Burgers equation with time-dependent coefficients and nonlinear forcing term: Linearization and exact solvability

Commun Nonlinear Sci Numer Simulat 22 (2015) 1068–1083

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Burgers equation with time-dependent coefficients and nonlinear forcing term: Linearization and exact solvability Axel Schulze-Halberg Department of Mathematics and Actuarial Science and Department of Physics, Indiana University Northwest, 3400 Broadway, Gary, IN 46408, USA

a r t i c l e

i n f o

Article history: Received 2 February 2014 Received in revised form 27 June 2014 Accepted 24 August 2014 Available online 30 August 2014 Keywords: Forced Burgers equation Linearization Point transformation Exceptional orthogonal polynomials

a b s t r a c t We construct and discuss a linearization method for solving Burgers equation with timedependent coefficients and a nonlinear forcing term. Our results are shown to contain and generalize recent findings (Miskinis, 2001; Buyukasik and Pashaev, 2013). As applications of our method we solve several initial- and boundary-value problems for Burgers equation with forcing of sinusoidal, polynomial, as well as X 1 -Laguerre exceptional orthogonal polynomial type. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Burgers equation is one of the most fundamental tools for describing nonlinear diffusion and dissipation phenomena. It was derived from the Navier–Stokes equations by dropping the pressure term and was proposed as a model of turbulence in hydrodynamic motion [4]. Since then Burgers equation has been found applicable to a wide variety of physical models, the most important of which include standing waves and resonance in opto-acoustic systems [21], non-steady-state forced vibrations in acoustic resonators [7], nonlinear standing waves in constant-cross-sectioned resonators [2], 1-D nonlinear dynamics of hydrodynamic-type fields [22]. Further applications concern soil–water flow in layered media [3] [12], the formation and propagation of soliton- and shock waves [23], acoustic streaming [10], population dynamics [17], and many more. For a detailed overview of modern applications and related mathematical methods the reader may refer to [11] or [25] and references therein. Due to the importance of Burgers equation, there is a general interest in particular cases, where solutions can be expressed in closed form. Such solutions can be constructed by several methods, for example through Lie symmetries [14], the Hirota method [18], the Backlund transformation [19], among others. In addition, one of the simplest and most popular schemes to solve Burgers equation is to linearize it by means of the Cole–Hopf transformation [6] [13] to an equation of Schrödinger type, including the heat equation as a particular case. This method of linearization has been used extensively in order to generate closed-form solutions of Burgers equation, in particular for cases where an external force field is included. Newer results on such cases include purely time-dependent [16] and linear forcing, see [8] and references therein. Very recently, the latter setting was extended to Burgers equation for time-dependent coefficients [5]. It turned out that linearizability to the heat equation persists if a certain interrelation between the coefficients in the equation holds. Now, it is well-known that Burgers equation for a nonlinear forcing term can only be linearized to a Schrödinger-type equation for a nonzero potential. Note that the expression ‘‘nonlinear’’ refers to the forcing term as being a nonlinear function in the spatial variable. In some cases, the latter equation can be further simplified by transforming it into its stationary counterpart, E-mail addresses: [email protected], [email protected] http://dx.doi.org/10.1016/j.cnsns.2014.08.029 1007-5704/Ó 2014 Elsevier B.V. All rights reserved.

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which is an ordinary differential equation [9]. It is therefore desirable to have a transformation that takes Burgers equation for time-dependent coefficients and nonlinear forcing term into a stationary Schrödinger equation. The construction of such a mapping, the discussion of its properties, and the presentation of applications are precisely the purpose of this note. In Section 2 we will introduce different point transformations for linearizing Burgers equation and relating it to its Schrödinger counterpart. Section 3 is devoted to the construction of the final transformation and the discussion of its properties. Applications of our method that involve different initial-value- and boundary-value problems, are presented in Section 4. One of these applications involves a forcing term expressed through exceptional orthogonal polynomials of X 1 -Laguerre type. 2. Point transformations The method that will be constructed in Section 3 is based on point transformations that convert the time-dependent Burgers equation to Schrödinger form. We particularly focus on the stationary Schrödinger equation, closed-form solutions of which are comparably easy to find. 2.1. Stationary and time-dependent Schrödinger equations We will now briefly review a point transformation that was first introduced in [9]. The purpose of this transformation is to interrelate a class of time-dependent Schrödinger equations to certain stationary counterparts. We start out by considering the time-dependent Schrödinger equation

iUt ðx; tÞ þ

1 Uxx ðx; tÞ  V 1 ðx; tÞUðx; tÞ ¼ 0; 2m

ð1Þ

where the indices stand for partial differentiation and m is a constant. Furthermore, V 1 represents the potential, which we assume to have the following form, introducing arbitrary differentiable functions A; B; C and a constant phase u:



4 m

V 1 ðx; tÞ ¼ exp

Z

t

0

Aðt 0 Þdt þ

     4u 2 2 V 0 ½uðx; tÞ þ A0 ðtÞ  A2 ðtÞ x2 þ B0 ðtÞ  AðtÞBðtÞ x þ CðtÞ: m m m

ð2Þ

The function V 0 that appears in this expression is assumed to be differentiable and have an argument u of the following form

uðx; tÞ ¼ exp



2 m

Z

t

0

Aðt 0 Þdt þ

" Z 0 #  Z 2u 1 t 2 t 2u 00 0 Bðt0 Þdt : xþ exp Aðt 00 Þdt þ m m m m

ð3Þ

Let us now assume that the function W is a solution of the time-dependent Schrödinger equation for the stationary potential V 0 , that is,

iWt ðx; tÞ þ

1 Wxx ðx; tÞ  V 0 ðxÞWðx; tÞ ¼ 0: 2m

ð4Þ

Define further the abbreviation

v ðtÞ ¼

Z

t

" Z exp 4

t0

# Aðt 00 Þ 00 0 dt dt ; m

then the function U, given by

(

# ) Z t" 0 Aðt Þ B2 ðt 0 Þ 0 0 Uðx; tÞ ¼ exp iAðtÞx  iBðtÞx þ i  iCðt Þ dt W½uðx; tÞ; v ðtÞ; m 2m 2

ð5Þ

provides a solution of the initial time-dependent Schrödinger equation (1) for the potential (2). In the particular case that W is a solution to the stationary Schrödinger equation

1 00 W ðxÞ þ ½E  V 0 ðxÞWðxÞ ¼ 0 2m

ð6Þ

for an arbitrary constant E, then

( 2

Uðx; tÞ ¼ exp iAðtÞx  iBðtÞx  iE

Z

t

"

4 exp m

Z

#

t0 00

Aðt Þdt

00

0

dt þ

Z t"

# ) Aðt 0 Þ B2 ðt0 Þ 0 0 i  iCðt Þ dt W½uðx; tÞ; m 2m

ð7Þ

solves our initial time-dependent Schrödinger equation (1) for the potential (2). Hence, if in the latter case a solution to the stationary equation (6) is known, then a corresponding solution for the fully time-dependent system (1) and (2) can be constructed by means of (7).

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2.2. Linearization of Burgers equation It is well-known that the conventional forced Burgers equation can be linearized to give a Schrödinger-type equation, the potential of which is related to the external force in the initial Burgers equation. We will now demonstrate that this is also true if the coefficients and the forcing term of Burgers equation are allowed to depend on both spatial and time variable. To this end, let us consider the following equation

U t ðx; tÞ þ

a0 ðtÞ Uðx; tÞ þ bðtÞU x ðx; tÞUðx; tÞ þ cðtÞU xx ðx; tÞ þ F x ðx; tÞ ¼ 0: aðtÞ

ð8Þ

This is a time-dependent, forced Burgers equation with differentiable coefficient functions a; b; c and an external force F x . Note that the coefficient of U is written as a ratio, because this will simplify subsequent calculations. For the same reason we state our forcedterm as the partial derivative of a function F ¼ Fðx; tÞ. Let us introduce one more assumption on the coefficients in our Burgers equation (8):

cðtÞ ¼ 

kbðtÞ ; 2aðtÞ

ð9Þ

where k is an arbitrary constant. Now, in order to linearize our time-dependent Burgers equation (8), we apply the following transformation of Cole–Hopf type

Uðx; tÞ ¼ 

kvx ðx; tÞ ; aðtÞvðx; tÞ

ð10Þ

introducing an unknown, differentiable function v. We substitute (10) into Burgers equation (8) and obtain a third-order equation for the function v that can be integrated with respect to the variable x. After some elementary manipulations, this integrated equation can be written in the form 2

2iaðtÞ 2ia ðtÞFðx; tÞ v ðx; tÞ  ivxx ðx; tÞ  vðx; tÞ ¼ 0: 2 kbðtÞ t k bðtÞ

ð11Þ

Note that we omit to state details of the calculations leading to the latter equation, as the general steps are well-known and straightforward to perform. It remains to match the linear equation (11) with the time-dependent Schrödinger form (1). To this end, we will apply the following three adjustments to (11). At first we observe that the coefficient of vxx can be interpreted as an imaginary mass function m ¼ i=2. Next, we redefine the coefficient of v as 2

Wðx; tÞ ¼

2ia ðtÞFðx; tÞ 2

k bðtÞ

:

The final adjustment consists in a scaling of the time variable in order to modify the coefficient of coordinate change

sðtÞ ¼

Z

t

kbðt 0 Þ 0 dt ; 2aðt 0 Þ

ð12Þ

vt . We introduce the ð13Þ

where the prime in t0 is part of the variable’s name and not to be confused with a derivative. Observe that s is a real function, because the coefficients a; b and the constant k are required to be real-valued. Let us rename the solution v of (11) in terms of the new time variable s as vðx; tÞ ¼ Uðx; sÞ. The effect of the time scaling (13) on the term involving vt in (11) is as follows:

2iaðtÞ 2iaðtÞ v ðx; tÞ ¼ Us ðx; sÞst ðtÞ ¼ iUs ðx; sÞ; kbðtÞ t kbðtÞ

ð14Þ

where we used the chain rule and the definition of s in (13). Let us now combine the adjustments m ¼ i=2, (12) and (13) by substituting them into Eq. (11):

iUs ðx; sÞ þ

1 Uxx ðx; sÞ  V 1 ðx; sÞUðx; sÞ ¼ 0; 2m

ð15Þ

observe that we introduced a function V 1 ðx; sÞ ¼ Wðx; tÞ, recall that W is defined in (12). Clearly, (15) has the form of a timedependent Schrödinger equation. Now that we have established a relation between the latter equation and its initial counterpart (8), let us state this relation explicitly, that is, without the use of intermediate expressions. At first, we want to express the forcing F x in our Burgers equation (8) through the potential V 1 that appears in (15). To this end, we solve (12) for F and insert the definition V 1 ðx; sÞ ¼ Wðx; tÞ, as well as (13). This gives the function F as follows:

Fðx; tÞ ¼ 

 Z t  2 2 2 ik bðtÞWðx; tÞ ik bðtÞV 1 ½x; sðtÞ ik bðtÞ kbðt0 Þ 0 : V x; dt ¼  ¼  1 2a2 ðtÞ 2a2 ðtÞ 2a2 ðtÞ 2aðt 0 Þ

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As in our previous Eq. (13) we are using the prime in t0 as part of the variable’s name. The forcing term in Burgers equation (8) is now determined by the partial derivative with respect to x: 2

F x ðx; tÞ ¼ 

ik bðtÞ @ 2a2 ðtÞ @x

 Z t  kbðt 0 Þ 0 : V 1 x; dt 2aðt 0 Þ

ð16Þ

This is the sought interrelation between the external force F x and the Schrödinger potential V 1 . Inspection of (16) shows that the latter force is in general complex-valued. We will comment on this issue further below. Let us now construct a relation between the solutions U and W of Burgers and the Schrödinger equation, respectively. On combining (10) and (13) and the above definition vðx; tÞ ¼ Uðx; sÞ, we obtain

h R i t kbðt 0 Þ 0   Z t  k U x; dt 0 x 2aðt Þ kvx ðx; tÞ kUx ½x; sðtÞ k @ kbðt 0 Þ 0 h R i : Uðx; tÞ ¼  log U x; dt ¼ ¼ ¼  t kbðt 0 Þ 0 aðtÞvðx; tÞ aðtÞU½x; sðtÞ aðtÞ @x 2aðt 0 Þ aðtÞU x; 2aðt 0 Þ dt

ð17Þ

Observe that the symbol t0 stands for the integration variable. Hence, formulas (16) and (17) characterize the interrelation between the forced Burgers equation (8) its Schrödinger counterpart (15). In particular, if a solution to the latter Schrödinger equation is known, a corresponding solution to the initial Burgers equation (8) can be constructed by means of (17). Before we proceed, let us mention an important point regarding the meaning of our Schrödinger equation (15). In the present context, the latter equation is not meant to describe quantum–mechanical models. In particular, the solutions of the Schrödinger equation that we will be using here, are not required to be normalizable in any L2 -space, nor to represent bound or scattering states. Instead, Eq. (6) stands for a purely mathematical vehicle that generates solutions to the time-dependent Burgers equation (8). 3. The time-dependent Burgers equation The point transformations discussed in Sections 2.1 and 2.2 can be combined to a single transformation that establishes an interrelation between the stationary Schrödinger equation (6) and the time-dependent Burgers equation (8). 3.1. Interrelation with the stationary Schrödinger equation Let us assume that our time-dependent Burgers equation (8) is defined on a domain D ¼ ðx0 ; x1 Þ  ð0; TÞ  R2 for real numbers x0 ; x1 and T > 0:

U t ðx; tÞ þ

a0 ðtÞ Uðx; tÞ þ bðtÞU x ðx; tÞUðx; tÞ þ cðtÞU xx ðx; tÞ þ F x ðx; tÞ ¼ 0; aðtÞ

ðx; tÞ 2 D;

ð18Þ

where we require the parameters a; b; c and F to be smooth functions on D or on its respective restrictions. Note that x0 or x1 are allowed to represent the infinities. Eq. (18) can be endowed with initial- and/or boundary conditions, a topic that we will discuss below in more detail. Now, in order to link (18) to a stationary Schrödinger equation, we first determine how a solution U of (18) is related to its counterpart W of (6) by substituting (7) into (17), recall that we must set m ¼ i=2:

( " # Z sðtÞ Z t0 k @ k @ 00 0 2 00 Uðx; tÞ ¼  log fU½x; sðtÞg ¼  log exp iA½sðtÞx  iB½sðtÞx  iE exp 8i Aðt Þdt dt aðtÞ @x aðtÞ @x   Z sðtÞ 0 þ  2iAðt 0 Þ  B2 ðt 0 Þ  iCðt0 Þdt Wfu½x; sðtÞg ( " # Z sðtÞ Z t0 k @ 00 0 ¼ iA½sðtÞx2  iB½sðtÞx  iE exp 8i Aðt 00 Þdt dt aðtÞ @x  Z sðtÞ k @ 0  2iAðt 0 Þ  B2 ðt 0 Þ  iCðt0 Þdt  log ðWfu½x; sðtÞgÞ: ð19Þ þ aðtÞ @x Let us point out that the symbol t 00 does not refer to any derivatives, but represents an integration variable’s name. We will continue to use the symbols t0 and t 00 throughout this note. Observe that we have not inserted the explicit form (13) of the function s due to the length of the resulting expressions. We can simplify (19) further by evaluating the derivative, which leads us to the compact representation

Uðx; tÞ ¼

  k @ 2iA½sðtÞx þ iB½sðtÞ  log ðWfu½x; sðtÞgÞ : aðtÞ @x

ð20Þ

Taking into account the definition (3) of the argument u, we can apply the remaining derivative, yielding the final result

Uðx; tÞ ¼

   0  Z sðtÞ k W fu½x; sðtÞg 0 : 2iA½sðtÞx þ iB½sðtÞ  exp 4i Aðt 0 Þdt  4iu aðtÞ Wfu½x; sðtÞg

ð21Þ

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Observe that the phase u appears here due to the derivative of W that acts on the first term of the argument u, as given in (3). It remains to determine the relation between the forcing in our Burgers equation (8) and the stationary potential of Eq. (6). We achieve this by plugging expression (2) into the explicit form of our external force (16). Keeping in mind that m ¼ i=2, this yields F x ðx; tÞ ¼ 

   Z sðtÞ 2 n o   ik bðtÞ @ 0 2 exp 8i Aðt 0 Þdt  8iu V 0 fu½x; sðtÞg þ A0 ½sðtÞ þ 4iA ½sðtÞ x2 þ B0 ½sðtÞ þ 4iA½sðtÞB½sðtÞ x þ C½sðtÞ : 2 2a ðtÞ @x

Recall that the function derivative.

F x ðx; tÞ ¼  

s is defined in (13). We take into account the definition (3) of our argument u, and apply the

  Z sðtÞ 2 2 o ik bðtÞ ik bðtÞ n 0 0 2 0 V 00 fu½x; sðtÞg  2 exp 12i Aðt Þdt  12i u 2A ½sðtÞ þ 8iA ½sðtÞ x 2 2a ðtÞ 2a ðtÞ 2  ik bðtÞ  0 B ½sðtÞ þ 4iA½sðtÞgB½sðtÞ : 2a2 ðtÞ

ð22Þ

This is the final form of the forcedterm for our Burgers equation (8). In summary, if a solution to the stationary Schrödinger equation (6) is known, then a corresponding solution of Burgers equation (18) is given by expression (21), where the forcing term F x can be found in (22). 3.2. Initial- and boundary conditions Initial- or boundary conditions that are imposed on solutions of Burgers equation (18), translate into corresponding conditions for the solution of the stationary Schrödinger equation (6). Such conditions can be useful to know, especially when choosing a particular Schrödinger solution to be plugged into (21). We will now illustrate this by equipping our Burgers equation (18) with the following initial condition

Uðx; 0Þ ¼ U 0 ðxÞ;

x 2 ðx0 ; x1 Þ;

ð23Þ

where we assume that U 0 is a smooth function. Let us now find out how condition (23) affects the solution W of our stationary Schrödinger equation (6). To this end, we substitute (20) for t ¼ 0 into (23):

U 0 ðxÞ ¼

  k @ 2iA½sð0Þx þ iB½sð0Þ  log ðWfu½x; sð0ÞgÞ : að0Þ @x

ð24Þ

At this point we would like to convert our initial condition (23) into a restriction for the function W. To this end, we first abbreviate the notation by setting

A½sð0Þ ¼ A0

B½sð0Þ ¼ B0

u½x; sð0Þ ¼ u0 ðxÞ:

ð25Þ

We plug these redefinitions into the initial condition (24) and solve for W:

W½u0 ðxÞ ¼ exp

Z

x



   Z að0Þ að0Þ x 0 0 U 0 ðx0 Þ þ 2iA0 þ iB0 dx ¼ exp  U 0 ðx0 Þdx þ ð2iA0 þ iB0 Þx ; k k

ð26Þ

where an irrelevant constant of integration was discarded. Next, we note that the function u0 is known explicitly, as it can be constructed from (3). Inverting the latter relation at t ¼ 0 and recalling m ¼ i=2 gives the result

" # )  Z sð0Þ ( Z sð0Þ Z t0 0 00 0 xðu0 Þ ¼ exp 4i Aðt0 Þdt þ 4iu u0 þ 2i exp 4i Aðt 00 Þdt  4iu Bðt0 Þdt :

ð27Þ

If we substitute this argument into (26) for x, then we obtain the condition for the solution W of the stationary Schrödinger equation (6) for m ¼ i=2 that is equivalent to the initial condition (23) of Burgers equation (18). We omit to state the explicit form of the combined general expression due to its length. Instead, we will evaluate the condition (26) in our application Section 4.2. 3.3. Reality of solution and external forcing term In order to be physically meaningful, both the forcedterm F x in our Burgers equation and the corresponding solution U should be real-valued functions. Starting out with the solution, we can see from (21) that U depends on the function W, which solves the stationary Schrödinger equation (6). As such, W will depend on the mass m. Since m must take imaginary values according to m ¼ i=2, the solution W of the Schrödinger equation (6) must in general be complex-valued. Moreover, the way W depends on the imaginary mass is different for each potential V 1 in (6), such that no generic reality condition for W can be given. Consequently, the same holds true for the solution (21) of our Burgers equation (8). If we make the following redefinition of parameters

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AðtÞ ¼ iaðtÞ BðtÞ ¼ ibðtÞ;

ð28Þ

where N is an integer and we assume the functions a and b to be real-valued, then our solution (21) takes the form

Uðx; tÞ ¼ 

  Z sðtÞ  0  k W fu½x; sðtÞg : 2a½sðtÞx þ b½sðtÞ þ exp 4 aðt0 Þdt0  4iu aðtÞ Wfu½x; sðtÞg

ð29Þ

Once the function W is known for a particular potential V 1 , we can use the phase u to render (21) a real-valued function. However, success of this procedure is not guaranteed. This situation is similar for the forcedterm F x in (22). We substitute (28) into the latter expression and expand:

 Z sðtÞ  2 h p i k2 bðtÞ  k bðtÞ  0 0 0 V 00 fu½x; sðtÞg þ 2 F x ðx; tÞ ¼ exp i þ 12u exp 12 a ðt Þdt a ½sðtÞ  4a2 ½sðtÞ x 2 2a ðtÞ a ðtÞ 2 2

þ

k bðtÞ 0 fb ½sðtÞ  4a½sðtÞb½sðtÞg: 2a2 ðtÞ

ð30Þ

It remains to verify the effect of our settings (28) to the argument u of the potential V 00 . According to (3), we have (recall m ¼ i=2)

(

" Z 0 # )  Z sðtÞ  Z sðtÞ t 0 00 0 0 00 0 u½x; sðtÞ ¼ exp ð4iuÞ exp 4 aðt Þdt x þ 2 exp 4 aðt Þdt bðt Þdt :

ð31Þ

If we assume that the potential derivative V 00 in (30) is a real function, then all quantities in the latter forcing term will be real-valued, except for the argument (31). Similarly to the solution (29), for each particular function V 0 that enters in (30), one must attempt to find a phase u that produces a real-valued function. Despite the absence of a general reality condition, we will see below that for a large class of applications we can render both solution and forcing term of our Burgers equation real-valued. 3.4. Reduction of order and a second solution We will now discuss a property of our solution (21) that is a direct consequence of the well-known reduction-of-order formula for linear ordinary equations. The latter formula applies to the stationary Schrödinger equation (6) in the sense that once a solution W is known, a second solution w can be generated, provided some integrability conditions are met. Consequently, each of the two Schrödinger solutions generates a corresponding solution of our Burgers equation. The reduction-oforder scheme renders the function w in the form

Z wðxÞ ¼ WðxÞ

x

1

W2 ðx0 Þ

0

dx ;

ð32Þ

where W is a solution of (6) that in practical applications is assumed to be known. If the integral exists, then linear independence of W and w is guaranteed. In such a case, the latter two functions induce two solutions of our Burgers equation (18) for the same forcedterm (22). The first solution is given by the expression (21), while the second solution reads according to (20) and (32)

( Z k @ Wðx; tÞ ¼ 2iA½sðtÞx þ iB½sðtÞ  log Wfu½x; sðtÞg aðtÞ @x

u½x;sðtÞ

!)

1

0

W2 ðx0 Þ

dx

ð33Þ

:

Since the two solutions W and w of the Schrödinger equation (6) are related by means of (32), we are able to relate the solution (33) of our Burgers equation (18) to its counterpart U. First we expand the logarithm as

( Z k @ @ Wðx; tÞ ¼ 2iA½sðtÞx þ iB½sðtÞ  log ðWfu½x; sðtÞgÞ  log aðtÞ @x @x

u½x;sðtÞ

1

W2 ðx0 Þ

!) 0

dx

:

Next, we substitute the explicit form (21) of our solution U into the latter expression, which renders W in the form

k @ Wðx; tÞ ¼ Uðx; tÞ  log aðtÞ @x

Z

u½x;sðtÞ

1

W2 ðx0 Þ

! 0

dx

:

ð34Þ

This is the sought relation between the two solutions U and W of the time-dependent Burgers equation (18) with forcedterm (22). Note that the function W is in general complex-valued, since it contains the complex functions W and the argument u. We will compute a second solution (34) when discussing our first example in the application Section 4. As a final remark in this paragraph, let us mention that the second solution (34) is unlikely to fulfill the same initial- or boundary conditions as its counterpart (21). For example, if we assume that U complies with (23), then W must satisfy

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Wðx; 0Þ ¼ U 0 ðxÞ 

k @ log að0Þ @x

Z

u0 ðxÞ

1

W2 ðx0 Þ

! 0

dx

;

ð35Þ

where we used abbreviations that were introduced in (25). Eq. (35) clearly shows that in general we have Wðx; 0Þ – Uðx; 0Þ ¼ U 0 ðxÞ. A similar statement can be made for boundary conditions. 4. Applications We will now present a series of examples, illustrating the applicability of our method to initial- and boundary-value problems. In the first of these examples we reobtain results found in the recent work [5] for linear forcing. The next application focuses on a generalization of a previously published result [16] for a purely time-dependent forcing term. Subsequent examples feature different aspects of solvable Burgers equations, such as solutions in polynomial and exceptional orthogonal polynomial form, as well as solutions for a sinusoidal forcing term that are related to Mathieu functions. 4.1. Construction of the solutions presented in [5] In the recent article [5] a solution method for linearly forced Burgers equations with time-dependent coefficients was presented, based on a generalization of the procedure introduced in [8]. The purpose of this section is to show that the findings made in [5] can be recovered as special cases of our method. In the latter reference, the following initial-value problem is considered on the domain D ¼ ð1; 1Þ  ð0; 1Þ:

U t ðx; tÞ þ

l0 ðtÞ 1 Uðx; tÞ þ U x ðx; tÞUðx; tÞ ¼ U xx ðx; tÞ  x2 ðtÞx; ðx; tÞ; 2 D; 2lðtÞ lðtÞ

Uðx; t0 Þ ¼ U 0 ðxÞ; x 2 ð1; 1Þ;

ð36Þ ð37Þ

where l; x and U 0 are smooth functions. Furthermore, t 0 is a real constant that for the sake of simplicity will be chosen here as t 0 ¼ 0. On comparison with our initial equation (8), we find that (36) is a special case, obtained from the identifications

aðtÞ ¼ lðtÞ bðtÞ ¼ 1 k ¼ 1 F x ðx; tÞ ¼ x2 ðtÞx:

ð38Þ

Observe that this renders the coefficient c, as given in (9), in the correct form c ¼ 1=ð2lÞ. We therefore expect that our solution method is applicable to Burgers equation (36), once all parameters have been set correctly. In the first step we have to adjust the parameters in (22), such that the latter forcing term matches its linear counterpart in (38). To this end, let us apply the following settings

BðtÞ ¼ 0 V 0 ðxÞ ¼ 0:

ð39Þ

Insertion of these values, in combination with (38), renders the forcing term (22) as

F x ðx; tÞ ¼



 0 4A½sðtÞ  iA ½sðtÞ x; l2 ðtÞ

ð40Þ

While this function is linear as desired, we must redefine our parameter A, such that (40) coincides precisely with the form given in (38), that is, the coefficients of the spatial variable x must be the same. Thus, we have to fulfill the condition 0

4A½sðtÞ  iA ½sðtÞ ¼ x2 ðtÞ; l2 ðtÞ

ð41Þ

by solving it for our parameter function A. This can be accomplished, because the argument s is known: substitution of the present settings (38) into the definition (13) of s gives the result

sðtÞ ¼

Z

t

1 0 dt : 2lðt0 Þ

We can now use this explicit representation of

1 d d A½sðtÞ ¼ 2lðtÞ A½sðtÞ; A0 ½sðtÞ ¼ 0 dt s ðtÞ dt

s to evaluate the derivative on the left-hand side of our condition (41): ð42Þ

where we used the chain rule and the inverse function theorem. Next, we redefine A in terms of a smooth parameter function r ¼ rðtÞ as follows

A½sðtÞ ¼ i

lðtÞr0 ðtÞ 2rðtÞ

:

ð43Þ

We now plug this expression and the derivative (42) into (41) and evaluate the resulting equation. A short calculation gives

A. Schulze-Halberg / Commun Nonlinear Sci Numer Simulat 22 (2015) 1068–1083

r00 ðtÞ þ

l0 ðtÞ 0 r ðtÞ þ x2 ðtÞrðtÞ ¼ 0: lðtÞ

1075

ð44Þ

This is precisely the constraint provided in the work [5]. Hence, finding a solution r of (44) is equivalent to providing a parameter A that complies with (41), such that the forcing term takes the form given in (38). Let us now assume that the constraint (38) is satisfied, it then remains to construct a solution of the initial-value problem (36) and (37). To this end, we plug our settings (40) and (43) into the general solution (21). Since evaluation of the latter expression is straightforward, we omit to state details of the calculations, but just present the result:

  0 exp ð4iuÞ exp ð 4i u Þ W x r 0 ðtÞ rðtÞ   : Uðx; tÞ ¼ x exp ð4iuÞ rðtÞ x rðtÞlðtÞW rðtÞ

ð45Þ

Here, the function W is a solution of the stationary Schrödinger equation (6) for a vanishing potential, as specified in (39). In order to recover the solution presented in [5], we first make the following observation regarding Section 2.1: instead of assuming that W ¼ WðxÞ solves the stationary Schrödinger equation (6), we can alternatively request W ¼ Wðx; tÞ to be a solution to the time-dependent counterpart (4), which for the present settings V 0 ¼ 0 and m ¼ i=2 reads

Wt ðx; tÞ  Wxx ðx; tÞ ¼ 0:

ð46Þ

Consequently, W must satisfy the heat equation. Introduction of the time-dependent function W into our solution (45) and applying the parameter definition r 0 ¼ expð4iuÞ for a nonzero constant r0 , our solution (45) then takes the final form

n h i o Rt R t0 00 0 r0 r0 W1 rðtÞ x; exp 8i Aðt 00 Þdt dt r 0 ðtÞ n h i o; Uðx; tÞ ¼ x Rt R t0 00 0 rðtÞ rðtÞlðtÞW r0 x; exp 8i Aðt 00 Þdt dt

ð47Þ

rðtÞ

where W1 refers to the partial derivative of W with respect to its first argument. Expression (47) represents the solution constructed in [5]. Note that in the latter reference the heat equation (46) contains an additional factor 1=2 in front of the second derivative term. This factor is missing here due to the definition of our mass m ¼ i=2. Before we conclude this section, let us construct the initial condition for the heat equation (46) by means of (26). Substitution of the present parameter values V 0 ¼ A0 ¼ B0 ¼ 0 and taking into account (38) gives



W½u0 ðxÞ ¼ exp lð0Þ

Z

x

 0 U 0 ðx0 Þdx ;

which coincides with the initial condition for the heat equation (46) found in the work [5]. 4.2. Purely time-dependent forcing term In our next application we will consider an initial-value problem (18) and (23) for a Burgers equation that features a purely time-dependent external force. This setting has been studied before in the work [16], where traveling-wave solutions were obtained. We will demonstrate that the results obtained in the latter reference can be recovered and generalized by means of our method. The stationary potential and the forcing term. The first step in constructing solutions of the problem (18) and (23) consists in identifying a stationary potential V 0 , such that the forcing term (22) becomes independent of the spatial variable x. Since the latter forcing term contains the derivative of V 0 , we choose a linear function

V 0 ðxÞ ¼ dx;

ð48Þ

where d is a real-valued, positive constant. This function renders the forcing term (22) in the form   Z sðtÞ 2 2 2 o  ik bðtÞ ik bðtÞ n 0 ik bðtÞ  0 0 2 F x ðx;tÞ ¼  2 exp 12i Aðt0 Þdt  12iu d   2 2A ½sðtÞ þ 8iA ½sðtÞ x  2 B ½sðtÞ þ 4iA½sðtÞgB½sðtÞ : 2a ðtÞ 2a ðtÞ 2a ðtÞ

ð49Þ Since we want this forcing term to be independent of x, we must make its coefficient vanish by a suitable choice of the parameter A. We are given the following two options for this choice:

AðtÞ ¼ 0 or AðtÞ ¼

1 ; 4it þ A0

ð50Þ

where A0 is an arbitrary constant. One can show that the second choice for A gives a real-valued forcing term F x , but renders the associated solution (21) complex-valued, no matter how the remaining parameters are chosen. Therefore, in this example we consider only the case A ¼ 0, which yields our forcedterm (49) in the form

F x ðx; tÞ ¼

ibðtÞ exp ð12iuÞd  B0 ðtÞ : 2a2 ðtÞ

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A. Schulze-Halberg / Commun Nonlinear Sci Numer Simulat 22 (2015) 1068–1083

We can turn this into a real-valued function by adopting (28) and setting the phase to u ¼ p=24, which gives

F x ðx; tÞ ¼

bðtÞ½d þ b0 ðtÞ : 2a2 ðtÞ

ð51Þ

Note that in order to obtain a purely-time-dependent function F x , we could have set d ¼ 0. This was not done in order to maintain as many free parameters as possible. The time-dependent solution. In order to determine a solution of our Burgers equation (18) for the external force (51), we must first solve the stationary Schrödinger equation (6) with potential (48). The general solution of this equation is wellknown and can be given in terms of Airy functions [1], note that we set E ¼ 0 and m ¼ i=2:

h

1

i

h

1

i

WðxÞ ¼ C 1 Ai ð2dmÞ3 x þ C 2 Bi ð2dmÞ3 x :

ð52Þ

Here, C 1 ; C 2 are arbitrary constants, while Ai, Bi stand for the Airy functions. Recall that the Schrödinger solution (52) is not required to represent any quantum–mechanical system, but provides a mathematical seed solution. We can now construct a solution of Burgers equation (18) by plugging (52) into the general expression (20) for the latter solution. In addition, we must incorporate the settings (28), as well as a ¼ 0; u ¼ p=24 and m ¼ i=2. This yields the result

Uðx; tÞ ¼ 

n h i h io 1 1 kb½sðtÞ k @  log C 1 Ai ðidÞ3 u½x; sðtÞ þ C 2 Bi ðidÞ3 u½x; sðtÞ : aðtÞ aðtÞ @x

ð53Þ

Let us now determine the argument of the Airy functions that appear in the latter expression. To this end, we insert our settings into (31) and evaluate.

  Z sðtÞ  1 1 ip ip 0 x þ 2 exp  ðidÞ3 u½x; sðtÞ ¼ ðidÞ3 exp  bðt 0 Þdt : 6 6

ð54Þ

1=3

We observe that the root i can take three different values, any of which renders the function (53) a solution of Burgers equation. Since we want expression (54) to take real values, we calculate

 1 1 1 ip ¼ ð1Þ6 ð1Þ6 ¼ 1: i3 exp  6 This simplifies (54) as follows 1

1

1

ðidÞ3 u½x; sðtÞ ¼ d3 x þ 2d3

Z sðtÞ

0

bðt 0 Þdt :

In the final step we plug this into our solution (53) and evaluate the derivative in front of the logarithm. We obtain

h 1 i h 1 i 1 R sðtÞ 1 R sðtÞ 0 0 0 0 1 bðt0 Þdt þ C 2 Bi c3 x þ 2c3 bðt 0 Þdt kb½sðtÞ kd3 C 1 Ai c3 x þ 2c3 h i h i:  Uðx; tÞ ¼  aðtÞ aðtÞ C 1 Ai c13 x þ 2c13 R sðtÞ bðt0 Þdt0 þ C 2 Bi c13 x þ 2c13 R sðtÞ bðt 0 Þdt 0

ð55Þ

This is a real-valued solution of our time-dependent Burgers equation (18) for the purely time-dependent forcing term (51). The initial condition (23) satisfied by the function (55) and the initial value U 0 cannot be given explicitly until more information is known about the free parameters a; b and b. Finally, let us point out that our solution (55) contains its counterpart constructed in [16] as a special case, if the parameters a; b; b are chosen to be constants and if C 2 ¼ 0. The second solution. Before we present examples and special cases of the solution U found in the previous paragraph, let us focus on determining a second solution of our Burgers equation (18) for the forcing term (51). To this end, we just need to substitute the solution of the stationary Schrödinger equation (6) for the potential (48) into expression (34). We obtain the result

0 k @ B Wðx; tÞ ¼ Uðx; tÞ  log @ aðtÞ @x

Z

u½x;sðtÞ

1 1

0C n h i h io2 dx A; 1 1 C 1 Ai ðidÞ3 x0 þ C 2 Bi ðidÞ3 x0

ð56Þ

where the functions U and u are given in (55) and (54), respectively. We were not able to obtain a closed-form representation of the integral that appears in the above solution W, except in the case that either C 1 or C 2 vanishes. An example. Let us now choose a particular set of parameters in order to visualize special cases of the solutions (55) and (56). We start by selecting the time-dependent coefficients in Burgers equation (18):

aðtÞ ¼ 

1 t

bðtÞ ¼ 

2 t

cðtÞ ¼ 1 bðtÞ ¼ 1:

Observe that the coefficient c was not chosen, but determined by the constraint (9) for k ¼ 1. Before we can state the initial-value problem (18) and (23) for the present case in explicit form, we need to determine its domain D and the function U 0 in the initial condition. Let us set D ¼ ð0; 1Þ  ð0; 1Þ and U 0 ðxÞ ¼ 0, then our initial-value problem reads

A. Schulze-Halberg / Commun Nonlinear Sci Numer Simulat 22 (2015) 1068–1083

1 2 U t ðx; tÞ  Uðx; tÞ  U x ðx; tÞUðx; tÞ  U xx ðx; tÞ þ dt ¼ 0; t t Uðx; 0Þ ¼ 0; x 2 ð0; 1Þ:

ðx; tÞ 2 D

1077

ð57Þ ð58Þ

We can now give a solution U to the time-dependent Burgers equation (57) by substituting our parameters settings into (55). This yields

n h 1 i h 1 io 1 0 0 d3 t C 1 Ai d3 ðx þ 2tÞ þ C 2 Bi d3 ðx þ 2tÞ h 1 i h 1 i Uðx; tÞ ¼ t þ : C 1 Ai d3 ðx þ 2tÞ þ C 2 Bi d3 ðx þ 2tÞ It is immediate to see that this function complies with the initial condition (58), because each term vanishes for t ¼ 0. As mentioned before, a second solution (56) to the time-dependent Burgers equation (57) cannot be given in general form due to the integral. This situation changes if we choose C 2 ¼ 0, rendering the first solution U as

h 1 i 1 0 d3 tAi d3 ðx þ 2tÞ h 1 i : Uðx; tÞ ¼ t þ Ai d3 ðx þ 2tÞ

ð59Þ

Now we can evaluate the integral in (56), leading to the following second solution W

Wðx; tÞ ¼ t þ

t 0 F1

h

1 1 ; dðx 3 9

ðx þ 2tÞ0 F1

h

þ 2tÞ3

4 1 ; dðx 3 9

i

þ 2tÞ3

i;

ð60Þ

where 0 F1 stands for the confluent hypergeometric function [1]. Note that the solution W does not have any singularities, because both variables x and t must be positive. Furthermore, it is interesting to observe that (60) satisfies the initial condition (58), such that both functions (59) and (60) are solutions to the initial-value problem (57) and (58). This, however, is a particularity of the present example rather than a general feature, see the discussion in Section 3.4 for details. The solutions U and W, as given in (59) and (60), respectively, are displayed in Fig. 1. 4.3. Fifth-order monomial forcing term In this example we will construct a polynomial solution to an initial-value problem of the form (18) and (23) for a monomial forcing term of order five. More precisely, we consider the following equation, defined on the domain D ¼ R  ð0; 1Þ as

U t ðx; tÞ þ

16t 1 x5 Uðx; tÞ  U ðx; tÞUðx; tÞ þ 4U xx ðx; tÞ þ ¼ 0; 2 x 2 2 2 4t 2 þ 1 6ð4t þ 1Þ 2ð4t þ 1Þ

ðx; tÞ 2 D;

ð61Þ

equipped with the initial condition

Uðx; 0Þ ¼ x3 ;

x 2 R:

ð62Þ

In order to find a solution to this initial-value problem, we will follow the same steps as in the previous application, starting out by considering a suitable potential of the Schrödinger equation. The stationary potential and the forcing term. In order to render the external forcing term (22) as a monomial of order five in the spatial variable, we choose the following potential V 0

Fig. 1. The two solutions (59) and (60) to the initial-value problem (57) and (58) for the parameter setting d ¼ 1.

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A. Schulze-Halberg / Commun Nonlinear Sci Numer Simulat 22 (2015) 1068–1083

V 0 ðxÞ ¼

2v 2 m 6 x þ v x2 ; 9

ð63Þ

where v is a real constant that will be determined later, and m stands for the constant mass in the Schrödinger equation (6) that we have to choose as m ¼ i=2. The particular form of the coefficients in the potential was chosen in order to obtain a simple solution of the associated Schrödinger equation, as will be explained in the next paragraph. In addition to (63), let us apply the following settings for the parameters that are involved in the general expression (22) for the forcing term:

AðtÞ ¼ aðtÞ ¼

it 4ðt 2 þ 4Þ 1

BðtÞ ¼ 0 bðtÞ ¼ 

2

48ð4t 2 þ 1Þ

ð64Þ

1

: 2 6ð4t 2 þ 1Þ

Observe that insertion of these functions a and b into (9) and (18) reproduces Eq. (61) except for the forcing term that we will compute now. Substitution of (64), together with m ¼ i=2; k ¼ 1 and (63) yields the following expression:

F x ðx; tÞ ¼ ½24iv expð16iuÞ þ 24x 

v 2 expð32iuÞ 2ð4t2 þ 1Þ

2

x5 :

Since we want this function to be real-valued, we choose the phase u and the constant

u¼

p 32

ð65Þ

v as

v ¼ 1:

ð66Þ

This setting removes the first term of (65) and renders the second term real-valued:

F x ðx; tÞ ¼

x5 2ð4t2 þ 1Þ

2

:

Recall that this is the desired form of our forcing term, as can be seen by comparison with the Burgers equation (61). The time-dependent solution. We can now construct a solution to the time-dependent Burgers equation (61) from the corresponding solution of the stationary Schrödinger equation (6), which reads in the present case

 i iW00 ðxÞ þ x2  x6 WðxÞ ¼ 0: 9

ð67Þ

Observe that we have employed the settings E ¼ 0, as well as m ¼ i=2 and v ¼ 1. The Schrödinger equation (67) has the particular solution



WðxÞ ¼ exp 

 i 4 x ; 12

ð68Þ

as can be verified by direct substitution. The reason for choosing (68) rather than a different solution is the compatibility with our initial condition (62). In order to see this, let us evaluate (26) and its argument (27). On plugging our settings (64) and (66) and U 0 ¼ x3 from (62) into the latter argument, we arrive at

xðu0 Þ ¼ 2 exp

 ip u0 : 8

In the next step we insert this into the right-hand side of (26) and obtain after simplification



Wðu0 Þ ¼ exp 

 i 4 u0 : 12

Comparison with (68) shows that both functions coincide. Consequently, we are guaranteed that the solution U of our Burgers equation (61) that is obtained through (68), will satisfy our initial condition (62). Let us now calculate the solution U. To this end, we plug our parameter settings (64) and (66), m ¼ i=2 and (68) into the expression (21). After some simplification we arrive at the following result

Uðx; tÞ ¼ x3  24tx  96t3 x:

ð69Þ

It can be easily verified that this polynomial is a solution of our time-dependent Burgers equation (61) and also satisfies the initial condition (62). Fig. 2 shows a plot of the solution, note that the initial condition Uðx; 0Þ ¼ x3 cannot be verified easily, as the values of U for t > 0 become much larger as compared to the case t ¼ 0. In principle, we can determine the second solution (34) for the present example. However, since the latter function is singular, we omit to show it here. Related solutions for a binomial forcing term. We will now present solutions of our time-dependent Burgers equation (8) that arise from the present example by a slight modification of the parameters. First, recall that in our preliminary form of the forcing term (65) a linear term in the spatial variable is contained that we eliminated by the settings (66). Let us now see what happens if we keep the parameter v arbitrary. We can repeat the steps that lead us to the solution (69), using

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A. Schulze-Halberg / Commun Nonlinear Sci Numer Simulat 22 (2015) 1068–1083

Fig. 2. The polynomial solution (69) of the initial value problem (61) and (62).

the stationary potential (63) for an undetermined constant v. For the sake of brevity, we omit to show the calculation, but only mention the result. We obtain the function

Uðx; tÞ ¼ v x3  24tx  96t3 x; which reduces correctly to (69) if

U t ðx; tÞ þ

ð70Þ

v ¼ 1 as shown in (66). Thus, the function (70) is a solution to the initial-value problem

16t 1 v 2 x5 Uðx; tÞ  U x ðx; tÞUðx; tÞ þ 4U xx ðx; tÞ þ ð24  24v Þx þ ¼ 0; 2 2 2 4t þ 1 2ð4t2 þ 1Þ 6ð4t 2 þ 1Þ

ðx; tÞ 2 D

ð71Þ

for the initial condition (62) and the same domain D that was used in (61). The latter equation is generalized by its counterpart (71), the forcing term is a binomial term with a linear and a fifth-order contribution. 4.4. Forcing term of X 1 -Laguerre type The purpose of this example is to present a particular case of our time-dependent Burgers equation (8), where both external force and solutions can be expressed through exceptional orthogonal polynomials (EOPs). To the best of our knowledge, this is the first example of Burgers equation containing EOPs. Let us start out by introducing a stationary Schrödinger equation that is solvable through EOPs. The stationary system and EOPs. The Schrödinger equation for the rationally extended radial oscillator model, defined on the positive real axis, is given by

"

# 1 2 2 lðl þ 1Þ 4x 8xð2l þ 1Þ W ðxÞ þ E  x x   þ WðxÞ ¼ 0; xx2 þ 2l þ 1 ðxx2 þ 2l þ 1Þ2 4 x2 00

ð72Þ

where E; x and l are constants. Unlike in the quantum–mechanical context, here we do not need to impose any initial constraints on the latter constants. In addition, boundary conditions that the solutions of (72) must fulfill in order to be physically meaningful, are for the present example irrelevant. It is well-known [20] that the Schrödinger equation (72) has an infinite number of solutions Wn ; n P 2 a nonnegative integer, each valid for a specific value of the parameter E ¼ En :

  xlþ1 1 1 lþ12 2 2 L x x x x exp  nþ1 4 2 xx2 þ 2l þ 1  3 : En ¼ x 2n þ l þ 2

Wn ðxÞ ¼

ð73Þ ð74Þ

Here, the symbol L stands for an exceptional Laguerre polynomial of X 1 -type, defined through associated Laguerre polynomials L as follows [20]

Lkn ðxÞ ¼ ðx þ k þ 1ÞLkn1 ðxÞ þ Lkn2 ðxÞ:

ð75Þ

We are now ready to use the Schrödinger equation (72) and its solutions (73) to generate corresponding solutions of a Burgers equation. The stationary potential and the forcing term. At first we must match Eq. (72) with our Schrödinger equation (6), where the mass is to be set as m ¼ i=2. To this end, we simply multiply (72) by the factor i, which gives the following stationary potential

" # 1 2 2 lðl þ 1Þ 4x 8xð2l þ 1Þ : V 0 ðxÞ ¼ i x x þ þ  xx2 þ 2l þ 1 ðxx2 þ 2l þ 1Þ2 4 x2

ð76Þ

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We will take the values (74) into account below, once we construct the solution of our Burgers equation. In order to construct the forcing term (22), let us choose our parameters as follows

AðtÞ ¼  aðtÞ ¼

i 2t

1 t4

2i p expðtÞ u ¼ 4 t2 2 1 bðtÞ ¼ 4 k ¼ 1 l ¼  : 2 t BðtÞ ¼

x¼1 ð77Þ

Let us briefly explain this particular choice, without going into too much detail. The functions A and B were chosen imaginary in order to render the forcing term and the solution of Burgers equation real-valued, according to our considerations in Section 3.3. This will be achieved in combination with our choice for the phase u. The particular functional form of A and B has the effect of removing singularities of the solution from the nonnegative x–t plane. The coefficients a; b of the Burgers equation were chosen such as to deliver the simple coordinate change s ¼ t, see (13), and to avoid further singularities in the solution to Burgers equation. While the values of x and k just simplify our calculations, the choice of l shortens the explicit form of the external forcing term, as well as of the solution to our Burgers equation considerably. Observe that in the quantum–mechanical context, l represents the angular momentum quantum number, which is not allowed to attain negative values. Substitution of (77), together with the derivative of the potential (76), leads to the following special case of (22):

 F x ðx; tÞ ¼ 

   t 10 þ 6 t 2 x 15t 10  2t t 9  t þ 4 expðtÞ þ

3 : 2 2 2 t x þ 4 expðtÞ

ð78Þ

Observe that this function does not have any singularities in the nonnegative x-t-plane, which is due to the choice of parameters (77). The time-dependent solution. Before we construct solutions associated with the forcing term (78), let us first set up the time-dependent Burgers equation and equip it with an initial condition. To this end, we substitute the parameter settings (77) and the forcing term (78) into Eq. (8):

 10    t þ 6 t2 x 4 2 15t 10 U t ðx;tÞ  Uðx; tÞ þ 4 U x ðx;tÞUðx; tÞ  U xx ðx;tÞ   2t t 9  t þ 4 expðtÞ þ

3 ¼ 0; t 2 t 2 t 2 x þ 4expðtÞ ðx; tÞ 2 ð0; 1Þ  ð0;1Þ:

ð79Þ

Furthermore, we define the following initial condition

Uðx; 0Þ ¼ 0:

ð80Þ

The solution to Burgers equation (79) can now be constructed by plugging the parameter values (77) into our general solution formula (21). After some simplification, we arrive at the following result

3t 6 1 t4 n

þ 2 expðtÞt2 ðt4  1Þ þ t3 xðt 5  2Þ  Uðx; tÞ ¼ 2

2 o 2 2 t x þ 4 expðtÞ L0nþ1 12 t2 x þ 4 expðtÞ  

2 @ 0 1 2 : Lnþ1 t x þ 4 expðtÞ  @x 2

ð81Þ

For each natural number n, this function is a solution of the time-dependent Burgers equation (79). In addition, the initial condition (80) is satisfied, such that (81) provides a infinite set of solutions to the initial-value problem (79) and (80). Note that all solutions are free of singularities in the nonnegative x-t-plane. Let us mention the two simplest special cases of (81), the first of which is obtained for n ¼ 1:

1 5t 6

: Uðx; tÞ ¼ t 2 expðtÞðt 4  1Þ þ t 3 ðt 5  2Þx  2 2 2 t x þ 4 expðtÞ

ð82Þ

If we set n ¼ 2 in (81), we get the particular solution



2t 6 t 2 x þ 4 expðtÞ 1 3 5 5t 6

 Uðx; tÞ ¼ 2t expðtÞðt  1Þ þ t ðt  2Þx  2 :

2 2 2 t x þ 4 expðtÞ t 2 x þ 4 expðtÞ  6 2

4

We omit to show more functions of the set (81), because the corresponding expressions become very long. Before we conclude this example, let us present a plot of the solution (82) in Fig. 3. 4.5. Sinusoidal forcing term In our last example we study our Burgers equation (8) for a time-dependent sinusoidal forcing. This type of model was found useful in the context of poloidal flows [24] and the interaction of dispersive waves [15], among other applications. Our governing equation, defined on the domain D ¼ ð0; 1Þ  ð0; 1Þ, has the form

A. Schulze-Halberg / Commun Nonlinear Sci Numer Simulat 22 (2015) 1068–1083

1 2 1 U t ðx; tÞ  Uðx; tÞ þ U x ðx; tÞUðx; tÞ  U xx ðx; tÞ þ t sinð2xÞ ¼ 0; t t 2

ðx; tÞ 2 D:

1081

ð83Þ

We equip this equation with both initial and Dirichlet-type boundary conditions:

Uðx; 0Þ ¼ 0;

x 2 ð0; 1Þ Uð0; tÞ ¼ Uð1; tÞ ¼ 0;

t 2 ð0; 1Þ:

ð84Þ

Our considerations that follow in the subsequent paragraphs will reveal that the solutions of the problem (83) and (84) can be expressed through Mathieu functions. The stationary potential and the forcing term. As in the previous examples, the first step of our construction consists in defining a stationary potential V 0 that generates the main part of the spatial dependence in the forcing term (22). Since we want the latter term to be sinusoidal, the potential V 0 must be a cosine function, which we choose in the form

V 0 ðxÞ ¼ 2i cosð2xÞ:

ð85Þ

In addition, we pick the following set of parameters that enter in our forcing term (22) and in the associated Burgers equation (8):

AðtÞ ¼ 0 BðtÞ ¼ 0 u ¼ 1 aðtÞ ¼ t

2 bðtÞ ¼ t

p 4

ð86Þ

k ¼ 1:

Substitution of these values into (22) and (8) gives precisely the equation (83). This can be verified by a straightforward calculation, which we omit to show here due to its length. Observe in particular that despite the potential (85) being imaginary, the resulting forcing term in Burgers equation (83) is real-valued. The time-dependent solution. In the first step we must solve the stationary Schrödinger equation (6) for the mass m ¼ i=2 and the potential (85). Insertion of the latter quantities and a parameter redefinition E ¼ ie for a real constant e gives the result

W00 ðxÞ þ ½e  2 cosð2xÞWðxÞ ¼ 0: This is a special case of the Mathieu equation [1], the general solution of which is well-known. It can be stated in the form

WðxÞ ¼ c1 Cðe; 1; xÞ þ c2 Sðe; 1; xÞ;

ð87Þ

where c1 ; c2 are constants. Furthermore, C and S stand for the even and odd Mathieu functions, respectively [1], sometimes referred to as Mathieu cosine and Mathieu sine functions. In order to keep the present example simple, we will restrict ourselves to the even solution, that is, we choose c2 ¼ 0 and c1 ¼ 1 in (87). This gives

WðxÞ ¼ Cðe; 1; xÞ:

ð88Þ

We obtain the corresponding solution of the time-dependent Burgers equation (83) after combining (3), (13), (86) and (88). Substitution into (21) leads to

Uðx; tÞ ¼ 

t @ Cðe; 1; xÞ: Cðe; 1; xÞ @x

ð89Þ

This function solves the time-dependent Burgers equation (83). In general, U has an infinite number of singularities in the x–t-plane, because the Mathieu function C vanishes at infinitely many points. In the present example, we will be able to render our solution (89) free of singularities, due to the bounded domain for the spatial variable and a suitable choice of the parameter e, which will be made below. Let us now verify that (89) is compatible with our initial- and boundary-value

Fig. 3. The solution (82) of the initial value problem (79) and (80).

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A. Schulze-Halberg / Commun Nonlinear Sci Numer Simulat 22 (2015) 1068–1083

Fig. 4. The solution (90) of the initial- and boundary-value problem (83) and (84).

conditions. To this end, we first observe that it satisfies the initial condition shown on the left side in (84), because t is an overall factor. Next, we take into account that (89) is an odd function with respect to the variable x, because the Mathieu function C is even. Since we assume our solution U to be continuous in its domain, we have Uð0; tÞ ¼ 0, which coincides with the first boundary condition in (84). In order to satisfy the remaining second boundary condition, we must use the parameter e. The latter condition is fulfilled, if the equation

@ Cðe; 1; xÞjx¼1 ¼ 0; @x holds. One can show that this equation has a solution at e  0:83017 that we denote as e0 . Substitution of this value into (89) yields the following solution of our initial- and boundary-value problem (83) and (84):

Uðx; tÞ ¼ 

t @ Cðe0 ; 1; xÞ: Cðe0 ; 1; xÞ @x

ð90Þ

The fact that this function complies with the conditions (84) can be verified by inspection of Fig. 4 that displays a plot of (90). 5. Concluding remarks We have devised a method of solution for the forced Burgers equation with time-dependent coefficients and nonlinear external forcing term. The solutions that become accessible by means of our method can be interpreted as generalized traveling-wave type, where the corresponding traveling-wave argument is provided by the function u, defined in (3). The examples and applications we have presented in Section 4 are far from being exhaustive, but are supposed to demonstrate the variety of problems that can be handled through our method, including some that have been studied before. Acknowledgement We thank J. Garcia-Ravelo for interesting discussions and for pointing out several mistakes in a previous version of this note. References [1] Abramowitz M, Stegun I. Handbook of mathematical functions with formulas, graphs, and mathematical tables. New York: Dover Publications; 1964. [2] Bednarik M, Cervenka M. Equations for description of nonlinear standing waves in constant-cross-sectioned resonators. J Acoust Soc Am 2014;135:EL134. [3] Broadbridge P, Srivastava R, Jim Yeh TC. Burgers equation and layered media: exact solutions and applications to soil-water flow. Math Comput Modell 1992;16:163–9. [4] Burgers JM. A mathematical model illustrating the theory of turbulence. Adv Appl Mech 1948;1:171–99. [5] Buyukasik S, Pashaev O. Exact solutions of forced Burgers equations with time variable coefficients. Commun Nonlinear Sci Numer Simul 2013;18:1635–51. [6] Cole JD. On a quasi-linear parabolic equation occurring in aerodynamics. Q Appl Math 1951:225–36. [7] Enflo BO, Hedberg CM, Rudenko OV. Resonant properties of a nonlinear dissipative layer excited by a vibrating boundary: Q-factor and frequency response. J Acoust Soc Am 2005;117:601. [8] Eule S, Friedrich R. A note on the forced Burgers equation. Phys Lett A 2006;351:238–41. [9] Finkel F, Gonzalez-Lopez A, Kamran N, Rodriguez MA. On form-preserving transformations for the time-dependent Schrödinger equation. J Math Phys 1999;40:3268–74. [10] Fokas AS, Stuart JT. The time periodic solution of Burgers equation on the half-line and an application to steady streaming. J Nonlinear Math Phys 2005;12(suppl. 1):302–14.

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