MANAS Journal of Engineering. Volume 5 (Issue 3) (2017) Pages Formulas for Solutions of the Riccati s Equation

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1 MANAS Journal of Engineering MJEN Volume 5 (Issue 3) (7) Pages 6-4 Formulas for Soluions of he Riccai s Equaion Avy Asanov, Elman Haar *, Ruhidin Asanov 3 Dearmen of Mahemaics, Kyrgy-Turkish Manas Universiy, Bishkek, Kyrgysan Dearmen of Mahemaics, Igdir Universiy, Igdir, Turkey 3 Dearmen of Alied Mahemaics and Informaics, Kyrgy sae Technical Universiy Bishkek, Kyrgysan Absrac: Keywords: Received: 3-6-7; Acceed: --7 In his aer we obained he formula for he common soluion of Riccai equaions. Here Riccai equaions was solved for common cases. Resuls obained have been comared wih he convenional ones and a commen has been made on hem. Riccai s equaion,aricular soluion, formula for he general soluaıon. Формулы о рещении уравнений Риккати Аннотация: Ключевые слова: На этом работе мы получили формулу для общего решения уравнений Риккати Для общего случае мы получили решений уравнения Риккати. Полученные результаты соответствует классическими результатами. Уравнение Риккати,частный решения, формулу для общего решения. Corresonding Auhor: Haar E., elmanaliyev@homail.com MJEN MANAS Journal of Engineering, Volume 5 (Issue 3) 7

2 Asano, Haar and Asanov, Formulas for Soluions of he Riccai s Equaion. INTRODUCTION We consider he equaion () ( ) a( ) b( ) ( ) c( ), I where I (, ), a (), b () and c () are known coninuous funcions, a ( ) for all I. Many works are dedicaed o he deerminaion of he common soluions of Riccai equaions[- 6]. Bu in common case any formulas for he decision of Riccai s equaıons have no obained. I is well known ha any equaion of he Riccai equaion can always be reduced o he linear Differenial equaions of he second order. In [7] was obained he formula for he general soluions of he linear Differenial equaion of he second order wih he variable coefficiens in he more common cases. In his heme he equaions () is invesigaed in he more common cases.. FORMULA FOR SOLUTION OF THE EQUATION () Deending on he correlaion beween a (), b () and c () formulas for he deerminaion of he arıcular soluıon and he common soluions of his equaion () were obained. Theorem. Le i be a (), b (), c( ) C( I), a( ) for all I, a () c( ) ( ( )) e ( ) ( ) d ( ) ( ) d a ( ) ( ) m ( )( ( )) m ( ) e [ b( ) ]( ( )) m( ) ( ) m( ) a( ) ( ) ( ( )) a( ) ( ) + ( )( ( )) ( ) ( )[ b( ) ]( ( )), I, a( ) a( ) ( ) () where () and () are he firs and second derivaives of he funcions (), a( ), ( ) and m() resecively, he derivaives of he funcions a( ), ( ) and m (), m ( ) and ( ) for all I. Then he arıcular soluion of he equaion ()is wrien as in he nex form. () ( ) ( ) d ( ) [ ( ) m( ) e ], I (3) a () and he general soluion of he Riccai equaion () is given by ( ) ( ) d ex{ ( )[ ( ) m( ) e ] b( )} ( ) ( ) ( ) ( ) d C a( )ex{ ( )[ ( ) m( ) e ] b( )} d (4) where I,C is an arbirary consan. Proof. Differeniaing (3), we obain MJEN MANAS Journal of Engineering, Volume 5 (Issue 3) 7 7

3 Asano, Haar and Asanov, Formulas for Soluions of he Riccai s Equaion a( ) ( ) ( ) ( ) ( ) d ( ) ( ) ( ) d ( ) [ ][ ( ) m( ) e ] [ ( ) m( ) e a () a( ) a( ) ( ) ( ) d m( ) ( ) ( ) e ]. (5) Hence aking ino accoun (3), (5) and () we have a( ) ( ) ( ) ( ) ( ) d ( ) ( ) ( ) d ( ) a( ) ( ) b( ) ( ) [ ][ ( ) m( ) e ] [ ( ) m( ) e a () a( ) a( ) ( ) ( ) d ( ( )) ( ) ( ) d ( m( ) ( ) ( )( ) e ] [ ( ) ( ) m( ) e ) ( ) ( ) ( ) m ( ) e d b ] [ ( ) a () a () ( ) ( ) d m( ) e ] c( ), I. Therefore, () will be a aricular soluion of he equaion (). I is known ha if one can find a aricular soluion () o he equaion (), hen he general soluion can be wrien as ( ) ( ), (6) u () where u () is he general soluion of an associaed linear differenial equaion Solving his equaion (7) we have u( ) [ a( ) ( ) b( )] u( ) a( ), I (7) ( ) ( ) u( ) e [ C e a( ) d], (8) where ( ) a( ) ( ) b( ) ( ){ ( ) m( )ex[ ( ) ( ) d]} b( ), I, (9) Taking ino accoun (8) and (9) from (6), we obain(4). Theorem has been roved. Corollary. Le he condiions of Theorem hold. Then he funcion given by he formula ( s) ( s) ds ( ( ))ex{ ( )[ ( ) m( ) e ] b( )} ( ) ( ), s ( s) ( s) ds ex{ ( )[ ( ) ( )] ( )} ( ( )) ( )ex{ ( )[ ( ) ( ) ] ( )} m b a s s s m s e b s ds is he soluion of equaion () wih iniial condiion ( ), where ( s) ( s) ds () ( ) [ ( ) ( ) ],,. m e I a () MJEN MANAS Journal of Engineering, Volume 5 (Issue 3) 7 8

4 Asano, Haar and Asanov, Formulas for Soluions of he Riccai s Equaion 3. COMPARISON WITH KNOWN RESULTS. According o he given sudy [5](..8,) for he equaion f ( ) a a f, a R, () () ahas been shown a aricular soluion. The equaion () rovides all requiremens of he our heorem for a( ), b( ) f ( ), c( ) a a f ( ), ( ), ( ), m( ) a, a I. Then from (3)we have () a.. In [5](..8,3) for he equaion f ( ) f ( ) () (), Ihas been shown a aricular soluion.the equaion () rovides all requiremens of he heorem for a( ), b( ) f ( ), c( ) f ( ), ( ), ( ), m( ), a I. Then from (3) we obain (), I. 3. In [5](..8,) for he equaion f f g g has been shown a aricular soluion. The equaion () rovides all necessary condiions of he Theorem for ( ) ( ) ( ) ( ), () Then from (3)we have a( ) f ( ), b( ) f ( ) g( ), a( ), c( ) g( ), m( ), ( ), I. ( ), I. 4. In [5](..8,3) for he equaion f f g g g ( ) ( ) ( ) ( ) ( ) (3) ( ) g( ) I, has been shown a aricular soluion. The equaion (3) rovides all necessary condiions of he heorem for MJEN MANAS Journal of Engineering, Volume 5 (Issue 3) 7 9

5 Asano, Haar and Asanov, Formulas for Soluions of he Riccai s Equaion Then from (3) we have ( ) g( ), I. a( ) f ( ), b( ) f ( ) g( ), c( ) g( ), ( ), ( ), m( ) f ( ) g( ), I. 5. In [5] (..8,4) for he equaion ( ) g( )( ( ) f ( )) f ( ) (4) ( ) f ( ) I, has been shown a aricular soluion. The equaion (4)rovides all necessary requiremens of he heorem for a( ) g( ), b( ) f ( ) g( ), c( ) f ( ) g( ) f ( ), ( ), ( ), I Then from (3) we have ( ) f ( ), I. 6. In [5](..8,5) for he equaion f ( ) g ( ) g( ) f ( ), (5) g () () I, has been shown a aricular soluion. The equaion (5) rovides all necessary condiions of he heorem for f ( ) g( ) f ( ) a( ), b( ), c( ), ( ), m( ), I. g( ) f ( ) f ( ) Then from (3) we have g () () I. 7. In [5] (..8,6) for he equaion ( ) ( ) ( )( ( )) (6) f f g f ( ) f ( ), I has been shown a aricular soluion. The equaion (6) rovides all necessary condiions of he heorem for Then from (3) we obain ( ) f ( ), I. 8. In [5] (..8,7) for he equaion f ( ) g( ) g( ) a( ), b( ), c( ), ( ), f ( ) f ( ) ( ), m( ), I MJEN MANAS Journal of Engineering, Volume 5 (Issue 3) 7

6 Asano, Haar and Asanov, Formulas for Soluions of he Riccai s Equaion, (7) ( ), I has been shown a aricular soluion. The equaion (7) rovides all necessary condiions of he heorem for a( ), b( ), c( ), ( ), ( ), m( ), I Then from (3) we obain ( ), I. 9. In [5] (..8,8) for he equaion ( ) ( ), (8) ae ae f f ( ), e I has been shown a aricular soluion. The equaion (8) rovides all a necessary condiions of he heorem for a( ) ae, b( ) ae f ( ), c( ) f ( ), ( ), ( ), m( ), I. Then from (3)we have ( ) e, I. a. In [5](..8.9) for he equaion, (9) ( ) f ae f ( ) ae ( ), ae I has been shown a aricular soluion. The equaion (9) rovides all necessary condiions of he heorem for a( ) f ( ), b( ) ae f ( ), c( ) ae, ( ), ( ), m( ) ae f ( ), I Then from (3)we obain ( ) ae, I.. In [5](..8.) for he equaion n n n f ( ) a g( ) an a ( g( ) f ( )) () a has been shown a aricular soluion. The equaion () rovides all necessary condiions () n MJEN MANAS Journal of Engineering, Volume 5 (Issue 3) 7

7 Asano, Haar and Asanov, Formulas for Soluions of he Riccai s Equaion of he heorem for a f b a g c an a g f n n n ( ) ( ), ( ) ( ), ( ) ( ( ) ( )), ( ), ( ), m( ) ae f ( ), I. n Then from (3)we have () a.. In [5](..8,4) for he equaion f ( ) g( ) ae ae g( ) a e f ( ), () () ae has been shown a aricular soluion. The equaion () rovides all necessary condiions of he heorem for a f b g c a e ae g a e f ( ) ( ), ( ) ( ), ( ) ( ) ( ), ( ), ( ), m( ) ae f ( ), I. Then from (3) we have () ae. 3. In [5](..8,35) for he equaion, () f ( ) ( a ln ) f ( ) aln a ( ) a ln has been shown a aricular soluion. The equaion () rovides all necessary condiions of he heorem for a( ) f ( ), b( ) f ( )( a ln ), c( ) a ln a, ( ), ( ), m( ) a(ln ) f ( ), I. Then from (3) we have ( ) a ln. 4. In [6,case] were invesigaed he equaion (), when d b( ) f ( ) b ( ) f () c ( ) [ ] d a( ) 4 a( ) f ( ) b( ) b( ) a( ) a( ) ( ) { [ b ( ) ] [ b( ) 4 f ( ) b ( ) a f( ) b ( )] f( )}, I 4 (3) MJEN MANAS Journal of Engineering, Volume 5 (Issue 3) 7

8 Asano, Haar and Asanov, Formulas for Soluions of he Riccai s Equaion where b( ), f ( ), a( ) C( I), a( ) for all I, f( ) b ( ), I.Then he general soluions of he Riccai equaion () were given by f ( ) b ( ) d e b( ) f ( ) b ( ) () f ( ) b ( ) d C ( ) a() e d a, where C is arbirary consan. In his case he equaion () rovides all necessary condiions of he heorem for, ( ), ( ), m( ) [ b( ) f( ) b ( )], I, where c () is defined by formulas (3).Then from(3) we have ( ) [ b( ) f ( ) b ( )], I. a ( ) 5. In [6,case7] were invesigaed he equaion (), when a( ) c( ) [ f 4 ( ) f4 ( ) f4 ( ) b( ) f4( )], I, (4) a( ) a( ) 4 where f 4 ( ), a( ), b( ) C( I), a( ) for all I. Then he general soluions of he equaion () were given by [ b( ) f4 ( )] d e f4 () () I C a( )ex{ [ b( ) f ( )] d} 4 a ( ) (5) where C is arbirary consan. In his case on he srengh of (4), he equaion () rovides all necessary condiions of he heorem for f4 Then from (3) we obain ( ) (), I. a ( ) ( ), ( ), m( ) f4( ), I. 6. In [6,case]were invesigaed he equaion (), when b ( ) 4 a ( ) f5 ( ) a( ) b( ) b( ) c( ) f 5( ), I, (6) 4 a( ) a ( ) a( ) MJEN MANAS Journal of Engineering, Volume 5 (Issue 3) 7 3

9 Asano, Haar and Asanov, Formulas for Soluions of he Riccai s Equaion where a( ), b( ), f 5( ) C( I), a( ) for all I. Then he general soluions of he equaion () were given by ex{ a( ) f5( ) d} b () ( ) f5 ( ), I, C c( )ex{ a( ) f ( ) d} d a ( ) 4 5 where C4 is arbirary consan. In his case on he srengh of (6) he equaion () rovides all necessary condiions of he heorem for ( ), ( ), m( ) b( ) a( ) f5( ), I. Then from (3) we obain b () ( ) f5( ), I. a ( ) 4.CONCLUSIONS In he resen aer,we have obained he inegrabiliy condiions of he Riccai Equaion (). The aricular soluion and he general soluions of he Riccai equaions () is resened in he common cases and will be wrien in he forms (3) and (4).The ossibiliy of he alicaion of he obained resuls is also considered. REFERENCES []. Basami A.A.L., Belic M.R.., Perovic N.Z.., Secial Soluions of he Riccai Equaion wih Alicaions o he Gross-Piaevskii Nolinear PDE, 66, (), -. []. Boyle P.P., Tian W., Guan F., The Riccai Equaion in Mahemaical Finance, Journal Symbolic Comuaion, 33,(), [3]. Egorov A.I., Ricca Equaions, Moskow, Nauka, (). [4]. Kamke E., Handbook of Ordinary Differenial Equaions, Moskow, Nauka, (976). [5]. Polyanin A.D., Zaysev V.F., Handbook of Exac Soluions for Ordinary Differenial Equaions, Boca Raon-NewYork, CRC Press, (3). [6]. Tiberin H., Francisco S.N., Lobo M.K.M., Analiical Soluions of he Riccai Equaion wih Coefficiens Saisfying Inegral or Differenial Condiions wih Arbirary Funcions, Journal of Alied Mahemaics,, (4), 9-8. [7]. Asanov A., Chelik M.H., Asanov R., Formulas for Soluion of he Linear Differenial Equaions of he Second Order wih he Variable Coefficiens, Journal of Mahemaics Research, 3, (), MJEN MANAS Journal of Engineering, Volume 5 (Issue 3) 7 4