Symbolic computations of exact solutions to nonlinear integrable differential equations

Citation
Symbolic computations of exact solutions to nonlinear integrable differential equations

Material Information

Title:
Symbolic computations of exact solutions to nonlinear integrable differential equations
Creator:
Grupcev, Vladimir
Place of Publication:
[Tampa, Fla.]
Publisher:
University of South Florida
Publication Date:
Language:
English

Subjects

Subjects / Keywords:
The tanh method
PDE
KdV
Solitary wave
Wave equation
Dissertations, Academic -- Mathematics -- Masters -- USF ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
ABSTRACT: In this thesis, first the tanh method, a method for obtaining exact traveling wave solutions to nonlinear differential equations, is introduced and described. Then the method is applied to two classes of Nonlinear Partial Differential Equations. The first one is a system of two (1 + 1)-dimensional nonlinear Korteweg-de Vries (KdV) type equations. The second one is a (3 + 1)-dimensional nonlinear wave equation. At the end, a few graphic representations of the obtained solitary wave solutions are provided, in correspondence to different values of the parameters used in the equations.
Thesis:
Thesis (M.A.)--University of South Florida, 2007.
Bibliography:
Includes bibliographical references.
System Details:
System requirements: World Wide Web browser and PDF reader.
System Details:
Mode of access: World Wide Web.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 34 pages.
Statement of Responsibility:
by Vladimir Grupcev.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
001917289 ( ALEPH )
181590952 ( OCLC )
E14-SFE0002025 ( USFLDC DOI )
e14.2025 ( USFLDC Handle )

Postcard Information

Format:
Book

Downloads

This item has the following downloads:


Full Text
xml version 1.0 encoding UTF-8 standalone no
record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd
leader nam Ka
controlfield tag 001 001917289
003 fts
005 20071119131546.0
006 m||||e|||d||||||||
007 cr mnu|||uuuuu
008 071119s2007 flu sbm 000 0 eng d
datafield ind1 8 ind2 024
subfield code a E14-SFE0002025
040
FHM
c FHM
035
(OCoLC)181590952
049
FHMM
090
QA36 (ONLINE)
1 100
Grupcev, Vladimir.
0 245
Symbolic computations of exact solutions to nonlinear integrable differential equations
h [electronic resource] /
by Vladimir Grupcev.
260
[Tampa, Fla.] :
b University of South Florida,
2007.
3 520
ABSTRACT: In this thesis, first the tanh method, a method for obtaining exact traveling wave solutions to nonlinear differential equations, is introduced and described. Then the method is applied to two classes of Nonlinear Partial Differential Equations. The first one is a system of two (1 + 1)-dimensional nonlinear Korteweg-de Vries (KdV) type equations. The second one is a (3 + 1)-dimensional nonlinear wave equation. At the end, a few graphic representations of the obtained solitary wave solutions are provided, in correspondence to different values of the parameters used in the equations.
502
Thesis (M.A.)--University of South Florida, 2007.
504
Includes bibliographical references.
516
Text (Electronic thesis) in PDF format.
538
System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
500
Title from PDF of title page.
Document formatted into pages; contains 34 pages.
590
Advisor: Wen-Xiu Ma, Ph.D.
653
The tanh method.
PDE.
KdV.
Solitary wave.
Wave equation.
690
Dissertations, Academic
z USF
x Mathematics
Masters.
773
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.2025



PAGE 1

by VladimirGrupcev Athesissubmittedinpartialfulllment oftherequirementsforthedegreeof MasterofArts DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:Wen-XiuMa,Ph.D. YunchengYou,Ph.D. AthanassiosG.Kartsatos,Ph.D. DateofApproval: April10,2007 Keywords:Thetanhmethod,PDE,KdV,Solitarywave,Waveequation cCopyright2007,VladimirGrupcev

PAGE 6

DierentialEquations VladimirGrupcev @t6@ @x+@3 @x3=@ @x+@3 @x3;@ @t6@ @x+@ @x+@3 @x3=k@ @x+@3 @x3; @t+@ @x+@ @x(r2)+@5 @x5=0;

PAGE 8

wherethesubscriptsdenotethepartialderivatives:ut(x;t)=@u(x;t)

PAGE 9

@t6@ @x+@3 @x3=@ @x+@3 @x3;@ @t6@ @x+@ @x+@3 @x3=k@ @x+@3 @x3; @t+@ @x+@ @x(r2)+@5 @x5=0;

PAGE 11

Theparametersk,mandnsymbolizethewavenumbersinx,yandzdirections,respectively,and!isthefrequency,whichisassumedtobeafunctionofthewavenumbersk,mandn.Theintroductionofthenewvariablebringsthefollowingchanges:@ @t!cd d;@ @x!cd d;@ @t!cd din(1+1)dimensionalcase;@ @t!!d d;@ @x!kd d;(2.3)@ @y!md d;@ @z!nd din(1+3)dimensionalcase:

PAGE 12

d!(1~y2)d d~y;d2 d~y+(1~y2)d d~y;(2.4)::: where~y=tanh()=8<:tanh[c(xt)];tanh(kx+my+nz!t):

PAGE 13

@t6@ @x+@3 @x3=@ @x+@3 @x3;@ @t6@ @x+@ @x+@3 @x3=k@ @x+@3 @x3;(3.1)where,,,,,andkarerealconstants.Weintroduce=c(xt);(x;t)=U()and(x;t)=W():(3.2)Becauseoftheintroductionofthenewvariables,weobservethefollowingchanges@ @x=cdU d;@3 @x3=c3d3U d3;@ @t=cdU d;@ @x=cdW d;@3 @x3=c3d3W d3;@ @t=cdW d:(3.3)Ifweputtheseformulasintothesystem(3.1),wegetthefollowingsystem:cdU d6cUdU d+c3d3U d3=cdW d+c3d3W d3;cdW d6cWdW d+cdW d+c3d3W d3=kcdU d+c3d3W d3:(3.4)Now,letusdividethetwoequationsin(3.4)bycandintegratethemwithrespectto7

PAGE 14

d2=W+c2d2W d2;W3W2+W+c2d2W d2=kU+c2d2U d2:(3.5)Now,weintroduce~y=tanh()asanewindependentvariable.Wealsomakethefollowingsubstitutionsandproposethefollowingseriesexpansiontobeasolutionfor(x;t)and(x;t):(x;t)=U()=F(~y)=R1Xr1=0ar1~yr1;(x;t)=W()=G(~y)=R2Xr2=0br2~yr2:(3.6)Becauseofthesubstitutions,weobservethefollowingchanges:dU d=~y0dF d~y;d2U d2=~y00dF d~y+~y02d2F d~y2;dW d=~y0dG d~y;d2W d2=~y00dG d~y+~y02d2G d~y2:(3.7)Sincewearegoingtouse~y0and~y00,letusexpressthemintermsof~y:~y0=[tanh(~y)]0=1~y2;~y00=2~y0~y=2~y~y0=2~y(1~y2):(3.8)8

PAGE 15

d~y+(1~y2)d2F d~y2]=G+c2(1~y2)[2~ydG d~y+(1~y2)d2G d~y2];(3.9)G3G2+G+c2(1~y2)[2~ydG d~y+(1~y2)d2G d~y2]=kF+c2(1~y2)[2~ydF d~y+(1~y2)d2F d~y2]: d~ywhichyieldsthedegreeofR1+2.Thesecondoneis2c2~y3dG d~ywhichyieldsdegreeofR2+2.Nowletusconsiderthesecondequationof(3.9).Thenonlineartermofhighestorderis3G2anditsdegreeis2R2.Thereareseverallineartermsthatcanbeconsideredofhighestorder.Therstoneis2c2~y3dG d~yanditsdegreeisR2+2.Thesecondoneis2c2~y3dF d~yanditsdegreeisR1+2.Now,letusbalancethelineartermofhighestorderwiththenonlinearoneinbothequationsof(3.9)aswecanseewehavetwoalternatives:1.R1>R2or2.R2>R1.BothofthemgivethevaluesforR1=2andR2=2.Sonowknowingthis,(3.6)becomes:(x;t)=U()=F(~y)=2Xr1=0ar1~yr1=a2~y2+a1~y+a0;(x;t)=W()=G(~y)=2Xr2=0br2~yr2=b2~y2+b1~y+b0:(3.10)Substituting(3.10)into(3.9)andorganazingitinorderofthepowersof~y,wecanhavethefollowingcomputations.9

PAGE 16

Comparingthecoecientsofeachpowerof~yin(3.11),weget:6a2c23a22=6c2b2;2c2a16a1a2=2c2b1;a26a0a23a218c2a2=b28c2b2;(3.13)a12c2a16a0a1=2c2b1+b1;3a20a0+2c2a2=2c2b2+b0:Comparingthecoecientsofeachpowerof~y,in(3.12)weget:3b22+6c2b2=6c2a2;6b1b2+2c2b1=2c2a1;b26b0b23b21+b28c2b2=ka28c2a2;(3.14)b16b0b1+b12c2b1=ka12c2a1;b03b20+b0+2c2b2=ka0+2c2a2:Aswecansee,in(3.13)and(3.14),wehaveasystemof10equationsand8variables,10

PAGE 17

Ifwecrossmultiplytheexpressionsin(3.16)anddividetheresultinequationbyb2,wegetthefollowingequation:(a2

PAGE 18

Goingbacktoa2=x2b2,weget:a2=2c2(x2) Now,fromthesecondequationsof(3.13)and(3.14)weexpressc2,equatetherighthandsidesofthetworepresentationforc2,crossmultiply,dividebyb2andthendividebyb1.Asaresultoftheseoperations,weget:x2x21+(x2)x1+=0;(3.21) wherea1

PAGE 19

6(x1+2c2x12c2+) Ifwenowpluga1=x1b1intothefourthequationin(3.14),weget:b16b0b1+b12c2b1=kx1b12c2x1b1; 6(+2c2kx1+2c2x1) Bynow,wehavea0,b0,a2andb2asfunctionsofandc2.Pluggingthesevaluesfora0,b0,a2andb2intothethirdequationof(3.13)andsolvethatfora1,wegeta1asafunctionofandc2:a1=s 3c2u1 whereu1=(x2)[(8c2)x1+(2c2)x26c2x1x2]: 3c2u2

PAGE 20

Wearegoingtousethepreviouslycomputedvaluesofx2,a2andb2presentedin(3.18),(3.20)and(3.19)respectivelyandplugthemintothethirdequationsin(3.13)and(3.14).Thisiswhatwegetasaresultofthat:from(3.13):2 (d2 3+12422d1 3)2(d2 3+12422d1 3+6d1 3(d2 312++42+2d1 34c2d2 3+48c216c228c2d1 3+24c2d1 3+6d1 3))=0;(3.28)14

PAGE 21

(d2 312+42+2d1 3)4(d2 312+42+2d1 36d1 3)(60k485+4k5d1 3+1443c222+85322c24++2kd2 34+4d1 34+123d132123d1 32+242c23d2 3+2d2 334d1 34+108k2222d2 33+7232+12965c222++324k224+6kd2 322522321642+216427232+5223+8k6243h+1222h+243h1222h+72c23d1 32163c22d1 3+36kd1 322+9kd2 32++93d2 393d2 3+363d1 32162c24d1 3363d1 354kd1 323+4kd1 32h+42d1 3h42d1 3h6kd1 32h202d1 3224k3d1 3432k23+132k238644c22++963c23+108321083236k2h+36k3h++12k3h483c2h+1444c2h+kd2 3h+2d2 3h2d2 3h24c23d1 3h72d2 38c22d2 32+202d1 32+72d2 39kd2 32+30k22d1 3)=0:(3.29) whered=36+1082+83+12h;h=p

PAGE 23

@t+@ @x+@ @x(r2)+@5 @x5=0;(4.1) wherer2=(@2 Becauseofthepreviousintroduction,weobservethefollowingchanges@ @t=!dU d;@ @x=kdU d;@ @y=mdU d;@ @z=ndU d;@2 @x2=k2d2U d2;@2 @y2=m2d2U d2;@2 @z2=n2d2U d2;(4.3)@3 @x3=k3d3U d3;@3 @x@y2=km2d3U d3;@3 @x@z2=kn2d3U d3;@5 @x5=k5d5U d5: d+kUdU d+k(k2+m2+n2)d3U d3+k5d5U d5=0:(4.4) Nowweintegrateoncethewholeequationandweget!U+1 2kU2+k(k2+m2+n2)d2U d2+k5d4U d4=0:(4.5)17

PAGE 24

Thisnewintroductionyieldsthefollowingnecessarychanges:dU d=~y0dS d~y;d2U d2=~y00dS d~y2;d3U d3=~y000dS d~y+3~y00~y0d2S d~y2+~y03d3S d~y3;(4.7)d4U d4=~y(4)dS d~y+4~y0~y000d2S d~y2+6~y02~y00d3S d~y3+~y04d4S d~y4: 2kS2+[k(k2+m2+n2)~y00+k5~yiv]dS d~y+[k(k2+m2+n2)~y02+k5(4~y0~y000+3~y002)]d2S d~y2+6k5~y02~y00d3S d~y3(4.8)+k5~y04d4S d~y4=0:

PAGE 25

2kS2++[24k5~y5+(2km2+2k340k5+2kn2)~y3+(2km22kn22k3++16k5)~y]dS d~y+[36k5~y6+(km2+k3+kn280k5)~y4+(52k52km22kn22k3)~y28k5+km2+k3+kn2]d2S d~y2++[12k5~y+36k5~y336k5~y5+12k5~y7]d3S d~y3++[4k5~y2+6k5~y44k5~y6+k5~y8+k5]d4S d~y4=0:(4.10) Letustakealookatthelinearandnonlineartermofhighestorderin(4.10).Wecanseethattherearemorethanonelineartermthatcanbeconsideredofhighestorderbutallofthemyieldthesamedegree.Consideroneofthem,say24k5~y5dS d~y.FromtheproposedsolutionforS(~y)in(4.6)itfollowsthatthetermofhighestorderinS(~y)hasadegreeofR,thusdS d~yhasadegreeofthehighesttermofR1.Thereforetheabovechosenlineartermofhighestorderin(4.10)hasadegreeofR+4.TheNonlineartermofhighestorder(andtheonlynonlinearterm)in(4.10)is1 2kS2.KnowingtheproposedsolutionforS(~y)in(4.6),itisobviousthatthedegreeofthenonlineartermis2R.BalancingtheLineartermofhighestorderwiththeNonlineartermofhighestorder,weget2R=R+4whichyieldsR=4.Sonow,bysubstitutingthisvalueforRintheequation(4.6),wegetthefollowingforthesolutionofS(~y):U()=S(~y)=4Xr=0ar~yr=a4~y4+a3~y3+a2~y2+a1~y+a0:(4.11) NowwendthederivativesofSwithregardsto~ydS d~y=4a4~y3+3a3~y2+2a2~y+a1;d2S d~y2=12a4~y2+6a3~y+2a2d3S d~y3=24a4~y+6a3;d4S d~y4=24a4:(4.12)19

PAGE 26

2ka24)~y8+(ka4a3+360k5a3)~y7+(1 2ka23+20km2a4+20k3a4+ka4a22080k5a4+20kn2a4+120k5a2)~y6+(12kn2a3+24k5a1+ka4a1+ka3a2+12km2a3+12k3a3816k5a3)~y5+(1 2ka22+1696k5a432k3a4+6km2a2240k5a2!a4+6k3a232kn2a4+ka4a0+6kn2a2+ka3a132km2a4)~y4+(40k5a1+2km2a118kn2a3+576k5a318km2a3+ka3a0+ka2a1++2kn2a1+2k3a118k3a3!a3)~y3+(ka2a0480k5a4+1 2ka21+12kn2a48km2a2+12km2a4+12k3a4++136k5a28k3a28kn2a2!a2)~y2+(6kn2a3+6k3a3+ka1a0!a12k3a1+16k5a12km2a1120k5a3++6km2a32kn2a1)~y+2km2a2+2kn2a2+2k3a216k5a2+24k5a4!a0+1 2ka20=0:(4.13) Comparingthecoecientsofeachpowerof~yinbothsides,wecangetthefollowingresult:From~y8840k5a4+1 2ka24=0(4.14) whichgivesusthesolutionfora4:a4=1680k4ora4=0:

PAGE 27

2ka23+20km2a4+20k3a4+ka4a22080k5a4+20kn2a4+120k5a2=0;(4.16) andknowingthevaluesfora4anda3,wegetthevalueofa2:a2=280 13m2280 13k2+2240k4280 13n2: Fromthis,thevalueofa1isa1=0: 2ka22+1696k5a432k3a4+6km2a2240k5a2!a4+6k3a232kn2a4+ka4a0+6kn2a2+ka3a132km2a4=0;(4.18) andwethenexpressa0asafunctionof!:a0=1 507(507!k37280k4n27280m2k431k431m4 k4:(4.19) From~y3weget0=0whichisTautology.21

PAGE 28

2ka21=0:(4.20) Whenweusethepreviouslycomputedvaluesfora4,a3,a2,a1anda0,then!cancancelout.Sothisequationleadstosomerestrictionsfortheconstantk.Factoringtheresultingequationforkandpluggingthepreviouslycomputedvaluesfora4,a3,a2,a1anda0,wesetallfactorsasfollows:280 6591k3;(52k4+n2+k2+m2);(4.21)(270402k81612k61612k4n21612m2k4+31k4++62m2k2+62k2n2+62m2n2+31m4+31n4) Wearegoingtousethesecondfactortoobtainrealsolutionsfork.Aftersolvingthefollowingequationfork:52k4+n2+k2+m2=0; 52q 52q 52q 52q 52q

PAGE 29

2ka20=0:(4.25) Now,everyvariableinthisequationiseitherknownorisafunctionof!,andthusinthisequationtheonlyunknownvariableis!.Since(4.25)isaquadraticequationof!,wegettwosolutionsfor!:!=1 507k3p whereh1=1377958400k12n23+451360m6k4+14423136k8m42+11532m2k2n4++14423136k8n42+28846272k10m22+1354080k6n4+1354080m2k8++1354080k4n4m2+28846272k8m22n2+1354080k4n2m4++2708160k6n2m2+451360k4n6+11532k4m2n2+1354080k8n2++1354080m4k61377958400k12m23+28846272k10n22++11532m4k2n2+961m8+961n8+310670563844k16+961k8++14423136k1221377958400k143+5766k4m4+3844k6m2+3844k6n2++5766k4n4+451360k10+3844m6k2+5766m4n4+3844m6n2++3844k2n6+3844n6m2; 507k3p

PAGE 30

507k4(7280k4n27280m2k431k431m462m2k262k2n2)1 507k4(31n462m2n2+2649922k87280k6);(4.27) where!andkhavethevaluespreviouslycomputed.So,nowbysubstitutingthepreviouslycomputedvaluesforallvariables,wegetthesolutionfor(4.1):(x;y;z;t)=1680k4tanh4(kx+my+nz!t)++h280 13(k2+m2+n2)+2240k4itanh2(kx+my+nz!t)++a0;(4.28) wherea0,!andkhavethepreviouslycomputedvalues.Ifweusetheothervaluefora4,whichis0,thenwegetthefollowingresultfortheothervariablesfromcomperingthecoecientsinfrontofeachpowerof~yin(4.13):a3=0;a2=0;a1=0:24

PAGE 31

2ka0)=0; 2ka0: 13(k2+m2+n2)+2240k4itanh2(kx+my+nz!t)+a0;

PAGE 32

Figure1:=1;=1;=30;=2;=2;=2;k=126

PAGE 33

Figure3:=5;=5;=10;=2;=2;=10;k=127

PAGE 39

@t6@ @x+@3 @x3=@ @x+@3 @x3+@5 @x5;@ @t6@ @x+@ @x+@3 @x3=k@ @x+@3 @x3+@5 @x5;


printinsert_linkshareget_appmore_horiz

Download Options

close
Choose Size
Choose file type
Cite this item close

APA

Cras ut cursus ante, a fringilla nunc. Mauris lorem nunc, cursus sit amet enim ac, vehicula vestibulum mi. Mauris viverra nisl vel enim faucibus porta. Praesent sit amet ornare diam, non finibus nulla.

MLA

Cras efficitur magna et sapien varius, luctus ullamcorper dolor convallis. Orci varius natoque penatibus et magnis dis parturient montes, nascetur ridiculus mus. Fusce sit amet justo ut erat laoreet congue sed a ante.

CHICAGO

Phasellus ornare in augue eu imperdiet. Donec malesuada sapien ante, at vehicula orci tempor molestie. Proin vitae urna elit. Pellentesque vitae nisi et diam euismod malesuada aliquet non erat.

WIKIPEDIA

Nunc fringilla dolor ut dictum placerat. Proin ac neque rutrum, consectetur ligula id, laoreet ligula. Nulla lorem massa, consectetur vitae consequat in, lobortis at dolor. Nunc sed leo odio.