The First Integral Method to Nonlinear Partial Differential Equations

Transcription

1 Avalable at Appl. Appl. Math. ISSN: Vol. 7, Issue (June 0), pp. 7 3 Applcatons and Appled Matheatcs: An Internatonal Journal (AAM) The Frst Integral Method to Nonlnear Partal Dfferental Equatons N. Taghzadeh, M. Mrzazadeh and A. Sae Paghaleh Departent of Matheatcs Unversty of Gulan P.O. Box 94, Rasht, Iran rzazadehs@gulan.ac.r Receved: Noveber 06, 00; Accepted: March 7, 0 Abstract In ths paper, we show the applcablty of the frst ntegral ethod for obtanng exact solutons of soe nonlnear partal dfferental equatons. By usng ths ethod, we found soe exact solutons of the Landau-Gnburg-Hggs equaton and generalzed for of the nonlnear Schrödnger equaton and approxate long water wave equatons. The frst ntegral ethod s a drect algebrac ethod for obtanng exact solutons of nonlnear partal dfferental equatons. Ths ethod can be appled to nonntegrable equatons as well as to ntegrable ones. Ths ethod s based on the theory of coutatve algebra. Keywords: Frst ntegral ethod; Landau-Gnburg-Hggs equaton; Generalzed for of the nonlnear Schrödnger equaton; Approxate long water wave equatons MSC 00: 35Q53; 35Q80; 35Q55; 35G5. Introducton Many phenoena n physcs and engneerng are descrbed by nonlnear partal dfferental equatons (NPDEs). When we want to understand the physcal echans of phenoena n nature, descrbed by nonlnear PDEs, exact soluton for the nonlnear PDEs have to be explored. Thus the ethods for dervng exact solutons for the governng equatons are portant and have to be developed. To study exact solutons of nonlnear PDEs has becoe one of the ost portant topcs n atheatcal physcs. For nstances, the nonlnear wave phenoena observed n flud dynacs, plasa, and optcal fber are often odeled by the bell-shaped sech solutons and the knk-shaped tanh solutons. The avalablty of these exact solutons for those 7

2 8 N. Taghzadeh, M. Mrzazadeh and A. Sae Paghaleh nonlnear equatons can greatly facltate the verfcaton of nuercal solvers on the stablty analyss of ther solutons. Nonlnear dfferental equatons have a wde array of applcatons n any felds. They could descrbe the oton of solated waves, localzed n a sall part of space. Ther applcatons could extend to agnetoflud dynacs, water surface gravty waves, electroagnetc radaton reactons, and on acoustc waves n plasas. Lookng for exact soltary wave solutons to nonlnear evoluton equatons has long been a ajor concern for both atheatcans and physcsts. These solutons ay well descrbe varous phenoena n physcs and other felds, such as soltons and propagaton wth a fnte speed, and thus they ay gve ore nsght nto the physcal aspects of the probles. In order to obtan the perodc wave and solton solutons of nonlnear evoluton equatons, a nuber of ethods have been proposed, such as tanh-sech functon ethod, extended tanh functon ethod, hyperbolc functon ethod, sne-cosne ethod, Jacob ellptc functon expanson ethod, F-expanson ethod, transfored ratonal functon ethod and the frst ntegral ethod. The frst ntegral ethod s a powerful soluton ethod for the coputaton of exact travelng wave solutons. Ths ethod s one of the ost drect and effectve algebrac ethods for fndng exact solutons of nonlnear partal dfferental equatons. Dfferent fro other tradtonal ethods, the frst ntegral ethod has any advantages, whch s the avodance of a great deal of coplcated and tedous calculatons resultng n ore exact and explct travelng soltary solutons wth hgh accuracy. In the poneer work, Feng (00) ntroduced the frst ntegral ethod for a relable treatent of the nonlnear PDEs. The frst ntegral ethod s wdely used by any such as n [Taghzadeh et al. (0), Taghzadeh and Mrzazadeh (0), Taghzadeh et al. (0), Moosae et al. (0) and by the reference theren]. Taghzadeh et al. (0) proposed the frst ntegral ethod to solve the odfed KdV KP equaton and the Burgers KP equaton. The ethod was utlzed to construct exact solutons of the nonlnear Schrödnger equaton. Taghzadeh and Mrzazadeh (0) used the frst ntegral ethod to obtan the exact solutons of soe coplex nonlnear partal dfferental equatons and Konopelchenko-Dubrovsky equaton. Moosae et al. (0) solved the perturbed nonlnear Schrödnger s equaton wth Kerr law nonlnearty by usng the frst ntegral ethod. Recently, t was successfully used for constructng the exact solutons of the Eckhaus equaton [Taghzadeh et al. (0)]. The paper s arranged as follows. In secton, we descrbe brefly the frst ntegral ethod. In secton 3, we apply ths ethod to the Landau-Gnburg-Hggs equaton and generalzed for of the nonlnear Schrödnger equaton and approxate long water wave equaton.. The Frst Integral Method Step. Consder a general nonlnear PDE n the for Euu (, x, ut, uxx, uxt,...) 0. () To fnd the travellng wave solutons to Equaton (), we ntroduce the wave varable x ct, ()

3 AAM: Intern. J., Vol. 7, Issue (June 0) 9 so that uxt (, ) u ( ). (3) Based on ths we use the followng changes (.) (.), x (.) c (.), t x (.) (.), (.) c (.), t x (4) and so on for the other dervatves. Usng (4) changes the PDE () to an ODE u u Hu (,,,...) 0, (5) where u u( ) s an unknown functon, H s a polynoal n the varable u and ts dervatves. Step. Suppose the soluton of ODE (5) can be wrtten as follows: uxt (, ) f ( ), (6) and furtherore, we ntroduce a new ndependent varable f ( ) X( ) f( ), Y( ). (7) Step 3. Under the condtons of Step, Equaton (5) can be converted to a syste of nonlnear ODEs as follows X( ) Y( ), Y( ) F( X( ), Y( )). (8)

4 0 N. Taghzadeh, M. Mrzazadeh and A. Sae Paghaleh If we can fnd the ntegrals to Equaton (8), then the general solutons to Equaton (8) can be solved drectly. However, n general, t s really dffcult for us to realze ths even for one frst ntegral, because for a gven plane autonoous syste, there s nether a systeatc theory that can tell us how to fnd ts frst ntegrals, nor a logcal way for tellng us what these frst ntegrals are. We wll apply the so-called Dvson Theore to obtan one frst ntegral to Equaton (8) whch reduces Equaton (5) to a frst order ntegrable ODE. An exact soluton to Equaton () s then obtaned by solvng ths equaton. Dvson Theore. Suppose that P ( w, z ) and Qw (, z ) are polynoals n Cw [, z ], and Pw (, z ) s rreducble n Cw [, z ]. If Qw (, z) vanshes at all zero ponts of Pw (, z ), then there exsts a polynoal Gw (, z ) n Cw [, z ] such that Qw (, z) Pw (, zgw ) (, z). The Dvsor Theore follows edately fro the Hlbert Nullstellensatz Theore. Hlbert Nullstellensatz Theore. Let K be a feld and L be an algebrac closure of K. Then: () Every deal of KX [, X,..., Xn] not contanng adts at least one zero n L n. n () Let x ( x, x,..., x n ) and y ( y, y,..., y n ) be two eleents of L. For the set of polynoals of KX [, X,..., Xn] zero at x to be dentcal wth the set of polynoals of KX [, X,..., Xn] zero at y, t s necessary and suffcent that there exsts a K autoorphs S of L such that y S( x) for n. () For an deal of KX [, X,..., Xn] to be axal, t s necessary and suffcent that there exsts an n x n L such that s the set of polynoals of KX [, X,..., Xn] zero at x. (v) For a polynoal Q of KX [, X,..., Xn] to be zero on the set of zeros n L n of an deal of KX [, X,..., X n], t s necessary and suffcent that there exsts an nteger 0 such that Q. 3. Applcaton of the Frst Integral Method to Npdes 3.. The frst ntegral ethod for obtanng exact solutons of NPDEs 3.. A. Landau-Gnburg-Hggs equatons Consder the Landau-Gnburg-Hggs equatons [Khur (008)]: u u u n u 3 tt xx 0, (9)

5 AAM: Intern. J., Vol. 7, Issue (June 0) where and n are real constants. By ake the transforaton uxt (, ) f( ) f( kx ( t)), where s the wave speed and k( xt), Equaton (9) becoes If we let f ( ) 3 k ( ) f( ) n ( f( )) 0. df ( ) X f( ), Y, the Equaton (0) s equvalent to the two densonal d autonoous syste X Y, n Y X X k ( ) k ( ) 3 ( ) ( ). (0) () Accordng to the frst ntegral ethod, we suppose the X( ) and Y ( ), are the nontrval solutons of () also qxy (, ) a( XY ) 0, 0 s an rreducble polynoal n the coplex doan C[ X, Y ], such that qx ( ( ), Y( )) a ( X( )) Y( ) 0, () 0 where a ( X )( 0,,..., ), are polynoals of X and a ( X ) 0. Equaton () s called the frst ntegral to (). Suppose that n (). Note that dq s a polynoal n d X and Y, and qx [ ( ), Y( )] 0 ples dq d there exsts a polynoal g( X) h( X) Y n CXY [, ] such that () 0. Accordng to the Dvson Theore, dq ( dq. dx dq. dy ) d dx d dy d () () n 3 ( a ( X) Y )( Y) ( a( X) Y ) ( X X ) k ( ) k ( ) 0 0

6 N. Taghzadeh, M. Mrzazadeh and A. Sae Paghaleh ( g( X) h( X) Y) a ( X) Y, (3) 0 where pre denotes dfferentaton wth respect to the varable X. By coparng wth the coeffcents of Y (,,0) of both sdes of (3), we have a( X) h( X) a ( X), (4) a ( X) g( X) a ( X) h( X) a ( X), (5) 0 0 n 3 a( X)[ X X ] g( X) a 0( X). k ( ) k ( ) (6) Snce a ( X )( 0,) are polynoals, then fro (4) we deduce that a ( ) X s constant and hx ( ) 0. For splcty, take a ( X ). Balancng the degrees of g( X ) and a ( ), 0 X we conclude that deg( g ( X )) only. Suppose that g( X) B0 AX, then we fnd a 0 ( X). a0( X) AX B0X A0, (7) a X a X and g( X ) where A 0 s the arbtrary ntegraton constant. Substtutng 0( ), ( ) n the last equaton n (6) and settng all the coeffcents of powers X to be zero, then we obtan a syste of nonlnear algebrac equatons and by solvng t, we obtan n B0 0, A0, A, (8) kn ( ) k ( ) where s an arbtrary constant. Usng the condtons (8) n (), we obtan Y ( ) ( nx ( )) 0. k ( ) n (9) Cobnng (9) wth (), we obtan the exact soluton to equaton (0) and then the exact soluton to Landau-Gnburg-Hggs equaton can be wrtten as

7 AAM: Intern. J., Vol. 7, Issue (June 0) 3 uxt (, ) tanh ( kx ( t) 0), n k ( ) (0) for. uxt (, ) tan ( kx ( t) 0), n k ( ) () for. 3.. B. The Non-ntegrable Equaton The nonntegrable equaton Mˆ D D v t x v v Mˆ D D D D () 3 ( t, x) (, ) 3, ( t, x) t x, s gven by Bakov and Khusnutdnova (996). We are nterested n the exact soluton to Equaton (). Substtutng vt (, x) f( ) f( x ct), nto Equaton (), we obtan (3) 3 ( c ) f ( ) f( ) 3 f ( ) f ( ) 0. If we let autonoous syste df ( ) X f( ), Y, the Equaton (3) s equvalent to the two densonal d X Y, 3 3 Y X( ) X ( ) X ( ). c c c (4) Accordng to the frst ntegral ethod, we suppose the X( ) and Y ( ), are the nontrval solutons of (4) also qxy (, ) a( XY ) 0, 0 s an rreducble polynoal n the coplex doan C[ X, Y ], such that

8 4 N. Taghzadeh, M. Mrzazadeh and A. Sae Paghaleh (5) 0 qx ( ( ), Y( )) a( X( )) Y( ) 0, where a ( X )( 0,,..., ), are polynoals of X and a ( X ) 0. Equaton (5) s called the frst ntegral to (0). Due to the Dvson Theore, there exsts a polynoal g( X ) h( X ) Y, n the coplex doan CXY [, ], such that dq dq dx dq dy.. ( g( X) h( X) Y) a ( X) Y. (6) d dx d dy d 0 Assung that, we have by coparng wth the coeffcents of Y (,,0) of both sdes of (6), a( X) h( X) a ( X), (7) a ( X) g( X) a ( X) h( X) a ( X), (8) a( X)[ X X X ] g( X) a 0( X). c c c (9) Snce a ( X )( 0,) are polynoals, then fro (7) we deduce that a ( ) X s constant and hx ( ) 0. For splcty, take a ( X ). Balancng the degrees of g( X ) and a ( ), 0 X we conclude that deg( g ( X )), only. Suppose that g( X ) AX B0, then we fnd a ( X ). 0 a0( X ) A0 B0X AX, (30) where A 0 s arbtrary ntegraton constant. Substtutng a0( X ), a( X ) and g( X ) n the last equaton n (9) and settng all the coeffcents of powers X to be zero, then we obtan a syste of nonlnear algebrac equatons and by solvng t, we obtan A0 0, B0, A, (3) ( c ) ( c ) where c s arbtrary constant. Usng the condtons (3) n (5), we obtan Y( ) ( X ( ) X( )) 0. (3) ( c )

9 AAM: Intern. J., Vol. 7, Issue (June 0) 5 Cobnng (3) wth (4), we obtan the exact soluton to Equaton (3) and then the exact soluton to the nonntegrable equaton () can be wrtten as ( xct0 ) c e vtx (, ), e ( xct0 ) c where c. Coparng our results wth Bakov s results [Bakov and Khusnutdnova (996)], t can be seen that the results are sae. 3.. The frst ntegral ethod for obtanng exact solutons of coplex NPDEs In ths secton we study the GNLS equaton [Moghadda et al. (009)]: u au bu u cu d u u ke (33) ( ( ) ) ( ) t t xx xxx x, a real s functon and abcd,,,,,, vare non-zero constants and where ( x vt) u u( x, t) s a coplex-valued functon of two real varables x,. t We use the wave transforaton uxt e f x x vt (34) ( ( ) t) (, ) ( ), ( ) 0, ( ), where,, v,, x0 are constants. By replacng Equaton (34) nto Equaton (33) and separatng the real and agnary parts of the result, we obtan the two followng ordnary dfferental equatons: ( ) ( 3 ) ( ) 3 ( ) ( ) 0, (35) 3 3 c f a v c f d f f ( a 3 c ) f ( ) ( v a c ) f( ) 3 ( b d ) f ( ) k 0. (36) Integratng Equaton (35) once, wth respect to, yelds: ( ) ( 3 ) ( ) ( ) 0, (37) 3 c f a v c f df R

10 6 N. Taghzadeh, M. Mrzazadeh and A. Sae Paghaleh where R s an ntegraton constant. Snce the sae functon f ( ) satsfes two Equatons (35) and (37), we obtan the followng constrant condton: c a v3c d R a 3c va c bd k If we let autonoous syste df ( ) X f( ), Y, the Equaton (37) s equvalent to the two densonal d X Y, 3c va d 3 R Y X( ) X ( ). c c c (38) Equaton () s called the frst ntegral to (38). Accordng to the Dvson Theore, there exsts a polynoal g( X) h( X) Y n CXY [, ] such that dq dq dx dq dy (.. ) d dx d dy d (38) (38) ( a ( X) Y )( Y) ( a( X) Y ) 0 0 3c va d 3 R ( X( ) X ( ) ) c c c ( g( X) h( X) Y) a ( X) Y, 0 (39) where pre denotes dfferentaton wth respect to the varable X. Assung that, by coparng wth the coeffcents of Y ( 3,,,0) of both sdes of (39), we have a ( X ) h( X ) a ( X ), (40) a ( X ) g( X ) a ( X ) h( X ) a ( X ), (4)

11 AAM: Intern. J., Vol. 7, Issue (June 0) 7 3c va d 3 R a 0( X) a( X) X X c c c g( X) a ( X) h( X) a ( X), 0 (4) 3c va d 3 R a( X) X X g( X) a 0( X). c c c (43) Snce a ( X)( 0,,) are polynoals, then fro (40) we deduce that a ( X ), s constant and hx ( ) 0. For splcty, take a ( X ). Balancng the degrees g( X ), a( X ) and a ( ), 0 X we conclude that deg( g ( X )), only. Suppose that g( X ) AX B0, then we fnd a ( X) 0 and a ( X ) as a( X ) A0 B0X AX, (44) R B0 A0A 3c va a0( X) pb0a0 X X c c A d AB X 8 c X, (45) a X a X a X and g( X ), n where p s arbtrary ntegraton constant. Substtutng 0 ( ), ( ), ( ) the last equaton n (43) and settng all the coeffcents of powers X to be zero, then we obtan a syste of nonlnear algebrac equatons and by solvng t wth ad Maple, we obtan cd (a 3 c v) cd B0 0, A0, A, R 0, cd c a ac4av9c 6cvv p, c d (46) where, and v are arbtrary constants. Usng the condtons (46) nto (), we get cd (a 3 c v dx ( )) Y ( ). cd (47)

12 8 N. Taghzadeh, M. Mrzazadeh and A. Sae Paghaleh Cobnng (47) wth (38), we obtan the exact soluton to (37) and then the exact solutons to the GNLS equatons can be wrtten as va 3c ( xvt) tx0 a 3c v uxt (, ) e tanh ( ( xvt) 0). d c 3.3. The frst ntegral ethod for obtanng exact solutons of systes of NPDEs Consder the followng systes of partal dfferental equatons: ( uvu,,, v, u, v, u, v, u, v,...) 0, t t x x tt tt xx xx ( uvu,,, v, u, v, u, v, u, v,...) 0. t t x x tt tt xx xx (48) We use the transforatons uxt (, ) f( ), vxt (, ) g( ), x ct. (49) Usng Equaton (4) to transfer the systes of NPDEs (48) to the systes of ODEs ( f, g, f, g,...) 0, ( f, g, f, g,...) 0. (50) Usng soe atheatcal operatons, the systes of ODEs (50) s converted nto a second-order ODE as ( f, f, f,...) 0. (5) If we let X( ) f( ), Y( ) f( ), the Equaton (5) s equvalent to the two densonal autonoous syste X Y, Y ( X, Y). (5) New, we wll apply Dvson Theore to obtan one frst ntegral to Equaton (5) whch reduces Equaton (5) to a frst order ntegrable ODE. An exact soluton to systes of NPDEs (48) s then obtaned by solvng ths equaton A. Now, we wll consder the approxate long water wave equatons [Wang et al. (008)]:

13 AAM: Intern. J., Vol. 7, Issue (June 0) 9 ut uux vx uxx 0, vt ( uv) x vxx 0. (53) Makng the transforaton uxt (, ) u( ), vxt (, ) v( ), kx lt, we change the ALWW syste (53) to the followng syste of ODEs lv k uv k v lu kuu kv k u ( ) 0. 0, (54) By ntegratng the frst equaton we have k (55) lu u kv k u R, where R s ntegraton constant. Rewrte ths equaton as follows l u R v( ) u ku. (56) k k Insertng Equaton (56) nto the second syste (54) and ntegratng the resultng equaton, we obtan l 3l k 3 3 ( R) u u u k u R, (57) k where R s ntegraton constant. If we let autonoous syste df ( ) X f( ), Y, the Equaton (57) s equvalent to the two densonal d X Y, 3 3l l R R Y X ( ) X ( ) X( ) k k k k k (58) Equaton () s called the frst ntegral to (58). Accordng to the Dvson Theore, there exsts a polynoal g( X) h( X) Y n CXY [, ] such that

14 30 N. Taghzadeh, M. Mrzazadeh and A. Sae Paghaleh dq dq dx dq dy (.. ) d dx d dy d (58) (58) ( a ( X) Y )( Y) ( a( X) Y ) l l R R ( ) X X X k k k k k ( g( X) h( X) Y) a ( X) Y, 0 where pre denotes dfferentaton wth respect to the varable X. By coparng wth the coeffcents of Y (,,0) of both sdes of (59), we have a( X) h( X) a ( X), (60) a ( X) g( X) a ( X) h( X) a ( X), (6) 0 0 (59) 3 3l l R R a( X) ( ) X X X g X a ( X). k k k k k (6) Snce a ( X )( 0,) are polynoals, then fro (60) we deduce that a ( ) X s constant and hx ( ) 0. For splcty, take a ( X ). Balancng the degrees of g( X ) and a ( X ), 0 we conclude that deg( g( X)) only. Suppose that g( X) B0 AX, then we fnd a 0 ( X). a0( X) AX B0X A0, (63) where A 0 s arbtrary ntegraton constant. Substtutng a 0 ( X ), a ( X ) and g( X ) n the last equaton n (6) and settng all the coeffcents of powers X to be zero, then we obtan a syste of nonlnear algebrac equatons and by solvng t, we obtan l B0, A, R k A0, R lka0, (64) k ka where kl, and A0 are arbtrary constant. Usng the condtons (64), we obtan l Y( ) A 0 X( ) X ( ). k k (65)

15 AAM: Intern. J., Vol. 7, Issue (June 0) 3 Cobnng (65) wth (58), we obtan the exact soluton to equaton (57) and then the exact soluton to the ALWW syste (53) can be wrtten as 3 3 l ka0 l ka0 l uxt (, ) tan ( kxlt 0), k k k l ka0 l ka0 l ka0l vxt (, ) ka 0 tan ( kxlt 0), k k k k for k A l l l k A 0 l k A 0 uxt (, ) tanh ( kxlt 0), k k k l ka0 l l ka 0 l k A 0 vxt (, ) ka 0 tanh ( kxlt 0), k k k k for k A l Concluson In ths paper, the frst ntegral ethod has been used to construct exact travelng wave solutons of nonlnear partal dfferental equatons, the Landau-Gnburg-Hggs equaton and generalzed for of the nonlnear Schrödnger equaton and approxate long water wave equatons. The perforance of ths ethod s found to be relable and effectve and t gves ore solutons. The ethod has the advantages of beng drect and concse. The ethod proposed n ths paper can also be extended to solve soe nonlnear evoluton equatons n atheatcal physcs. Acknowledgeent The authors are very grateful to the professor Alakbar Montazer Haghgh and the referees for ther detaled coents and knd help. REFERENCES Bakov, V.A. and Khusnutdnova, K.R. (996). Foral lnearzaton and exact solutons of soe nonlnear partal dfferental equatons, Nonlnear Matheatcal Physcs, Vol. 3, NO.-, pp Feng, Z.S. (00). The frst ntegral ethod to study the Burgers-Korteweg-de Vres equaton, J. Phys. A. Math. Gen. Vol. 35, No., pp

16 3 N. Taghzadeh, M. Mrzazadeh and A. Sae Paghaleh Khur, S.A. (008). Exact solutons for a class of nonlnear evoluton equatons: A unfed ansatze approach, Chaos, Soltons and Fractals, 36, Moosae, H., Mrzazadeh, M., and Yldr, A. (0). Exact solutons to the perturbed nonlnear Schrödnger s equaton wth Kerr law nonlnearty by usng the frst ntegral ethod, Nonlnear Analyss: Modellng and Control, Vol. 6, No. 3, pp Taghzadeh, N. and Mrzazadeh, M. (0). Exact Travellng Wave Solutons for Konopelchenko-Dubrovsky Equaton by the Frst Integral Method, Appl. Appl. Math. Vol. 6, pp Taghzadeh, N. and Mrzazadeh, M. (0). The frst ntegral ethod to soe coplex nonlnear partal dfferental equatons, J. Coput. Appl. Math. Vol. 35, pp Taghzadeh, N., Mrzazadeh, M. and Farahrooz, F. (0). Exact solton solutons of the odfed KdV KP equaton and the Burgers KP equaton by usng the frst ntegral ethod, Appl. Math. Model. 35, Taghzadeh, N., Mrzazadeh, M. and Farahrooz, F. (0). Exact solutons of the nonlnear Schrödnger equaton by the frst ntegral ethod, J. Math. Anal. Appl. Vol. 374, pp Taghzadeh, N., Mrzazadeh, M. and Tascan, F. (0). The frst-ntegral ethod appled to the Eckhaus equaton, Appl. Math. Letters, Vol. 5, pp Wang, W., Zhang, J. and L, X. (008). Applcaton of the ( G / G) expanson to travellng wave solutons of the Broer-Kaup and the approxate long water wave equatons, Appl. Math. Coput. Vol. 06, pp Yaghob Moghadda, M., Asgar, A. and Yazdan, H. (009). Exact travellng wave solutons for the generalzed nonlnear Schrödnger (GNLS) equaton wth a source by Extended tanhcoth, sne-cosne and Exp-Functon ethods, Appl. Math. Coput. Vol. 0, pp