Lie Group Analysis of Second-Order Non-Linear Neutral Delay Differential Equations ABSTRACT

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1 Malaysian Journal of Mahemaical Sciences 0S March : Special Issue: The 0h IMT-GT Inernaional Conference on Mahemaics Saisics and is Applicaions 04 ICMSA 04 MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: hp://einspem.upm.edu.my/journal Lie Group Analysis of Second-Order Non-Linear Neural Delay Differenial Equaions Laheeb Muhsen and Normah Maan * Deparmen of Mahemaics Faculy of Science Al-Musansiriya Universiy Baghdad Iraq Deparmen of Mahemaical Sciences Universii Teknologi Malaysia Johor Bahru Malaysia normahmaan@um.my *Corresponding auhor ABSTRACT Lie group analysis is applied o second order neural delay differenial equaions NDDEs o sudy he properies of he soluion by he classificaion scheme. NDDE is a delay differenial equaion which conains he derivaives of he unknown funcion boh wih and wihou delays. I urns ou ha in many cases where rearded delay differenial equaion RDDE fail o model a problem NDDE provides a soluion. This paper eends he classificaion of second order non-linear RDDE o solvable Lie algebra o ha for second order non-linear NDDE. In his classificaion he second order eension of he general infiniesimal generaor acing on second order neural delay is used o deermine he deermining equaions. Then he resuling equaions are solved and he solvable Lie algebra is obained saisfying he inclusion propery. Finally one-parameer Lie groups which are corresponding o NDDEs are deermined. This approach provides a heoreical background for consrucing invarian soluions. Keywords: Neural delay differenia equaion Lie group analysis Lie group Lie algebra one-parameer Lie group.. Inroducion Delay differenial equaion DDE was iniially inroduced in he 8h cenury by Laplace and Condorce see Gorecki e al I arises when ordinary differenial equaions ODEs fail o eplain some naural

2 Laheeb Muhsen & Normah Maan phenomena. Then DDEs have been successfully used in he mahemaical formulaion of real life phenomena in a wide variey of applicaions especially in science and engineering such as populaion dynamics infecious disease conrol problems secure communicaion raffic conrol elecrodynamics and economics Bellen and Zennaro 00; Bazel and Tran 000; Nagy e al. 00. In conras wih ordinary differenial equaions ODEs where he unknown funcion and is derivaives are evaluaed a he same insan in a DDE he evoluion of he sysem a a cerain ime depends on he sae of he sysem a an earlier ime. The delay however adds era compleiies and generally DDEs are difficul o solve. When here is no direc way o solve i we ry o arrive a suiable soluion by analyzing he properies of DDEs. The bes way o sudy he properies of he soluion of delay differenial equaion is by Lie group analysis. Lie group analysis was inroduced by Sophus Lie see Oliveri 00 i is considered o be an effecive mehod for sudying he properies of differenial equaion DE. Lie developed heories on coninuous groups which are called Lie groups. Since hen Lie group analysis has been widely eploied Bluman and Kumei 989; Hill 98; Olver 99. Tanhanuch and Meleshko 004 defined an admied Lie group for funcional differenial equaion FDDE which helped Pue-on 009 o inroduce group classificaion for specific cases of second order delay differenial equaion. Mos researchers deal wih Lie group analysis of DEs by making a change in space variables bu DDEs do no possess equivalen ransformaions o change he dependen and independen variables. Because of his some researchers Pue-on 009 failed o classify DDEs as Lie algebras. Recenly Muhsen and Maan 04a inroduced a classificaion of second order linear delay differenial equaion o solvable Lie algebra wihou changing he space variables. Then hey eend he classificaion o second order non-linear RDDEs o solvable Lie algebra Muhsen and Maan 04b. This paper eends he classificaion mehod o second order nonlinear neural delay differenial equaion o solvable Lie algebra and obains he one-parameer Lie group of he corresponding NDDEs. This resul is useful o sudy he properies of many naural phenomena which are described by non-linear NDDEs. The conen of he presen paper is as follows. Secion gives he principal deails abou Lie algebra and delay differenial equaions. The classificaion of second order non-linear neural delay differenial equaions o solvable Lie algebra wih main resuls are described in Secion. Secion 8 Malaysian Journal of Mahemaical Sciences

3 Lie Group Analysis of Second Order Non-Linear Neural Delay Differenial Equaions 4 concludes wih commens on he robusness and versailiy of our approach.. Preliminaries This paper proposes a classificaion of second order non-linear neural delay differenial equaions o solvable Lie algebra. We firs give some informaion on Lie algebra and delay differenial equaions. Definiion. Andreas 009: A Lie algebra L is an n-dimensional solvable algebra if here eis a sequence ha yields L L... L Here Lk is called k-dimensional Lie algebra and L k is an ideal of L k k...n in which wo dimensional Lie algebra are solvable. Definiion. Bluman and Kumei 989: Le Qi s and Q j s i j... r and s... n be wo infiniesimal generaor. s The commuaor Q i Q j of Q i and n L Q j is he firs order operaor n n s s Qi Q j QiQ j Q jqi m m s Definiion. Humi and Miller 988: A finie se of infiniesimal generaor Q Q... Q } is said o be a basis for he Lie algebra L if Q i L and { r. Q Q... Qr form a basis of he vecor space L m. Qi Q j cijkqk. m m s. s. The coefficiens cijk consans of he Lie algebra. i j k... r are called he srucure Malaysian Journal of Mahemaical Sciences 9

4 Laheeb Muhsen & Normah Maan Definiion.4 Andreas 009; Koláȓ 99: A Lie groups G is a smooh manifold and a group such ha he muliplicaion : GG G is smooh. The inversion : G G is also smooh. Theorem.5 Second Fundamenal Theorem of Lie Bluman and Kumei 989: Any wo infiniesimal generaors of an r-parameer Lie group saisfy commuaion relaion of he form Q Q c Q where i j k... r. The commuaor and he Jacobi ideniy ogeher wih he capabiliy o form real or comple linear combinaions of he Qi gives hese infiniesimal generaor he srucure of he Lie algebra associaed wih he Lie group. Noe ha in such a case he infiniesimal generaors Q Q... Q r form a basis for a Lie algebra. Theorem.6 Ibragimov 999: For any variable he funcion F is an invarian under he Lie group of ransformaion if and only if XF 0 where X is an infiniesimal generaor. Theorem.7 Bluman and Kumei 989: The one-parameer Lie group F is equivalen o Q e Q Q Q... his is called Lie series. This! heorem gives an approach o find a one-parameer group. Lemma.8 Muhsen and Maan 04a: The second order neural delay differenial equaion conaining he infiniesimal generaor ha obeys periodic propery is given by. Lemma.8 implies ha does no depend on. Now le f where and. i j ijk k 0 Malaysian Journal of Mahemaical Sciences

5 Lie Group Analysis of Second Order Non-Linear Neural Delay Differenial Equaions Malaysian Journal of Mahemaical Sciences Equaion is a second order NDDE he general infiniesimal generaor of is X where and. By Lemma.8 he deermining equaion for is of he form 0 f X where X and 4 Algorihm.9 Muhsen and Maan 04b Classificaion of second order non-linear RDDEs: This algorihm is used o classify second order non-linear rearded delay differenial equaion o solvable Lie algebra. i. Wrie he delay differenial equaion in he solved form. ii. Wrie he general infiniesimal generaor of he delay differenial equaion.

6 Laheeb Muhsen & Normah Maan iii. Eend he infiniesimal generaor acing on. iv. Apply he eended infiniesimal generaor o he given delay differenial equaion o obain invariance condiion. v. Subsiue Equaions 4 in he invariance condiion. vi. Spli up invariance condiions by powers of he derivaives o give deermining equaions for he infiniesimal symmery group. vii. Then hese deermining equaions are solved in he following seps a. Find he general soluion of and. b. Subsiue hese resuls in an equaion ha does no depend on he derivaives o obain a polynomial of. c. Solve he polynomial by comparing coefficien mehod. d. Find he soluion of. Then subsiue he resul o obain he specific soluion of and. viii. Subsiue he infiniesimals and in he general infiniesimal generaor. i. Span Lie algebra of he given equaion by he hree infiniesimals generaors corresponding o each c i i... n where ci are arbirary consans.. Compue he commuaor able of he basis for Lie algebra. i. If he basis for Lie algebra saisfies he inclusion propery hen he solvable Lie algebra is obained.. Classificaion of Second Order Non-Linear NDDES o Solvable Lie Algebra In his secion a classificaion of non-linear homogenous and nonhomogenous second order neural delay differenial equaion o solvable Lie algebra is presened. We use Algorihm.9 o complee he classificaion of second order non-linear NDDEs wih eension in seps iii. Eend he infiniesimal generaor acing on second order neural delay differenial equaion insead of rearded delay. vi. Spli up invariance condiions by powers of he derivaives o give deermining equaions for he infiniesimal symmery group. vii. b. Subsiue hese resuls in an equaion ha does no depend on he derivaives o obain a polynomial of. Malaysian Journal of Mahemaical Sciences

7 Lie Group Analysis of Second Order Non-Linear Neural Delay Differenial Equaions Malaysian Journal of Mahemaical Sciences To find he one-parameer group one need o add anoher sep ii. Applied Theorem.7 on he resuls o ge he one-parameer Lie group of he corresponding equaion. Eample: Consider he second order non-linear homogenous NDDE 0. 5 The general infiniesimal generaor associaed wih Equaion 5 is. X 6 The second order eension of 6 ha is acing on neural delay is X X. 7 We ge he invariance condiions by applying 7 o 5 0 where and. Now subsiuing he formula from 4 o obain 0 where. Equaing he coefficiens of he various monomials in he firs second orders of and we ge he following deermining equaions lis in Table for he symmery group of Equaion 5.

8 Laheeb Muhsen & Normah Maan TABLE : The deermining equaions for he symmery group of Equaion 5 MONOMIAL COEFFCIENT NUMBER OF EQUATION 0 a 0 a 0 a 0 a 4 0 a 5 0 a 6 0 a 7 0 a 8 0 a 9 0 a 0 0 a From a 5 does no depend on. From a is linear in so g h where g and h are arbirary funcions of. From a 6 g. From a 0 is independen of. From Lemma.8 so and g. From a g. This implies ha g k where k is an arbirary funcion. If 0 hen k h. Now from a g h g k g h g k 0. Equaing he coefficiens of he various erms we obain g Malaysian Journal of Mahemaical Sciences g g k 0 9 h k h 0 0 which means ha g h and k are he soluions of 5. From 8 g 0 hen g c. Since g his implies ha c c where c c. From 9 k 0 so k c 4. From 0 h h. This

9 Lie Group Analysis of Second Order Non-Linear Neural Delay Differenial Equaions implies ha h h c 5. From above c h c5 and c c 4 where c i i...5 are arbirary consans. Recall from Equaion 6 ha he infiniesimal generaor of Equaion 5 is X. Le u hen X c c c h c5 c c4 u Thus he Lie algebra of Equaion 5 is spanned by he following hree infiniesimal generaors corresponding o each c i. Q Q u Q Q4 Q5 u wih infinie dimensional Lie subalgebra Q6 h. The commuaor able is given in Table. Thus he algebra L5 { Q Q Q Q4 Q5} spanned by Q Q Q Q4 Q5 is Lie algebra of Equaion 5. The subspaces L } L Q } { Q { Q L { Q Q Q} L4 { Q Q Q Q4} are Lie subalgebras of L 5 of dimensions one wo hree four and five respecively. Furhermore hese Lie subalgebras saisfies he inclusion propery: L L L L4 L5 hence by Definiion. L5 is a solvable Lie algebra of Equaion 5. TABLE : The commuaor able for he generaors of he symmery group of Equaion 5 Qi Q j Q Q Q Q 4 Q 5 Q Q4 Q5 Q 0 0 Q 0 0 Q 0 Q Q Q 5 Q4 Q Malaysian Journal of Mahemaical Sciences 5

10 Laheeb Muhsen & Normah Maan Now by applying Theorem.7 on Q Q Q Q4 Q5 Q 6 one can ge he one-parameer Lie groups generaor by hese space. Q : n n n u n u. n0 n n0 Q : n n n u n u. n n0 n0 : u u. Q Q : 4 u u 4. Q 5 : 5 u u. Q : 6 h u u. 6 Here are he parameers of he one-parameer groups generaed by Q Q Q Q4 Q5 Q 6 respecively. Remark: Suppose Equaion 5 is non-homogenous i.e. r where r is arbirary non-inegrable funcion. Then he Lie algebra of Equaion is spanned by Q u Q Q Q u 4 Q h 5 where Q 5 is infinie dimensional Lie subalgebra and he commuaor able is shown in Table. Then he algebra L4 { Q Q Q Q4} spanned by Q Q Q Q4 is solvable Lie algebra of Equaion. 6 Malaysian Journal of Mahemaical Sciences

11 Lie Group Analysis of Second Order Non-Linear Neural Delay Differenial Equaions TABLE : The commuaor able for he generaors of he symmery group of Equaion j Q i Q Q Q Q Q 4 Q Q 0 0 Q 0 Q 0 Q 0 0 Q Applied Theorem.7 o hese space o ge Q : n n n u n u. n0 n n0 Q : n n n u n u. n n0 n0 : u u. Q Q : 4 Q : 5 u u 4. h u u. 5 Where 4 5 are he parameers of he one-parameer groups generaed by L 4 wih Q 5 respecively. 4. Conclusion This paper eends he classificaion of second order non-linear RDDEs o he classificaion of second order non-linear NDDEs as solvable Lie algebras. Then he one-parameer Lie group are obained by Lie series corresponding o NDDEs which can be used for general analysis of he equaions. These resuls and he successful implemenaion form he basis for he classificaion of non-linear delay differenial equaions of neural ype o solvable Lie algebra. Thus he classificaion of second order nonlinear DDEs o solvable Lie algebra is compleed. Resuls of his paper could be eended o higher order non-linear delay differenial equaions. Malaysian Journal of Mahemaical Sciences 7

12 Laheeb Muhsen & Normah Maan Acknowledgmens The auhors hank he Research Managemen Cener UTM and he Minisry of Higher Educaion MOHE Malaysia for financial suppor hrough research grans of voe 06H49. Laheeb is graeful o Al- Musansiriya Universiy and he Minisry of Higher Educaion Iraq for providing sudy leave and a fellowship o coninue docoral sudies. References Andreas Ĉ Lie algebras and represenaion heory. Universiä Wien: Nordbergsr. Bazel J. J. and H. T. Tran H. T Sabiliy of he human respiraory conrol sysem I. Analysis of a wo-dimensional delay sae-space model. J Mah Biol. 4: Bellen A. and Zennaro M. 00. Numerical mehods for delay differenial equaions. Numerical mahemaics and scienific compuaion. New York Clarendon Press Oford Universiy Press. Bluman G. W. and Kumei S Symmeries and differenial equaions. New York: Sprinder. Gorecki H. Fuksa S. Grabowski P. and Koryowski A Analysis and synhesis of ime delay sysems. New York: John Wiley and Sons. Hill J. M. 98. Soluion of differenial equaion by means of oneparameer groups. London. Humi M. and Miller W Second course in ordinary differenial equaions for scieniss and engineers. New York: Springer. Ibragimov N. H Elemenary Lie group analysis and ordinary differenial equaions. London: Wiley. Koláȓ I. and Slovák J. 99. Naural operaions in differenial geomery". Deparmen of Algebra and Geomery. Faculy of science. Masaryk Universiy. Berlin:Springer. 8 Malaysian Journal of Mahemaical Sciences

13 Lie Group Analysis of Second Order Non-Linear Neural Delay Differenial Equaions Muhsen L. and Maan N. 04a. New approach o classify second order linear delay diferenial equaions wih consan coeficiens o solvable Lie algebra. Inernaional journal of mahemaical analysis. 0: Muhsen L. and Maan N. 04b. Classificaion of second order non-linear rearded delay differenial equaions o solvable Lie algebra. Proceeding in Simposium Kebangsaan Sains Maemaik ke-. Shah Alam: Malaysia. Nagy T. K.Sépán G. and Moon F. C. 00. Subcriical Hopf bifurcaion in he delay equaion model for machine ool vibraions Nonlinear Dyn. 6: 4. Oliveri F. 00. Lie symmeries of differenial equaions: classical resuls and recen conribuions Symmery Journal. : Olver P. J. 99. Applicaion of Lie groups o differenial equaions. nd ed. New York. Springer Verlag. Pue-on P Group classificaion Of second-order delay differenial equaions. Commun Nonlinear Sci Numer Simul. 5: Thanhanuch J. and Meleshko S. V On definiion of an admied Lie group for funcional differenial equaions. Commun Nonlinear Sci Numer Simul. 9: 7-5. Malaysian Journal of Mahemaical Sciences 9