47
ON THE CONNECTION OF LAMBERT FUNCTIONS AND CLASSES OF
SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS
Alfred Huber
Institute of Theoretical Physics, Technical University Graz, Austria.
ABSTRACT
In this paper a new algebraic procedure is introduced to compute new classes of solutions of (1+1)-nonlinear partial differential equations (nPDEs) of scientific and technical relevance. The crucial step of the method is the basic assumption that the unknown solution of the nPDE under consideration satisfies an ordinary differential equation (ODE) of the first order that can be integrated completely. A further important aspect of this paper however is the fact that we have the freedom in choosing some parameters bearing positively on the algorithm. So the solution-manifold of any nPDE under consideration is augmented naturally. Since Lambert function involves several classes of unknown solutions in terms of further special functions could obtain. The present algebraic procedure can widely be used to study several nPDEs and is not only restricted to time-dependent problems. We note that no numerical methods are necessary and so closed-form analytical classes of solutions result. The algorithm works accurately, is clear structured and can be converted in any computer language. On the contrary it is worth to stress out the necessity of such sophisticated methods since a general theory of nPDEs does not exist.
Key words: Nonlinear partial differential equations; Evolution equations; Special function methods. PACS-Code: 02.30Jr, 02.20Qs, 02.30Hq.
AMS-Code: 35L05, 35Q53, 14H05.
1. INTRODUCTION
This article introduces an important concept in the study of nPDEs especially of higher order. The important outcome is that one is able to derive solutions by a pure algebraic approach.
Many models in physics and applied sciences can be described by nPDEs of the general form ut K
u,ux,uxx,....
where K[u] means a nonlinear operator in general. Explicit solutions are of basic interest especially those with physical relevance, e.g. the propagation of traveling waves. It is still of interest to evaluate new or improve known methods for finding closed-form solutions since the calculation of the general solution-manifold fails.
Many powerful methods such as the Inverse Scattering Transform Method [1], the Darboux transformation [1], [2],
Hirota’s bilinear method [1], [2], the Painlevé expansion [3] and the Homogeneous Balance Method [4] are appropriate to handle such problems effectively.
The Jacobian elliptic function method described in [5] and [6] is also suitable to generate solutions; a technique using pure series of sine and cosine functions can be found in [7]. More analytical solutions derived by further algebraic methods are listed in [8], [9] and [10]. Further applications of algebraic procedures in which the solution is assumed in terms of hyperbolic tangent functions can be found in [11].
A generalized approach, the so-called improved projective Riccati-equation method is derived in [12] as well as the Weierstrassian elliptic-function method [13]. In recent papers, [14] and [15] we applied the method both to a combined KdV-mKdV equation and moreover to an evolution equation of fourth order for the first time successfully.
Further ‘classical ansatz-methods’ are discussed in [16], [17] and [18]. Similarity reductions dealing with Lie symmetries (using the invariance of a nPDE under point transformations) are also appropriate to calculate solutions as shown in recent papers of the author, e.g. [19].
2. GENERAL DESCRIPTION – BASIC ASSUMPTIONS
Consider a nPDE in its two variables x and t which describes the dynamical evolution of a wave form u( tx, ),
:
48
, , , , ,..., 1, 0
1
2 2
n n
n n
x u x
u t
u x u x
u t u x u u
P . (1)
We convert eq.(1) into a nODE by using a frame of reference u(x,t) f(), f C1, xt and R is a constant to be determined. So eq.(1) becomes
Q
f(),f'(),f''(),....,f(n1)(), f(n)()
0, (2) with the prime denoting differentiation w. r. t. .) (
f represents a localized wave solution which travels with speed (in context of solitary theory it means the
soliton’s velocity). It exemplifies a stationary wave with a characteristic width of L1. We integrate the nODE (if possible) as long as
all terms contain derivatives and further the associated integration constants can taken to be zero in view of the localized solution
we seek. This is a necessary (but not sufficient) condition that f() tends to zero for.
The next step is that the solution we seek assumes in form of the following series representation with some coefficients aj and bj to be determine:
n
j
j
j w b w
a w j a
w f f
1
0 exp ( 1)( ( ) exp( ( )) exp( ( ))
) ) ( ( )
( , (3)
containing the auxiliary function ww() which satisfies a solution of the first order ODE:
kw
e b
a w d
w d
' , k
b
wln( ) , ( ba, ) R, k Z . (4)
For the ODE (4) we require: Let D be a complex domain and D
for all holomorphic functions and :
so that
w,w',w'',....,w(n)
0 and the prime means d /d.We further require that the ODE (4) has at least one solution and the solution is unique. Solutions of eq.(4) are bounded and n-fold differentiable. The solution develops completely in an interval I so that
,w() I
DI holds.So we ensure complex-valued solutions of the ODE (4) if necessary. General solutions of the ODE (4) are given as
b b
c k a
W b c k a b w
) ( exp )
1 ( 1 ) (
1
1 , (4.1)
where W() means the Lambert function and c1 is an arbitrary integration constant. Let . Then, in the interval 1/e0 we have the following possibilities:
The function W() has two branches in the closed intervals (0,] and (,0] and W() is a complex-valued function in general, not injective on
,0
and has two branches on
1 ,0
e . Here we restrict the analysis to the
main branch for W()1, that is W0() or simply W().
If necessary any solutions of arbitrary studied nPDEs can be expressed for the branch W()1or W1(). There
exist a singularity of the first derivative of W() at the point 1/e.
We refer to the following fundamental property, e.g. [21], [22], and [23]: We restrict us to the positive domain (with respect to concrete applications) of the Lambert function. Hence this defines a solution of a general transcendental equation:
W:
(x,y) xyey, x,y0
, (4.2) and the same is true in the complex domain so that we have49
We further note that the function has a branch cut along to1/e. For further calculations it is convenient to introduce a general closed-form series representation:
n
n n
x n n x
W
1 1
! ) ( )
( , (4.4)
with the radius of convergence being R1/e. The series converges at least for ME(,)1/e with
) (
) ,
(ab 12 a2 b2
ME is the Euclidian measure from a and b.
The balancing parameter n in series eq.(3) computes by balancing the highest order nonlinear term with the highest order partial derivative term in the relevant nODE (2).
So, n must be a positive integer since it represents the number of terms in the series so that a closed analytical solution results.
In case of fractions one can take transformations as shown in [20] or see Example 2. Therefore the first suitable
‘ansätze’ promoted by the parameter
n
are given by:
.: 2 ,
: 1 ,
2 2 1 2 2 1 0
1 1 0
w w
w w
w w
e b e b e a e a a f n II Case
e b e a a f n I Case
(5)
Since the functional dependence of the function
w
(
)
should be clear we drop the argument for simplicity. Now the transformed expressions are introduced in the series (3). Then substitute it into the relevant nODE together with their derivations relating to the given ODE (4).All terms with the same power in the exponential terms are collected and set to zero their coefficients to get a nonlinear homogeneous algebraic system (NHAS). Both they are over- and under-determined in some cases and making difficult the solubility.
Now we solve for the unknowns
a0,....,aj,b1,...,bj, a,b,
in a consistent way whereby it may happen thatsolutions are of trivial form only and therefore useless for the solution-manifold of the nPDE, eq.(1) under consideration.
In the following we shall study some examples of various equations to clarify the new algebraic method. We point out that the algorithm is explained in detail at Example 1 since the method is self-explanatory. Hence only solutions of various equations are given in a most general form whereby we find it useful to point out some properties of interest.
3. EXAMPLES OF VARIOUS NPDES
Example 1: A nonlinear system of PDEs of the third order
Consider the coupled system of third order [11], [17] looking for the unknown functions uu( tx, ) and vv( tx, )
with uC2(,), vC3(,) and positive time t0:
. 0 3
0
2 2 3
3 3 2
x u x v x
v x
v t v
t v x v t x
u
(3)
In [11], the tanh-approach was applied to eq.(3) leading to solitary solutions (regular as well as irregular solutions) and furthermore, elliptic functions of the first kind are solutions of eq.(3), [17]and further, the Painlevé-property of eq.(3) was also proven in [17].
By now only solitary solutions are known and no direct physical connection of the system is realizable. Following the procedure, substituting u(2)v'2 and introducing hv' as a new dependent variable, one obtains a nODE of second order. Balancing leads to n1 and the linear ‘ansatz’, eq.(5), Case I is appropriate. We derive a NHAS with eight equations and six unknowns which give two possible solutions:
Case(i): a0 0, b1 a, Case(ii): b1 a,
2 0
2 a
.
50 b e W b c a a a a h b c a( )/ 1 1 0 1 ) ( exp ) ( )
( . (3.1)
This function can not be integrated in a closed-form so we use a series of W() breaking off the linear term:
2 2
] [ )
( O
W . Then, integrating once with c1 0 (for simplicity) we derive at
2 2 1 0 / ) ( ) ( c a b b e Exp b a a a
v
, (3.2) with arbitrary chosen coefficients a,a0,b,c2. Calculating the limiting behaviour it is seen that the relations
a b ea
aev( ) ( 1) /
lim 2
0
and lim v () hold.
The first, the second and the third derivations exist and a further analysis shows that the function is stable. For practical calculations a series representation up to order three is useful:
43 2 2 2 6 ) 1 ( ) 1 ( 1 ) 1 ( 1 ) ( O e b a a e a b a e a b a
v , (3.3)
and for large values of the argument, that is an asymptotic representation is derived to:
2 32
41 1 3 2 2 1 1 ~ ) ( O e e
v . (3.4)
The function u(), after integrating twice and using the abbreviation a / b we get:
2 8 ,2 2 2 a -a 2 4 1 ) ( 4 0 3 1 2 0 3 2 4 2 1 2 c b e Ei a b e Ei b a a a c a b e Exp b b e Exp a u (3.5)
where Ei(.) means the exponential-integral function and the involved constants are assumed to be arbitrary numbers.
For the following wet set a11, ac3a0 1, c40 for simplicity. The term containing the Ei-function can
be rewritten by using the general connection (0,x)Ei(x), where (.,.) means the incomplete gamma function:
[2 ]
4 [2 ](2 (2 ) 4( 8 (0, ) 2 (0, ))
41 )
( Exp e Exp e e e2
u , (3.6)
and is the Euler-Mascheroni constant.
In addition, by using the connection of the incomplete gamma function with those of the Laguerre polynomials one can write for the two last terms by considering the relation L0n(x)Ln(x) alternatively in a compact fo
0 ( 1)
) 2 ( ) ( 4 )) 2 ( 4 ) 2 ( 2 ( ] 2 [ 4 ] 2 [ 4 1 ) ( n n n n e L e L e e Exp e Exp u
. (3.6a)
51
, )) 1 ( 160 ) 2 ( 40 20 9 ( 18 ( 2 ) ) ) ) 1 ( 32 ) 2 ( 8 (
) 1 ( 4 ) 2 ( 2 ( ( ) (
4 3
2 2 4
2
O Ei
Ei e
e Ei
Ei
e Ei Ei
e e u
(3.6b)
and here Ei(-1) and Ei(-2) are pure numbers, e.g. Ei(1)0,219383....
In Figure1 we give a qualitative picture for both functions using different values of the parameters. Obviously one can see a certain domain defined by a straight line in which solutions can not exist.
-6 -4 -2 0 2 4
-5 -2.5 0 2.5 5 7.5
-2 -1.5 -1 -0.5 0 -500
0 500 1000 1500 2000 2500 3000
Fig.1 The solution functions u() and v(), eq.(3.2) and eq.(3.6), respectively of the nPDE (3). Left: The behaviour of the function v()
with a0 a1c2 1, Right: the function u() with a11and a0 c31. The parameters a and b varies in the domain
1(a,b)4. A straight line limits the existence of solutions (left).
Example 2: The nonlinear heat conduction equation
The equation under consideration is given in the following form with p0,q0, [25]:
(2 ) 2 0
2 2
qu pu x
u t u
, uu( tx, ) , uC2(,) , t0. (3.7)
Putting u(x,t) f(),xt, one derives at f'(f2)''pf qf20.
This is an example in which the balancing number is not an integer (n1). Therefore the following procedure is necessary: Let f V1, then eq.(3.7) transforms into a nODE of the second order in the new depending variable
) (
V where the prime means derivation w.r.t. :
pV36V'2V2(qV')2VV''0. (3.8)
Balancing V2V' and VV'' results in n1and the linear polynomial ‘ansatz’ is suitable. We have to solve a NHAS with six equations and six unknowns to arrive two useful solutions (two of them are trivial): Case(i): a0 q/p,
a
b1 and Case(ii): a0 0, b1a. From here we choose the first solution deducing new solutions in terms of
the Lambert function:
1
1
1 1
) ( exp )
( )
( )
, (
b b
c a W
b c t x a Exp a a p q t x
u . (3.9)
The parameters a,a1,b,c1 and can be chosen arbitrary. For a further qualitative discussion it is sufficient to choose the following values of the parameters: pqbc11,a10 and a1to derive a closed-form
52
) ( 1
1 )
( 1 1
e W e
u , R . (3.9a)
It is seen that this analytical function is positive and real-valued, u()0 as 0 and the function takes indeterminate as . Higher derivations exist for all R and the functions are stable as one can see in Figure 2.
-6 -4 -2 0 2 4 6 0
0.2 0.4 0.6 0.8 1
-4 -2 0 2 4
-0.2 -0.15 -0.1 -0.05 0 0.05
0.1 0.15
-4 -2 0 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35
Fig.2 Left: Solution curves of the nonlinear heat equation, solution eq.(3.9a) generated by the parameter choice pqb1,a10 and a1. The constant c varies between 0ci 5. Middle and right: the first and the second derivative of
the solution eq.(3.9a), both functions are stable in the considered domain.
It is of interest that the function shows a typical kink-like profile for different values of the constants
c
i. After transformation a representation for u() with c11 reads
24 132 167 32
1
1
. 1921
1 8
15 4 16
1 1 2 1 1 1 ) (
4 3
3 2 4
2 2
3 2
O e
e e e
e e e
e e e u
(3.9b)
Taking into account the properties of the exponential function it is proven that the radius of convergence is R
that means (3.9b) converges on the interval I(1,].
Therefore solutions of this special kind of the nonlinear heat equation with a polynomial nonlinearity derived by this method are suitable to describe any physical situations relating to heat transfer and due to the equivalence of heat, diffusion mass transfer the given algorithm is suitable to predict real-valued solutions of advanced character. So the function u()depends linear, quadratic or otherwise by a power law depending on how accuracy is required.
Example 3: The viscous Burgers equation (BE)
The equation plays an important role in fluid dynamics with 0 as the viscosity parameter:
t u x u u x
u
2 2
, uu( tx, ) , uC2(,) , t0. (3.10)
Eq(3.10) is a simplified model for turbulence, boundary layer behaviour, shock wave formation and mass transport [26], [27]. With the linear ‘ansatz’, Case I we end up by an NAHS with four equations and six unknowns. The NAHS admits only one possible solution, that is b1a; the remaining parameters left arbitrary. We found a
53 . / 1 , ) ( ! ) 1 ( 2 1 ) ( ) ( ) ( exp ) ( ) ( ) ( 1 1 1 1 1 0 1 1 1 0 e b b c a Exp k b c a Exp a a a b b c a W b c a Exp a a a u k k k
(3.10a)Again, for a qualitative discussion we set a0b1, a12and a1 for simplicity to derive
u()1ec1W(ec1), ( ,)
1
c R. (3.10b)
Figure 3 shows a sequence of planar solution curves of eq.(3.10b) by assuming different values for the constants
c
i and one can see that the solution function eq.(3.10b) is similar to those of the heat conduction equation. Therefore an explicit analysis is not necessary. It further follows that
lim0u( ) 2 W(1) R
+
and by considering limiting laws it is shown that the function eq.(3.10b) remains infinite since we have
limW(e ) W lime W( ) . (3.10c)
Consider now c10 in (3.10a) we derive at a compact written form:
1
.192 1 6 1 16 1 2 1 2 1 ! 1 ! ) 1 ( ! 2 ) ( 4 3 2 1 1 1 1
O e e e e e e n n n u e n n n n n n n (3.10d)Here, the closed-form expression above represents an analytic function which converges analogues to the function eq.(3.9b) on the convergence interval I(1,].
-2 0 2 4 6 0 5 10 15 20 25 30 35
Fig.3 Some planar solutions of the Burgers equation (3.10). The solution function (3.10b) is generated by the choice of the parameters
a0ab1and a12. For the constants c we chose0ci 7. The functions tend to a real value as
0
and take
infinite as and is stable in the considered domain.
4. SUMMARY AND CONCLUDING REMARKS
54
The unknown function of the nPDE under consideration acting as an auxiliary function is assumed to be a solution of the given ODE. The generalization is possible at once by choosing the factor k in the exponential function to be arbitrary so that the general case in the exponent is nw().
All examples are performed for the case k1. So we showed that by applying the procedure completely new solutions in terms of special functions result. The success of algebraic methods depends strongly on the solubility of the NHAS and in fact this can not predicted in general.
The experience shows, as mentioned above that most of the NHAS are over-determined and ways influencing the solutions are restricted. However, increasing the number of unknowns as shown in [24] make the situation tractable and is a suitable starting point in handling such problems efficiently (how far a numerical determination is useful
should depend on the user’s decision).
Moreover we point out some remarkable facts exemplified on some counter examples by applying the procedure: The algorithm tested on some other standard nPDEs resulted in the following:
The Korteweg de Vries equation admits a suitable solution a1b1, a2b2, a0 for the NHAS. This means
that a solution of the KdV constructed by the series (3) is only a constant.
The same property appears by applying the procedure to the Regularized Long Wave equation (RLW) whereby this behaviour is not a specific peculiarity of the given method.
On the contrary such behaviour in applying algebraic procedures is well-known and one may not conclude the unrestricted application of algebraic methods to any nPDEs in general.
But they are a valuable contribution in solving nPDEs in a reasonable time and calculating expense.
Since the algorithm works accurately and allows one to determine exact solutions the application is therefore suitable to a wider class of equations and is not only restricted to any time-dependent problems.
Finally the approach is done without any numerical methods, works fast and easy and is practicable for translation into appropriate computer languages.
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