solving some kinds of linear third order P.D.Es.

Transcription

1 orna o o o ation or Pr Sins Vo. No. U th assmption or soin som kins o inar thir orr P..s. i assan Mohamm Ka Unirsit o o ation or irs Math. partmnt raq aabass85@ahoo.om Sattar Nasr Ktab Ka Unirsit Rsarh an normatis opmnt ntr raq s_a@ahoo.om STRT: Th main aim o this rsarh is to in th ompt sotion o som kins o inar partia irntia qations o thir orr with onstant oiints whih ha th nra orm = th assmption This assmption wi transorm th abo qation to th non-inar son orr orinar irntia qation with two inpnnt ntions whih ha th nra orm. Not :- w s insta o aso insta o.. kwors: inar thir orr P..s inpnnt ntions non-inar son orr orinar irntia qation الجزئية الخطية من الرتبة الملخص : الهذف الرئيسي لهزا البحث هى إيجاد الحل التام لبعض أصناف المعادالت التفاضلية الثالثة رات المعامالت الثابتة والتي صيغتها العامة = باستخذام الفرضية :- هزه الفرضية سىف تحىل المعادلة أعاله إلى معادلة تفاضلية اعتيادية ال خطية من الرتبة الثانية بذالتين مستقلتين والتي صيغتها العامة. 7

2 Th Proins o th th onrn o o o ation or Pr Sins - NTROUTON Man o rsarhs tri to in nw mthos or soin P..s. Th rsarhr Kar 56 sti th inar son orr O..s whih ha th orm P Q an s th assmption to in th nra sotion o it an th sotion pns on th orms o P an Q. Th rsarhr b -Saa 6 sti th inar son orr P..s with onstant oiints an whih ha th orm =. Whr an ar arbitrar onstants an s th assmption to in th ompt sotion o it an th sotion pns on th as o an. Th rsarhr ani 8 sti th inar son orr P..s whih ha thr inpnnt ariabs an it s nra orm + + t + + t + tt t += Whr an w t t t ar arbitrar onstants an s th assmption to in th ompt sotion o it. Th sotion pn on th as o an. Th rsarhr anon 9 sti th inar son orr P..s with ariab oiints whih ha th orm = Whr som o an ar ntions o or or both an. To so this kin o qations sh s th assmptions an Ths assmptions i th ompt sotion o th abo qation an th sotion pns on th orms o an. ina th rsarhr Moh 6 sti th noninar son orr P..s o homonos r whih ha th nra orm Whr an ar inar ntions o pnnt ariab an partia riatis o pnnt ariab with rspt to th inpnnt ariabs an b th oowin assmptions 8

3 orna o o o ation or Pr Sins Vo. No. an to in th ompt sotions o th abo kin o qations n or work w sarh ntions an sh that th assmption is th ompt sotions o inar thir orr partia irntia qations whih ha th nra orm in b = an it pns on th as o an This assmption transorms th abo P..s to th noninar son orr orinar irntia qation with two inpnnt ntions an. - Soin Spia Tps o th Linar Thir Orr Partia irntia qations with onstant oiints : Th nra orm o ths qations ar in b: = To obtain th ompt sotion o ths qations w assi th abo qation to th oowin ass : as : = i.. ==== == s.t. an ar not intia zro. as : = i.. === == s.t. an ar not intia zro. as : + + +=. i.. === == s.t. an ar not intia zro. as : =. i.. == == s.t. an ar not intia zro. as 5: = i.. == s.t. an ar not intia zro. as 6: = i.. = s.t. an ar not intia zro.. sription o th sst mtho : Lt s onsir th inar thir orr partia irntia qations whih ha th nra orm = n orr to in th ompt sotion o qation w sarh two inpnnt ntions an sh that th assmption in th ompt sotion o it this assmption wi transorm qation to noninar son orr orinar irntia qation. inin an rom th qation w t 9

4 Th Proins o th th onrn o o o ation or Pr Sins an b sbstittin an into th qation w t } { Sin so... qationis non-inar son orr orinar irntia qation an ontains two inpnnt ntions an.. Th ompt Sotion o th Linar P..s o thir orr : To in th ompt sotion o th inar partia irntia qation o thir orr with onstant oiints w o bak to th prios ass on th binnin o this papr :- as :- ==== == so th P.. is in b : = thn th qation boms

5 orna o o o ation or Pr Sins Vo. No. n th ompt sotion is in b :- i os ; = os = Whr an ar arbitrar onstants ii ; Whr an ar arbitrar onstants. Proo : Sin r aso w ann t sparat th ariabs in this qation so w sppos that = whr is an arbitrar onstant thn th ast qation boms i w t : ; tan tan ; = so n os tan os ; n th ompt sotion is in b :-

6 Th Proins o th th onrn o o o ation or Pr Sins i os ; = os = Whr an ar arbitrar onstants omain : - < < - < < ii w t : thror n n th ompt sotion is in b :- ; Whr an ar arbitrar onstants omain : - < < - < < amp : To so th P.. : hr Sin thn b th orma as in as-- i W t th ompt sotion whih is orm os 6 8 Whr an ar arbitrar onstants as :- === ==so th P.. is in b : = thn th qation boms n th ompt sotion is in b :- ; = os =. i os

7 orna o o o ation or Pr Sins Vo. No. Whr an ar arbitrar onstants ii ;. Whr an ar arbitrar onstants. Proo : Sin r aso w ann t sparat th ariabs in this qation so w sppos that = whr is an arbitrar onstant thn th ast qation boms i w t : b b tan ; =b b so b b b n os tan b os ; n th ompt sotion is in b :- os ; = os = Whr an ar arbitrar onstants omain : - < < - < < ii w t : thror n n th ompt sotion is in b :- ; Whr an ar arbitrar onstants

8 Th Proins o th th onrn o o o ation or Pr Sins omain : - < < - < < amp : To so th P.. : hr Sin thn b th orma as in as-- i W t th ompt sotion whih is orm 9 9 os Whr an ar arbitrar onstants as : === == so th P.. is in b : + + += thn th qation boms n th ompt sotion is in b i os ; = os = Whr an ar arbitrar onstants ii ; Whr an ar arbitrar onstants. Proo : Sin r w an sparab th ariabs an w an writ so = an

9 orna o o o ation or Pr Sins Vo. No. 5 i w t : tan k k ; =k k so k k k n os tan k os ; n th ompt sotion is in b :- os ; = os = whr an ar arbitrar onstants omain: - < < - < < ii w t : thror n n th ompt sotion is in b :- ; Whr an ar arbitrar onstants omain : - < < - < <. amp : To so th P.. : hr Sin thn b th orma as in as-- i W t th ompt sotion whih is orm ; os 6 Whr an ar arbitrar onstants as :- == == so th P.. is in b : = thn th qation boms n th ompt sotion is in b :-

10 Th Proins o th th onrn o o o ation or Pr Sins 6 i os ; = os = whr an ar arbitrar onstants. ii ; Whr an ar arbitrar onstants. Proo : r w ann t sparat th ariabs in this qation so w sppos that = whr is an arbitrar onstant thn th ast qation boms i w t : r r tan ; =r r so r r r n os tan r os ; n th ompt sotion is in b :- os ; = os = Whr an ar arbitrar onstants omain : - < < - < < ii w t : thror n n th ompt sotion is in b :-

11 orna o o o ation or Pr Sins Vo. No. 7 ; Whr an ar arbitrar onstants omain : - < < - < < amp : To so th P.. : hr = = = = - = - Sin thn b th orma as in as-- i W t th ompt sotion whih is orm ; 6 6 os Whr an ar arbitrar onstants as 5 : - === so th P.. is in b : = thn th qation. boms n th ompt sotion is in b :- i os ; = os =. Whr an ar arbitrar onstants ii ;. Whr an ar arbitrar onstants. Proo:Sin r aso w ann t sparat th ariabs in this qation so w sppos that = whr is an arbitrar onstant thn th ast qation boms

12 Th Proins o th th onrn o o o ation or Pr Sins 8 i w t : t t tan ; =t. t so t t t n os tan t os n th ompt sotion is in b :- os ; = os = omain : - < < - < < Whr an ar arbitrar onstants ii. w t : thror n n th ompt sotion is in b :- ; Whr an ar arbitrar onstants omain : - < < - < <. Not : w sppos that w t th ompt sotion b th sam mtho. amp : To so th P.. : hr = = = - = = - = - = -

13 orna o o o ation or Pr Sins Vo. No. 9 Sin thn b th orma as in as-5-i W t th ompt sotion whih is orm os Whr an ar arbitrar onstants as 6 : = so th P.. is in b : = thn th qation boms n th ompt sotion is in b :- i os = os = Whr an ar arbitrar onstants. ii ; Whr an ar arbitrar onstants. Proo : r w ann t sparat th ariabs in this qation so w sppos that = whr is an arbitrar onstant thn th ast qation boms

14 Th Proins o th th onrn o o o ation or Pr Sins i w t : tan w w ; =w w so w w w n os tan w os ; n th ompt sotion is in b :- os ; = os = Whr an ar arbitrar onstants omain : - < < - < <. ii w t : thror n n th ompt sotion is in b :- ; Whr an ar arbitrar onstants

15 orna o o o ation or Pr Sins Vo. No. omain : - < < - < <. amp : To so th P.. : hr Sin thn b th orma as in as-- i W t th ompt sotion whih is orm 6 7 os ; 8 Whr an ar arbitrar onstants Rrns b -Saa N.. "Th ompt Sotion o Linar Son Orr Partia irntia qations" Ms thsisunirsit o Ka o o ation or irs partmnt o Mathmatis 6. ar. ointon "n ntrotion to Orinar irntia qations " Nw hi-9. ani N.N. "On Sotions o Partia irntia qations o son orr with onstant oiints " Ms thsis Unirsit o Ka o o ation or irs partmnt o Mathmatis8. anon W.. "On Sotions o Partia irntia qations an Thir Phsia ppiations " Ms thsis Unirsit o Ka o o ation or irs partmnt o Mathmatis 9. 5 Kar R.. "Soin Som Kins o Linar Son Orr Non-omonos irntia qations with Variab oiints" Ms thsis Unirsit o Ka o o ation or irs partmnt o Mathmatis 6. 6 MohL.."Soin Spia Tps o Son Orr Orinar irntia qations" Ms thsis Unirsit o Ka o o ation or irs partmnt o Mathmatis.