilde{u}_{0} \lt 0 $|⁠, monotonically decreasing stationary solution plays an important role while constructing the large time asymptotic behavior of the solution of Burgers’ equation in the positive quarter plane.

3. Exact solution of the IBVP (1.1)–(1.3) via Cole–Hopf transformation

In this section, we obtain the exact solution of the IBVP (1.1)–(1.3) using the Cole–Hopf transformation

and then obtain its large time asymptotic behavior by analyzing the integrals involved in different regions of the |$ xt- $| plane. The transformation (3.1) transforms (1.1) and (1.3) to

respectively. Integrating both sides of |$ u=-2V_{x}/V $| with respect to x from |$ 0 $| to x and simplifying, we get

In view of (1.2) and (3.4), we arrive at

where |$ U(x)=-\int_{x}^{\infty}u_{0}(s)ds $| and |$ \bar{B}=V(0,0)\exp(U(0)/2) $|⁠. The solution of the IBVP (3.2), (3.3) and (3.5) is given by

See (20, p. 265). Now |$ V(x, t) $| in (3.6) can be rewritten as

where |$ d=xt^{-1/2}/2 $|⁠. Differentiating (3.7) with respect to x, we get

Using (3.7) and (3.8) in (3.1), we obtain the exact solution of the IBVP (1.1)–(1.3):

where

The influence of the boundary condition |$ u(0, t)=\tilde{u}_{0} $| can only be seen in |$ \tilde{J} $| and |$ \tilde{K} $|⁠, whereas the influence of the initial condition can be seen in all the terms. It is not easy to evaluate explicitly the integrals involved in the solution (3.9). Hence we use the asymptotic expansions (in different limits) of the integrals to study the asymptotic behavior of u in different regions of xt-plane.

Now we shall analyze the integrals |$ \tilde{\hat{I}},\tilde{\bar{I}},\tilde{\hat{L}} $|⁠, |$ \tilde{\bar{L}},\tilde{J} $| and |$ \tilde{K}. $| To obtain an asymptotic expansion of |$ \tilde{\hat{I}} $|⁠, we expand |$ \tilde{\hat{I}} $| as follows:

This equation can be rewritten as

where

|$ n\,=\,0,~{}1,~{}2,~{}\cdots $|⁠. Using the transformation |$ z\,=\,l+(\hat{u}_{0}/2)t^{1/2} $| in (3.14), we get

where |$ b(x, t)=(\hat{u}_{0}t-x)t^{-1/2}/2 $|⁠. Recall that

In the region |$ x \gt\hat{u}_{0}t $|⁠, |$ U(2z\sqrt{t}+x-\hat{u}_{0}t)\to 0 $| as |$ t\to\infty $|⁠. Consequently, for |$ x \gt\hat{u}_{0}t $| and |$ t\to\infty $|⁠, we have |$ \tilde{I}_{0}(x, t) \gt \gt\tilde{I}_{n}(x, t) $| for |$ n\in\mathbb{N} $| and in turn |$ \tilde{\hat{I}}(x, t)\sim\tilde{I}_{0}(x, t) $|⁠. Using the asymptotic expansion of the |$ \mathrm{erfc} $| function, we have

Observe that for |$ x \lt\hat{u}_{0}t $| and |$ t\to\infty $|⁠, we have |$ b(x, t)\to\infty $|⁠. Using the transformation |$ z\,=\,l+(\hat{u}_{0}/2)t^{1/2} $| in (3.10), we get

Since |$ U(x) $| is bounded on |$ [0,~{}\infty) $|⁠, we have |$ \exp\left(-U(x)/2\right)~{}\leqslant~{}\tilde{B} $| for |$ 0~{}\leqslant~{}x \lt\infty $| and

as |$ (\hat{u}_{0}t-x)t^{-1/2}\to\infty $|⁠. Using the transformation |$ z\,=\,l+(\hat{u}_{0}/2)t^{1/2} $| in (3.11), we get

Since |$ u_{0}(x) $| is integrable function on |$ [0,~{}\infty] $|⁠, |$ u_{0}(x)\to 0 $| as |$ x\to\infty $|⁠. Consequently, from (3.17) and (3.19), we have |$ \tilde{\hat{I}}(x, t) \gt \gt\tilde{\bar{I}}(x, t) $| in the region where |$ x \gt\hat{u}_{0}t $| and |$ t\to\infty $|⁠. Following closely the calculations presented in (3.17) and (3.18), one may obtain

where |$ u_{0}(x)\exp\left(-U(x)/2\right)~{}\leqslant~{}\bar{\tilde{D}} $| for |$ 0~{}\leqslant~{}x \lt\infty $|⁠. We express |$ \tilde{\hat{L}} $| as

where

|$ f(x, t)=(\hat{u}_{0}t\,+\,x)t^{-1/2}/2 $|⁠. It is easy to see that

and |$ \tilde{\hat{L}}(x, t) \gt \gt\tilde{\bar{L}}(x, t) $| in the region |$ 0~{}\leqslant~{}x \lt -\hat{u}_{0}t $| and |$ t\to\infty $|⁠. Further, |$ \tilde{\bar{L}} $| satisfies the inequality (3.20) in the region |$ x \gt -\hat{u}_{0}t $| and |$ t\to\infty $|⁠.

The function |$ \tilde{K}(x, t) $| can be simplified as

It can be verified that |$ \tilde{K}(x, t)=4\tilde{\hat{L}}(x, t) $| using the transformation |$ z=(x\,+\,s+\hat{u}_{0}t)t^{-1/2}/2 $| in (3.23).

We expand |$ \tilde{J}(x, t) $| as

where

The transformation |$ z=(x\,+\,s+\eta-\tilde{u}_{0}t)t^{-1/2}/2 $| transforms |$ \tilde{J}_{n} $| to

Using the asymptotic expansion of the |$ \mathrm{erfc} $| function for |$ x \lt\tilde{u}_{0}t $| and |$ t\to\infty $| in (3.26), we get

Observe that the integration involved in |$ \tilde{J}_{0} $| is finite when |$ \tilde{u}_{0}+\hat{u}_{0} \gt 0 $|⁠. This suggests us to discuss three different cases (i) |$ \tilde{u}_{0}+\hat{u}_{0} \gt 0 $|⁠; (ii) |$ \tilde{u}_{0}+\hat{u}_{0}=0 $|⁠; and (iii) |$ \tilde{u}_{0}+\hat{u}_{0} \lt 0 $|⁠.

Note that |$ \tilde{\hat{I}} \gt \gt\tilde{\bar{I}} $| and |$ \tilde{\hat{L}} \gt \gt\tilde{\bar{L}} $| as |$ t\to\infty $|⁠. Therefore, we need to compare the terms |$ \tilde{J},~{}\tilde{\hat{I}} $| and |$ \tilde{\hat{L}} $|⁠. Since |$ \tilde{u}_{0}+\hat{u}_{0} \gt 0 $|⁠, |$ \tilde{\hat{I}} \gt \gt\tilde{J} \gt \gt\tilde{\hat{L}}=O(\tilde{K}) $| in the region |$ (\tilde{u}_{0}+\hat{u}_{0})t/2\,\lt\,x \lt\tilde{u}_{0}t $| and |$ t\to\infty $| (see (3.28), (3.16), (3.21)). Similarly, for |$ x \lt (\tilde{u}_{0}+\hat{u}_{0})t/2 $|⁠, |$ \tilde{J} \gt \gt\tilde{\hat{I}} \gt \gt\tilde{\hat{L}}=O(\tilde{K}) $| as |$ t\to\infty $| and |$ \tilde{J}=O(\tilde{\hat{I}}) \gt \gt\tilde{\hat{L}}=O(\tilde{K}) $| when |$ x=(\tilde{u}_{0}+\hat{u}_{0})t/2 $| and |$ t\to\infty $|⁠. Therefore, |$ \tilde{J} $| and |$ \tilde{\hat{I}} $| are the dominant terms. Keeping the most dominant terms and ignoring all other terms of (3.9) and using the asymptotic expansions of |$ \tilde{J} $| and |$ \tilde{\hat{I}} $| presented in (3.28) and (3.16), respectively, we get

when |$ x \lt\tilde{u}_{0}t $| and |$ t\to\infty $|⁠. Here |$ k^{*}=(\tilde{u}_{0}-\hat{u}_{0})/2 $| and |$ c^{*}=(\tilde{u}_{0}+\hat{u}_{0})/2 $|⁠. Therefore, the traveling wave solution |$ U_{\rm{tw}} $| given by (2.2) of Burgers’ equation describes the large time asymptotic behaviour of the solution of the IBVP (1.1)–(1.3) for |$ \tilde{u}_{0}+\hat{u}_{0} \gt 0 $|⁠.

In this section, we have presented the asymptotic expansions of the solution to the IBVP (1.1)–(1.3) in specific regions. Our next objective is to derive the error terms that describe the rate of convergence of the solution to its large time asymptotic forms. This requires revisiting the previously neglected terms involving undetermined integrals. By selecting a specific function, |$ u_{0}(x)=u_{\infty}\mathrm{e}^{-\sigma x} $|⁠, we can further refine the asymptotic expansions of the functions |$ \tilde{\hat{I}},\tilde{\bar{I}},\tilde{\hat{L}} $|⁠, |$ \tilde{\bar{L}},\tilde{J} $|⁠, |$ \tilde{K} $|⁠, which are essential for accurately determining the error terms.

4. Asymptotic solutions of the IBVP (1.1)–(1.3) with |$ u_{0}(x)=u_{\infty}\mathrm{e}^{-\sigma x} $|

In this section, we study the large time asymptotic behavior of the solution of the IBVP:

where |$ \tilde{u}_{0} $| and |$ \sigma $| are positive real numbers and |$ \hat{u}_{0}\in\mathbb{R} $|⁠. Further |$ \tilde{u}_{0} \gt\hat{u}_{0} $|⁠. In view of (3.9), the solution of (4.1)–(4.3) is given by

where

Here |$ d\,=\,x/(2\sqrt{t}) $|⁠. Now we shall analyze the integrals |$ J, K, \hat{I},\bar{I},\hat{L}, $| and |$ \bar{L}. $|

In view of (3.12)–(3.14), we have

where

|$ n\,=\,0,~{}1,~{}2,~{}\cdots $|⁠. Using the transformation |$ z\,=\,l+(\hat{u}_{0}/2+n\sigma)t^{1/2} $| in (4.5), we arrive at

Similarly,

where

The transformation |$ z=(x\,+\,s+\eta-\tilde{u}_{0}t)t^{-1/2}/2 $| transforms |$ J $| to

To obtain the asymptotic expansion of |$ J $| given in (4.10), we expand |$ J $| in the form:

where

Using integration by parts, we have

when |$ (\hat{u}_{0}+\tilde{u}_{0}+2n\sigma)\neq 0 $| for all integers |$ n~{}\geqslant~{}0. $| Now

In view of (4.12)-(4.14), we have

when |$ (\hat{u}_{0}+\tilde{u}_{0}+2n\sigma)\neq 0. $|

Thus, in view of (4.9), (4.11) and (4.15), we rewrite |$ J $| in the form

where

and

Now |$ K(x, t) $| can be written as

(4.18) can be rewritten as

where

In view of (4.9), |$ K_{n}=2L_{n} $|⁠, |$ n\,=\,0,~{}1,~{}2,~{}\cdots $|⁠, and thus |$ K\,=\,4\hat{L} $|⁠. Therefore, in view of (4.4), the solution of the IBVP (4.1)-(4.3) is

where

for |$ (\hat{u}_{0}+\tilde{u}_{0}+2n\sigma)\neq 0 $| and |$ n~{}\geqslant~{}0 $| is any integer.

Suppose that there exists an integer |$ N~{}\geqslant~{}0 $| such that |$ \hat{u}_{0}+\tilde{u}_{0}+2N\sigma\,=\,0 $|⁠. Then |$ J_{N}(x, t) $| is given by

Using |$ z=(x\,+\,s-\tilde{u}_{0}t)t^{-1/2}/2 $|⁠, |$ J_{N} $| takes the form

In view of (3.31), we have

Therefore

where

and

Thus, for |$ N~{}\geqslant~{}1 $|⁠, the solution of the IBVP (4.1)–(4.3) becomes

where

Note that |$ L_{N}=\hat{J}_{0} $| as |$ \hat{u}_{0}+\tilde{u}_{0}+2N\sigma\,=\,0 $|⁠. One may obtain the solution for |$ \tilde{u}_{0}+\hat{u}_{0}=0 $| using (4.24). In this case |$ N\,=\,0 $|⁠. Using |$ N\,=\,0 $| and |$ L_{0}=\hat{J}_{0} $| in (4.24), we have

where

and

One may note that |$ \bar{L}_{n},\bar{I}_{n} $| and |$ \bar{A} $| are obtained from |$ L_{n},I_{n} $| and |$ \hat{B}, $| respectively, using |$ \hat{u}_{0}=-\tilde{u}_{0} $|⁠. Further, observe that |$ \bar{L}_{0}=\hat{J}_{0}. $|

In view of asymptotic expansions of |$ \mathrm{erfc} $| function, we have following asymptotic expansions for |$ \hat{J}_{0},~{}I_{n},~{}L_{n} $|⁠, which will be useful to obtain the asymptotic expansions of |$ u: $|

4.1 Asymptotic expansions of the solution u of the IBVP (4.1)–(4.3) in different regions of xt-plane

In this subsection, we derive the asymptotic expansions of u in different regions of the quarter plane |$ x~{}\geqslant~{}0, t~{}\geqslant~{}0. $|

Let us consider the region |$ R_{1} $|⁠: |$ x\,=\,O(1),t\rightarrow 0 $|⁠. We rewrite (4.19) in the form

where

Observe that |$ I_{n} $|’s are dominant when compared to |$ L_{n} $|’s and |$ \hat{J}_{0} $| in the region: |$ x\,=\,O(1) $|⁠, |$ t\to 0 $| (see (4.28), (4.31) and (4.32)). Retaining the most dominant terms in the solution u given in (4.34), we arrive at

as |$ t\to 0 $|⁠.

Now we consider the region |$ R_{2} $|⁠, where |$ t\to 0 $| and |$ x \gt \gt 1 $|⁠. Following closely, the calculations presented for the region |$ R_{1} $|⁠, we arrive at the form of u presented in (4.35). Simplifying (4.35), we have

as |$ x\to\infty $|⁠. The form of u in (4.36) may be written in the form:

as |$ x\to\infty $| and |$ t\,=\,o(1) $|⁠. One may note that the asymptotic representation (4.37) is valid for the region |$ R_{3}: $|  |$ t\,=\,O(1),x \gt \gt 1 $| also.

The discussion of the asymptotic behavior of u requires the study of u in the limits |$ x\rightarrow\infty $| and |$ t\rightarrow\infty $| also. We introduce the new coordinate |$ y\,=\,x/t. $| The asymptotic behaviors of the functions |$ \hat{J}_{0},I_{n} $| and |$ L_{n} $| are presented below.

In view of (4.32) and (4.33), the function |$ \hat{J}_{0} $| has the following asymptotic representations:

as |$ t\to\infty $|⁠. In terms of the variables y and t, the asymptotic representations presented in (4.28)–(4.31) become:

as |$ t\to\infty $|⁠.

The asymptotic behaviors of the functions |$ \hat{J}_{0},I_{n} $| and |$ L_{n} $| suggest us to consider the cases: (i) |$ \tilde{u}_{0}+\hat{u}_{0} \gt 0 $|⁠, (ii) |$ \tilde{u}_{0}+\hat{u}_{0} \lt 0 $| and (iii) |$ \tilde{u}_{0}+\hat{u}_{0}=0 $|⁠. Here, we shall discuss the case (i) in detail. Other cases can be dealt with similarly.

4.2 Large time asymptotic behavior of the solutions of the IBVP (4.1)–(4.3) when |$ \tilde{u}_{0}+\hat{u}_{0} \gt 0 $|

In this section, it will be shown that the traveling wave solution |$ U_{\rm tw} $| of (4.1) describes the large time asymptotic behavior of the solution of (4.1)–(4.3) as |$ I_{0} $| and |$ \hat{J}_{0} $| are the most dominant terms in the solution (4.19) when |$ \hat{u}_{0} \lt y \lt\tilde{u}_{0} $|⁠. To obtain the rate of convergence to the traveling wave solution, we need to look at the asymptotic behaviors of the left out terms of the solution (4.19). We discuss the asymptotic behaviors of the solution u in detail when |$ \hat{u}_{0} \gt 0 $| and the asymptotic behaviors of u for |$ \hat{u}_{0} \lt 0 $| are presented in Table 2.

It is clear that if |$ \tilde{u}_{0} \lt\hat{u}_{0}+2\sigma $|⁠, then |$ I_{1} $| is the dominant term in the region |$ 2\sigma+\hat{u}_{0} \lt y \lt\infty $|⁠, |$ I_{0},~{}\hat{J}_{0} $| are the dominant terms in the region |$ \hat{u}_{0} \lt y \lt\tilde{u}_{0} $|⁠, and |$ \hat{J}_{0} $| is the only dominant term in the region |$ 0\,\lt\,y \lt\hat{u}_{0} $|⁠. On the other hand, both |$ I_{1} $| and |$ \hat{J}_{0} $| are significant in the region |$ 2\sigma+\hat{u}_{0} \lt y \lt\tilde{u}_{0} $| when |$ 2\sigma+\hat{u}_{0} \lt\tilde{u}_{0} $|⁠. Therefore, we expect different correction terms for the cases (A) |$ \tilde{u}_{0} \lt\hat{u}_{0}+2\sigma $| (B) |$ 2\sigma+\hat{u}_{0} \lt\tilde{u}_{0} $|⁠.

Region |$ y\,\gt\,2\sigma+\hat{u}_{0} $| and |$ t\to\infty $|⁠:

In this region, the functions |$ \hat{J}_{0} $| and |$ L_{n} $| have the asymptotic behaviors given by (4.38) and (4.40), respectively. The functions |$ I_{n} $| have either (4.42) or (4.43) depending on the value of y. Clearly, |$ I_{0} $| has the asymptotic behavior given by (4.43) with |$ n\,=\,0 $|⁠. Further, |$ I_{0} $| is the most dominating one in (4.34) and hence |$ u(y, t)\sim\hat{u}_{0} $| as |$ t\to\infty $|⁠. Therefore, the form of u in (4.34) will be useful. Using the definitions of |$ I_{n},L_{n}, $| and |$ \hat{J}_{0} $| in (4.34) and simplifying, we arrive at the following equation:

where

For a given |$ y\neq\hat{u}_{0}+2N\sigma $| for any natural number |$ N $| in this region, there exists a natural number |$ N_{1} $| such that |$ 2N_{1}\sigma+\hat{u}_{0} \lt y\,\lt\,2(N_{1}+1)\sigma+\hat{u}_{0} $|⁠. Retaining only the dominant terms and using their asymptotic behaviors as |$ t\rightarrow\infty $|⁠, (4.44) gives us

as |$ t\to\infty $|⁠.

In particular, in the region |$ (2\sigma+\hat{u}_{0}) \lt y \lt (4\sigma+\hat{u}_{0}) $|⁠, one may observe that |$ I_{1} $| (see (4.43) with |$ n\,=\,1 $|⁠) dominates all other terms |$ I_{n},~{}n\,=\,2,~{}3,~{}\cdots $| (see (4.42)) and |$ L_{n} $| (see (4.40)). Therefore, we obtain the following asymptotic expansion of u from (4.44):

as |$ t\to\infty $|⁠.

For |$ y=\hat{u}_{0}+2(N_{1}+1)\sigma $|⁠, |$ N_{1}=1,2, \cdots, $| the asymptotic behavior of the solution u is obtained from (4.45) by two modifications: (i) run the summations up to |$ (N_{1}+1) $| in both numerator and denominator and (ii) the coefficients of the last terms corresponding to |$ n=(N_{1}+1) $| in both the summations should be multiplied with (1/2).

One may note that the terms of order |$ O\left(t^{-1/2}{\rm e}^{-t(y-\hat{u}_{0})^{2}/4}\right) $| are contributing to the error term. Further, we used only the most dominant term in the asymptotic expansion of the complementary error function.

Note 1:

We prove now that a solution of the linear partial differential equation

which is obtained from the equation

using the transformation |$ y\,=\,x/t $|⁠, describes the asymptotic behavior of the solution u of Burgers’ equation in the region |$ y\,\gt\,2\sigma+\hat{u}_{0} $| as |$ t\to\infty $|⁠. Retaining the most dominant terms |$ I_{0} $|⁠, |$ I_{1} $| and using the fact that |$ \mathrm{erfc}\{\sqrt{t}(\hat{u}_{0}-y)/2\}\rightarrow 2 $| as |$ t\rightarrow\infty $| in (4.44), we arrive at the asymptotic behavior of u in terms of a complementary error function:

as |$ t\to\infty $|⁠. Note that the first two terms of u (see (4.48)) constitute an exact solution of the linear partial differential equation (4.46).

Region |$ \tilde{u}_{0} \lt y\,\lt\,2\sigma+\hat{u}_{0} $| and |$ t\to\infty $|⁠:

In this region |$ \hat{J}_{0} $|⁠, |$ I_{n} $|⁠, |$ n~{}\geqslant~{}1 $| and |$ L_{n} $| satisfy (4.38), (4.42) and (4.40), respectively. Thus, (4.44) becomes

as |$ t\to\infty $|

Note 2:

In the interval |$ (\hat{u}_{0},\tilde{u}_{0}) $|⁠, the most dominant terms are |$ I_{0} $| and |$ \hat{J}_{0} $|⁠. Observe that |$ I_{0}/\hat{J}_{0}\sim{\rm e}^{k^{*}t(y-c^{*})} $| as |$ t\rightarrow\infty $| in this interval. Here |$ k^{*}=\frac{\tilde{u}_{0}-\hat{u}_{0}}{2} $|⁠. This implies that |$ I_{0} \gt \gt\hat{J}_{0} $| for |$ y\,\gt\,c^{*} $| and |$ I_{0} \lt \lt\hat{J}_{0} $| for |$ y\,\lt\,c^{*}. $| In view of these observations, we divide the interval |$ (\hat{u}_{0},\tilde{u}_{0}) $| into two subintervals |$ (\hat{u}_{0},c^{*}),(c^{*},\tilde{u}_{0}) $| and study the asymptotic behaviors of u as |$ t\rightarrow\infty $| in these two subintervals separately.

Region |$ c^{*} \lt y \lt\tilde{u}_{0} $| and |$ t\to\infty $|⁠:

In this region, we show that the large time asymptotic behavior of the solution u is described by the traveling wave solution |$ U_{\rm{tw}} $| given by (2.2) of Burgers’ equation (4.1). Further, we also obtain the rate of convergence of the solution u to the traveling wave solution |$ U_{{\rm tw}} $| as |$ t\rightarrow\infty. $|

Rewriting u given in (4.19), we have

In this region, the functions |$ I_{0} $|⁠, |$ \hat{J}_{0} $|⁠, |$ I_{n} $|⁠, |$ n~{}\geqslant~{}1 $| and |$ L_{n} $| have the asymptotic behaviors given by (4.43), (4.39), (4.42) and (4.40), respectively.

Because |$ I_{0} $| is the most dominant term, we divide the numerators and denominators of (4.50) with |$ I_{0} $|⁠, use its asymptotic behavior (4.43) and simplify to get

as |$ t\to\infty $|⁠, where |$ k^{*}=(\tilde{u}_{0}-\hat{u}_{0})/2 $| and

(4.51) can be rewritten in the form

where |$ \tilde{E}_{1}(y, t)=\max[U_{{\rm tw}}(y, t)T(y, t),~{}T_{1}(y, t),~{}T_{2}(y, t)] $|⁠. Using the asymptotic behavior of |$ I_{n} $|⁠, |$ n~{}\geqslant~{}1 $| given in (4.42) and |$ L_{n} $| given in (4.40), we can easily observe that |$ T,~{}T_{1}, $| and |$ T_{2} $| are of order |$ O\left(t^{-1/2}\exp(-(y-\hat{u}_{0})^{2}t/4)\right) $|⁠. Since |$ \hat{u}_{0}~{}\leqslant~{}U_{{\rm tw}}~{}\leqslant~{}\tilde{u}_{0} $|⁠, the function |$ U_{{\rm tw}}T $| is also of order |$ O\left(t^{-1/2}\exp(-(y-\hat{u}_{0})^{2}t/4)\right) $|⁠. Clearly |$ \tilde{E}_{1}(y, t)=O\left(t^{-1/2}\exp(-(y-\hat{u}_{0})^{2}t/4)\right) $|⁠. Therefore, we can write (4.51) in the form

as |$ t\to\infty $|⁠.

Region |$ \hat{u}_{0} \lt y\,\lt\,c^{*} $| and |$ t\to\infty $|⁠:

In this region, the functions |$ I_{0} $|⁠, |$ \hat{J}_{0} $|⁠, |$ I_{n} $|⁠, |$ n~{}\geqslant~{}1 $| and |$ L_{n} $| have the asymptotic behaviors given by (4.43), (4.39), (4.42) and (4.40), respectively. Observe that |$ \hat{J}_{0} $| is the most dominant term. Dividing the numerators and denominators of (4.50) with |$ \tilde{u}_{0}\hat{A}\hat{J}_{0}, $| using the asymptotic behavior of |$ \hat{J}_{0} $|⁠, and simplifying we arrive at

as |$ t\to\infty $|⁠, where

(4.56) can be rewritten in the form

where |$ \tilde{E}_{2}(y, t)=\max[U_{{\rm tw}}(y, t)R(y, t),~{}R_{1}(y, t),~{}R_{2}(y, t)] $|⁠. Using the asymptotic behaviors of |$ I_{n} $|⁠, |$ n~{}\geqslant~{}1 $| given in (4.42) and |$ L_{n} $| given in (4.40), we observe that the functions |$ U_{{\rm tw}}R,~{}R_{1}, $| and |$ R_{2} $| are of order |$ O\left(t^{-1/2}\exp(-(y-\tilde{u}_{0})^{2}t/4)\right) $| as |$ t\rightarrow\infty $|⁠. In view of these observations, (4.60) becomes

as |$ t\to\infty $|⁠. Note that the first term of equation (4.55) takes the constant value |$ (\tilde{u}_{0}^{2}\hat{A}+\hat{u}_{0})/(\tilde{u}\hat{A}+1) $|

when |$ y\,=\,c^{*} $|⁠.

Region |$ 0\,\lt\,y \lt\hat{u}_{0} $| and |$ t\to\infty $|⁠:

In this region, the functions |$ \hat{J}_{0},~{}I_{n} $| and |$ L_{n} $| have the asymptotic expansions (4.39), (4.42) and (4.40), respectively. Clearly, |$ \hat{J}_{0} $| is the most dominant term compared to the functions |$ I_{n} $| and |$ L_{n} $| for all n. This implies that |$ u(y, t)\sim\tilde{u}_{0} $| as |$ t\to\infty $| (see(4.19)). Thus, following (4.44)–(4.49), we get

as |$ t\to\infty $|⁠.

One may follow the calculations presented above for |$ \tilde{u}_{0} \lt y \lt\hat{u}_{0}+2\sigma $| to study the large time asymptotic behaviors of IBVP (4.1)-(4.3) for other parametric regions. We consider two particular parametric regions to show different correction terms due to |$ I_{1} $| and |$ L_{0} $|⁠. In Table 1, we summarize asymptotic expansions of u for |$ c^{*} \gt\hat{u}_{0}+2\sigma $| with |$ \hat{u}_{0} \gt 0 $|⁠. Observe that |$ I_{1} $| contributes significantly to the error term. Table 2 presents the asymptotic behaviors for |$ \tilde{u}_{0} \lt\hat{u}_{0}+2\sigma $| with |$ c^{*} \lt -\hat{u}_{0} $| and |$\hat{u}_0<0 .$| Here |$ L_{0} $| also contributes significantly to the error term. Hanaç (9) studied the IBVP (4.1)–(4.3) using the method of matched asymptotic expansions when |$ u_{\infty}=0 $|⁠. Table 3 presents the comparison of the asymptotic solutions of the Burgers’ equation constructed here and the asymptotic solutions of Hanaç (9).

Large time asymptotic behavior of the solutions of the IBVP (4.1)–(4.3) when |$ \tilde{u}_{0}+\hat{u}_{0} \lt 0 $|⁠:

In this case, the solution of the IBVP (4.1)–(4.3) is given by (4.19) and (4.24) for |$ \tilde{u}_{0}+\hat{u}_{0}+2n\sigma\neq 0 $| for all integers |$ n~{}\geqslant~{}0 $| and |$ \tilde{u}_{0}+\hat{u}_{0}+2n\sigma\,=\,0 $| for some |$ n\in\mathbb{N} $|⁠, respectively. Note that the asymptotic expansion (4.35) describes the asymptotic behavior of u in the region |$ R_{1} $| (⁠|$ x\,=\,O(1) $|⁠, |$ t\to 0 $|⁠). Further, (4.37) describes the asymptotic behavior of u in both the regions |$ R_{2} $| (⁠|$ x\to\infty $|⁠, |$ t\,=\,o(1) $|⁠) and |$ R_{3} $| (⁠|$ x\to\infty $|⁠, |$ t\,=\,O(1) $|⁠). We need to find the dominating terms in the region |$ 0\,\lt\,y\,=\,x/t \lt\infty $| and |$ t\to\infty $| to obtain the large time asymptotic expansions of u. We found that |$ L_{0} $| and |$ I_{0} $| are the most dominant terms in the region |$ y\,=\,o(1) $| and |$ t\to\infty $|⁠. Using the asymptotic expansions of |$ L_{0} $| and |$ I_{0} $| presented in (4.41) and (4.43), respectively, we arrive at

as |$ t\to\infty $|⁠. Therefore, a stationary wave describes the large time asymptotic behavior of the solution of IBVP (4.1)–(4.3) when |$ \hat{u}_{0}+\tilde{u}_{0} \lt 0 $|⁠.

Asymptotic behavior of the solution of the IBVP (4.1)–(4.3) when |$ \tilde{u}_{0}+\hat{u}_{0}=0 $|⁠:

In this case, the solution is given by (4.25). We present below the asymptotic behaviors of |$ \hat{J}_{1} $|⁠, |$ \bar{L}_{n} $| and |$ \bar{I}_{n} $|(see (4.22), (4.26) and (4.27)). These asymptotic expansions are used to obtain the asymptotic expansions of u in different regions of xt-plane.

Large time asymptotic behavior of u:

Let us consider the case |$ \tilde{u}_{0} \gt 2\sigma $| and the region |$ 0\,\lt\,y \lt\sigma $|⁠. Recall that the functions |$ \bar{I}_{0},\hat{J}_{1},\bar{L}_{0},\hat{J}_{0}, $| and |$ \bar{I}_{1} $| are the most dominant functions and satisfy the ordering |$ \bar{I}_{0} \gt \gt\hat{J}_{1} \gt \gt\bar{L}_{0}=\hat{J}_{0} \gt \gt\bar{I}_{1} $| as |$ t\rightarrow\infty. $| We shall show that the large time asymptotic behavior of u is given by the slow moving traveling wave solution (2.7) as |$ t\rightarrow\infty $| and the dominant term in the error is given by |$ \bar{I}_{1}/\bar{I}_{0} $| as |$ t\rightarrow\infty. $|

Using (4.33), (4.63) and (4.67) with |$ n\,=\,0 $|⁠, we get

as |$ t\to\infty $|⁠. The asymptotic expansion (4.68) can be written as |$ u(y, t)\sim U_{\rm stw}(y, t) $| as |$ t\to\infty $|⁠, where

Observe that |$ U_{\rm stw} $| is the slow moving wave of (4.1) with

(see (2.7)). Therefore, a slow moving wave describes the large time asymptotic behavior of the solution of IBVP (4.1)–(4.3) for |$ \hat{u}_{0}+\tilde{u}_{0}=0 $|⁠.

5. Conclusions

In this paper, we studied the asymptotics for the solution of the IBVP (1.1)–(1.3) in the quarter plane. The exact solution was obtained using the Cole–Hopf transformation. Identifying the most dominant terms of the exact solution of the IBVP (1.1)–(1.3), we could get the large time asymptotics of the solution. We also presented the order of the error committed when we dropped the less dominant terms as |$ t\rightarrow\infty $| for |$ u_{0}(x)=u_{\infty}\mathrm{e}^{-\sigma x} $|⁠. Our study shows the evolution of the solution of the IBVP (1.1)–(1.3) to the special solutions—Traveling wave solution, Slow traveling wave solution and Stationary solution as |$ t\rightarrow\infty. $|

Acknowledgements

The comments of the referees are gratefully acknowledged.

References