Solving Parabolic Partial Delay Differential. Equations Using The Explicit Method And Higher. Order Differences

Transcription

1 Jornal of Kfa for Maemacs and Compe Vol. No.7 Dec pp 77-5 Solvng Parabolc Paral Delay Dfferenal Eqaons Usng e Eplc Meod And Hger Order Dfferences Asss. Prof. Amal Kalaf Haydar Kfa Unversy College of Edcaon for Women Ma. Deparmen Absrac In s paper we se e ger order dfferences for second order dervave n solvng parabolc paral delay dfferenal eqaons by sng e eplc meod and we ge resls are more closer o e eac vales an e resls wc can be obaned f e famlar second order dervave form s sed. Fnally we mae a comparson sng Malab beween e resls rog wo ables of vales form ] wc resls from aylor seres epansons of a fncon cenered on e grd pon. In s paper we se e ger order dfferences for second order nsead of e famlar form for solvng e parabolc delay dfferenal eqaons.. Inrodcon Delay dfferenal eqaons play an mporan role n many lfe applcaons say problems of mng of lqds poplaon grow conrol sysems mecancal and elecrcal sysems. In addon many pyscal and engneerng problems can be modeled maemacally n e form of paral delay dfferenal eqaons PDDE's. Ineres n eqaons of s ype see for eample ] and ] ave conned o grow as as become apparen a ey are also of mporance n area of bomedcal modelng especally pysologcal and ormonal conrol sysems. I s well nown a PDDE s are dfferenal eqaons n wc e nnown fncon depends on wo or more ndependen varables and s paral dervaves w several vales of e delay. Mos researces dscss e meods for solvng paral delay dfferenal eqaons of second order by sng e famlar second order. Classfcaon of Second Order PDDE's Consder e followng dfferenal eqaon paral delay a b c d e f g. were a b c d e f g and are nown fncons of and. e classfcaon of second order paral delay dfferenal eqaon. s smlar o a n paral dfferenal eqaons. If b ac en eqaon. s sad o be of e yperbolc ype.. If b ac en eqaon. s sad o be of e parabolc ype.. b ac en eqaon. s sad o be of e ellpc ype. 7

2 Amal Kalaf Haydar 7. Eplc Meod for Solvng PDDE s e eplc meod ]5] and 6] s one of e nmercal meods a se for solvng paral dfferenal eqaons.in s secon we sall se s meod n solvng e ea eqaon w consan delay. s eqaon aes e form L. sbec o e bondary condons L. and nal condon f L.. Also nal delay condon s gven by L. were s a consan and s e delay erm. o appromae e solon of s problem we frs selec wo mes consans and w L m s an neger nmber.e grd pons are were... m and... e ea eqaon. can be wren a e neror grd pon... - m and... as..5 Dervaon of e dfference eqaon n or meod s obaned by sng e forward dfference qoen of ].6 sc a and e dfference qoen of e for order formla of ] sc a. Applyng.6 and.7 n eqaon.5 gves ] 6 6 ]..8 Solvng eqaon.8 for gves. ] If we ave. 5

3 Jornal of Kfa for Maemacs and Compe Vol. No.7 Dec pp were... m.... If we neglec e error erm we ge. 5 w w w w w w. were w s e appromae solon of. a e grd pon m Now e nal condon f L mples a w f... m and e bondary condons L mply a w w.... e eplc m nare of e dfference meod a epressed n eqaon. can be wren n erms of marces form as W sc a AW B W w w w m....5 W w w w m....6 W f f f m A B w w... m m wm m wm ].7 I s clear a for fndng e solon of. sng eqaon. we ms ave w and wm.....erefore we sall se e eplc meod ] wo sng ger order dfferences o fnd ese vales as follows.8 W A... W B sc a A W w w... w m.9 W w w... wm. 7

4 Amal Kalaf Haydar f f... f W m. B... ]. m and A 8 8 and. Eample.. Consder e parabolc paral delay dfferenal eqaon. 5 A w e bondary and nal condons. and nal delay condon.5 e..6 e eac solon of s eample as e form e..7 For s PDDE f we sppose a. and. en we ave Frs we ms fnd e vales m w and w wen...99 sng e above mar A and e relaons.8 and.-.. en we can se e mar A and e relaons o fnd w were... m and...99 gven n able.. e nmercal resls wen.. and s comparson w e eac solon of e eqaon. are gven n able. sc a e colmns nmercal ] and nmercal represen e nmercal solons of e eqaon. sng e eplc meod wo and w ger order dfferences respecvely. Smlarly we can fnd e nmercal resls gven n able. wen.and. 5

5 Jornal of Kfa for Maemacs and Compe Vol. No.7 Dec pp 77-5 Eac Sep leng. =. and = Nmercal Error Nmercal Error able. 5

6 Amal Kalaf Haydar Eac Sep leng. =. and = Nmercal Error Nmercal Error able. 5

7 Jornal of Kfa for Maemacs and Compe Vol. No.7 Dec pp 77-5 I s clear form able. and able. a e resls n e colmn nmercal s more closer o e eac vales an e resls n e colmn nmercal ]. In oer words we can ge a beer resls f we se ger order dfferences for second order nsead of e famlar second order dfferences. References. M. Bodnar e Nonnegavy of Solons of Delay Dfferenal Eqaons Appled Maemacs Leers Vol. pp S. Y P.W. Nelson and A.G.Ulson 7 Delay Dfferenal Eqaons Va e Mar Lamber W Fncon and Bfrcaon Analyss : Applcaon o Macne ool Caer Maemacal Boscences and Engneerng Vol. No. pp G. Jasm7 On e Solons of Lnear Paral Delay Dfferenal Eqaons M.sc. ess Unversy of Bagdad Bagdad-Iraq.. H. C. Saena 8 Fne Dfferences and Nmercal Analyss Ram Nagar New Del. 5. L. V. Fase 8 Appled Nmercal Analyss Usng Malab nd ed. Pearson Prence-Hall New Yor. 6. W. F. Ames977 Nmercal Meods For Paral Dfferenal Eqaons nd ed. Academc press Inc. حل المعادالت التفاضمية التباطؤية الجزئية المكافئة باستخدام الطريقة الصريحة والفروقات من الرتب العميا أ.م.آمال خمف حيدر جامعة الكوفة كمية التربية لمبنات قسم الرياضيات المستخمص في هذا البحث نستخدم الفروقات من الرتب العميا لمرتبة المشتقة الثانية في حل المعادالت التفاضمية التباطؤية الجزئية المكافئة باستخدام الطريقة الصريحة وحصمنا عمى نتائج تكون اقرب إلى القيم الحقيقية من النتائج التي يمكن الحصول عميها عند استخدام الصيغة المألوفة لمرتبة المشتقة الثانية. أخي ار عممنا مقارنة بين النتائج من خالل جدولين من القيم باستخدام. Malab 5