Nonlinear Hyperbolic Problems: Theoretical, Applied, and Computational Aspects

Proceedings of the Fourth International Conference on Hyperbolic Problems, Taormina, Italy, April 3 to 8,1992

Edited by Andrea Donato and Francesco Oliveri

Notes on Numerical Fluid Mechanics, Volume 43 (Vieweg, Braunschweig 1993)

The title of the book series is allowed to be abbreviated as NNFM.

II V1ew eg

I Reprint I

The BURGERS EQUATION: EXPLICIT SOLUTIONS OF AN INITIAL BOUNDARY

VALUE PROBLEM

Sandra CARILLO Dipartimento eli Metoeli e lvlodelli l'vlatematici

perle Scienze Applicate Universita, "La Sapienza.I

'

I 00161 Roma, Italy

Abstract

Here, on the basis of the results obtained ill [9J, we construct the solution of an initial boundary value problem for t he Burgers equation,

OUT method is based on t he application of the well know n Cole-Hopf t ransforma­lion which relates t.he nonlinear Burgers equation to the linear heat C{luat.ion. Thus, lhe initial boundary value problem for the Burgers equation we are interested in is trasfOflllcd into an initial bound ary value prohlem for a linear diffu sion eq uation (heat equation).

The latter is solved in terms of a series expansion of repeated integra ls of error functions. The convergence of t.hi s se ries is provcd co rres po nding to initi,,1 data which are a nalyt.ic in 11/2.

Rcmarkably, the solution here presented can be easily expressed ill terms of the initial cond itions on the boundary.

1. INTRODUCTION

The nonlinear Burgers equation has been widely studied since it represent.s one of the most simple examples of nonlinear part.ial differential equations, Many interesting properties have been proved to be enjoyed by the Burgers equation snch "s to admit a hel'cciitay recursion operator 18] [12], to be amenable t.o t.he Inverse Scattering Transform (I.S.1'.) [3], lo posses lhe Painlev" Properly [18], and lo admil similarily solulions [13] 114] and to possess inllni tely many sY lllmet.ries 11 2]12], only to mention the most well known ones,

On t.he ot.hel' hand , a great deal of interest. in t he study of t.he Burgers equation is due to the fact that this equat.ion turned out t.o represent a IIgooo modeP' in the description of physical phenomena. Specifically, it.s import.ance in the field of fluid dynamics is well known [19].

Here, on t.he basis of t.he result.s obtained in [9], we present the solution of an initial boundary vallie problem which is relatcd to a problem of infiltration of wate r in soils with prescribed boundary concent.ration. Indeed, the nonlinear partial differential equa­t.ion governing nOll -hysteric inlllt.ration in non-swelling soil turned out to be t.he Burgers equal ion ([5], [17!).

""

The subsequent application of Lrasformations in volving the dependent as well as the independent variables is proved to link thc or iginal problcm to a problcm for the linea r heat equat ion. Specifica llYI a first transformatio n produces another initial boundary valu(! problem for a Burgers eCluatioll wherei n only dimension less variables appear; thus! the succcssive applicat ion of the Cole- Hopf Transformation I"L [I O]! delivers an ini t ial bound­ary value problem for t.he heal equation. The laU er is! t hell l solved via a se ries expan sion of repea t.ed in t.egrals of error fUllct. iolls; its convcrgence is proved .

In part.icular l corresponding to smoot,h data the solut.ion converges for all x > 0 and o < t < t2 where 12 > 0; furthermorc 1 t,he smoot.her the t.he initia l data are! t he greater t2 is.

2. THE PROBLEM

According to Hogers and Shadwick ([l7] ! 151L the nonlinear evolut.ion equation which call be adopted to model the non-hysteric vertical infiltration of water into a non-swelling soil is given by:

DO =!!... (DDO) _ f('(0)DO DI D" Dx Dx

(2. 1 )

where D and J( represent! respective ly! t he \\ moisture diffusivityU and the "hydrau li c conductiv ity!'. It provides the evolution of the moist.ure content 0 as a fun ction of depth x and time t.

The assumpt.ion:

D( 0) = const.ant., A ,

f((0 ) = '2 (0 - 0,,) (2.2)

wherein On is t he anteccdent watcr content and A > 0 is a constant! has been adopted by Clothicl\ Knigh t and \Vhi te [5] invest igat. ing the problem of const.a nt rate infiltration of water in to a deep bed of Dungedore fine sand .

Here, under the same assumptions expressed by (2.1) and (2.2), we a re concerned about the case of an ass igned law for the t,ime dependence of the moisture content on the soil surface. Thus, the Burgers problem reads:

(

DO D [ DO A , ] - + - -D-+ - (O - O") = 0 DI Dx Dx 2 0(0, I) = G(l) + 0" 0(", l) --> 0" O(x,O) = 0"

x> 0, t > 0;

t > 0; x -+ 00, t 2: 0; x~O

(2.3)

where O( t.) is a given fun ct ion which represents the time dependence of the moisture content at x = 0 (on the soil surface) .

The other two condi tons express, respect ively! that , in the limit of infini te depth, the moisture content is equa l to its reference value and the coincidence of the moisture content with its reference va lue at t = 0 in the whole soil.

Such a nonlinear differential problem models the sa me phys ica l situation stud ied by Clot.hier et al. 15] except that.! here! the moisture content Oil t he ground surface is assumed to be expressed by a known fun ction of t ime.

First of all , the introduction of dimcnsionless viHiablcs:

(2.4)

wherein OJ > On represents a refercnce moisture con t.ent, tra ll sfo rms the problem (2,3) int.o the following:

where:

lao' ao' a'o' Dr + 20' 7i[ - a{' = 0

O' (O , T) = C'(T) O ' ({,T)~ O

O' ({,O ) = 0

< > 0, T > OJ

< = 0, T > OJ ~ -l oo , T 2:: 0; ~ > 0

G"( ) = C(t ) 1 T OJ _ On '

On applicat.ioll of the celebrated Cole-Hopf transformat.ion (4 ], I IOJ :

O. = _~ au U a~ ,

T he Burgers equa t. ion is transformed into

where B( T) is an arbi trary function of T,

Subsc<luentiy [9J, [,IJ , 1101, since (2.7) impl ies lhal

U(~,T) = exp ( - l' O' (S,T)"S + C(T))

where c( T) is an arbi trary funct ion of T. Then, lett ing:

I/=ex r ( - [ B(a)da) U -I

wherein

J.~ ao' d

B(T ) = - -a (S,T)+ -, C(T) o T (. T

produces the heat equation.

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

(2.10)

(2.11 )

T hus, the solution of the original problem (2,3) is ob t.ained by solving the fo llowing problem:

wherein

~ > 0, T > 0;

T > OJ

{ ........ 00 , T 2:: 0; ~ > 0

C" (T) = ..2J.!:L. OJ - On

(2. 12)

(2. 13)

3. THE SERIES EXPANSION SOLUTION

In t.his sectioll} it is shown that. t.he problem (2.12) admits a series expansion solution. The solut.ion is looked for in t.he form of a series expansion of repeated integrals of

error functions:

II(C T) = f lI"r G + I) (4JT)"i"er/c C~) 11=1

which introducing t.he Ilew variable:

'I = JT call be wriltcJl as:

11((, 'I) = f lI"r G + I) (4,,)"i"<,!c UJ· n=1

The repealed integrals of error functions arc defined by [1):

alld, therefore (II:

i"er{c(O)= 1 . 2"r(~+I)

~i"er.rc(z) = -i."-'<r/c(z). <iz

(3 .1 )

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

Tints} the problem , now} is to show t.hat. there exist. suitable lin such that. (2.12) is satisfied by II expressed in by (3.1) and that. t.he series expansion therein converges.

First of aliI we show t.hat the ansat.z (3.1) is compat.ible wit.h t.he init.ial boundary condit.ions assigned in (2.12). Indeed:

a) II(C 0) = 0; ( > 0 (3.7)

follows illlll1cdiatly 011 subst.it.ution of 'I = 0 in (3.1) provided we set. 110 = 0;

b) 11((, 'Il ---> 0; ( ---> 00, 0 <:: 'I <:: 00 (3.8)

follows from t.he definit.ion (3.4) of t.he repeated int.egrals of the error function;

c) the furt.her condition which, expressed in t.he variables ~ and 'I, reads:

(3.9)

wherein Y"(IJ) := 0"(111), implies t.hat.:

r ( " + 1 ) 00 -2-+1

-2L",,+,(2'1l" , (" ) = 9'('I}(1I + I). ,,=0 I 2" + 1

(3.10)

T his rc lat.ion fo llows dcrivat ing with resped to ~ the solu tion U(~,11 ) given by (3 .1 ). P recisely, (3. 1 L undcr pa rtial diffe rcnt iation with resped to C gives [I ]:

a ( ) 2~ 1,(/ +1 )( )" '" f ( ~ ) nile" ;;;:;-' L-Un+ 1 --+ 1 "'I I er c - . o~ ,,=0 2 2,/

(3. 19)

\\'hf' l'ci ll (:3.6) has hecn used. The suhsequcnt subst itut.ion of (3.5L gives:

(3. 12)

T hus, the condi t. io ll cxpressed by (3.9), de li vers (3. 10).

Secondly, we obscrve that t. iw mOllotonicity of the repeated int.egra ls or erro l' functions

II I (3.5) , g ives:

lo" r G + I) (4 ,/ )"i"er.rc (2~,) 1 s lo" r G + I) (,,,/)"i"eI'fc(O) 1 s 10,,1(2,/)". (3.13)

lIenec, the convergence of the series ex pansion (2. 12) is proved ilS SOOIl as is proved t hatL::= 1 lunl(2,,)n convcrges.

Let. us rUl't lier ass llme that:

i) y"(,/) is cont inuous at '/ ~ 0;

ii) 9"('/) is a n ana lyt ic fnnctioll for 0 < '/ < '/1;

o < '11 ~ +00, and:

"() I:= 9"(2,/)" 9 'I ;;;:; - --; III

n:::O

iii ) g" 2: 0 ( or 9" S 0) VII;

(3. 14)

Lemma 3.1. If the hypoth esis i),ii), iii) (Ire flt lfilled, thell for 0 ::; '1 ~ '/0 < '11 , Iunl ~ lin where Ci" E 01+ U{ O). Proof. Under t. he hypot.es is i) and ii), on subst. it. ut.ion of (3.1) and (3. 14) into (3.7) implies 90 = O.

Moreover, on s uhst.it.u t. ion of (3. 14 ) int.o (3.12), it. fo llows:

/J r' (':.) " ___ --'-;2C'-.,---:, [9" I: 0,9,,- , I 2(11 + I )r (" ~ I) ,, 1 + '=0 (" - k) ! .

\Ve observe 19] t. hat the hy pot.hes is i), ii) and iii) imply that

3Mo > 0 slIch that 19,, [ Mo -<--. II! - (2,/)"

(3.15)

(3. 16)

Indeed, since g-(IJ ) is supposed to be analytic for any '1 E (0, +00) and continuous in 'I = 0, then on any compact set 10, "o}, where '10 is an arbi tra ry posit ive real number, there exists a constant Mo(I10) E lR. + such that.:

(3.17)

Thus:

19'; + t t'~'".; ,1 ~ Mol!1" + t lu, I!1"-'1 ,1. 11. k=O k=O

(3. 18)

where (J = 1/21JO, which, substit.uted into (3.15), gives:

1\10 n l~ (~) 1\

111,,+11 ~ () 1!1" + '" 11I,1!1"-'1 , 11+ 1 L-2{II + 1)1 -2- '=0

(3.19)

Application of the Stirli ng formula II 1 implies that

(3.20)

and, thus:

(3.21 )

where , = Mo../i . By induction, it ca ll be further proved 19] that.

(3.22)

and, t hus, lu,,1 .s: (I" fo r all n .•

Lemma 3.2. The series expa nsion which represents th e solution ;'1 {3.1} is absolutely CO ll ucl'gellt

Proof.

Vf. > 0 alld 0 < 'I < '/2; '12 = mill {I/OI +-} 2,., + I

From (3.22) it follows that 00 00

Llu,,1 ~ La,,; n=O n=O

hence. t he convergence of the latler implies t.he absolu te convergence of the first one and, theil, of the series expansion in (3,1). On applicat ion of thc ratio test to such a series deli vers the result. •

Combi nation of thc results already proved , produces the following:

Theorem 3.3. If g-(l}) sa tisfies the hypoth esis i) , ii) and iii), th en the solution of the pl'Oblem (2.12) COli be expressed ill Ihe jO,.,11 (3.1) V~ > 0 olld O<1}<1}2'

4. CONCLUDING REMARKS

As well as most of t.he au thors interested in solving Burgers problems, our procedure is based 011 t.he nonlinear link between such a nonlinear evolution equation and t.he linear difrusion equation first.ly observed by Cole [4] and Hopf [10]. Among t he many interest ing studies on t he Burgers equat ions, recent.iy, solutions to init.ial boundary va lue problems have been obt.a ined , on li se of the Inverse Scattering t.ransform in (/6]), on use of st ream functions (/ 15]) similarity solut.ions to generalized Burgers problems have been present.ed ;11(17]).

The advant.age of the solution we const.ruct relies in it s easy applicabilil.y. Indeed, t here exist.s t.ables of iterated integrals of elTor fu nct ions (see for inst.ance ([1])

and, t.hus , approximat.ed solutions ca n be obt.ained. Finally, we observe t ha t. t. he sol ution he re presented is consis tent wit.h t.he classical

one ([19)) , si nce it is easy t.o verify t.hat. IIrJ := l/rJirJerfc((/21}) represents a solution of t. he heat equat.ion and , thus , t.he series in (3. 1) which is a se ries expa nsion in t.erms of t.he hlJ is such.

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