GROUP CLASSIFICATION OF LINEAR SECOND-ORDER DELAY ORDINARY DIFFERENTIAL EQUATION

Transcription

1 Proceedings of the nd IMT-GT Regional onference on Mathematics Statistics and Applications Universiti Sains Malasia GROUP LASSIFIATION OF LINEAR SEOND-ORDER DELAY ORDINARY DIFFERENTIAL EQUATION Prapart Pue-on School of Mathematics Institute of Science Suranaree Universit of Technolog Thailand Abstract The linear dela ordinar differential equation ( ) + a ( ) ( ) + b ( ) ( ) + c ( ) ( ) + d ( ) ( ) = g ( ) is studied where the coefficients a ( ) b ( ) c ( ) and d( ) and function g( ) are arbitrar In this manuscript group analsis is applied to find equivalent smmetries of the equation Introduction Let us consider a linear second-order dela ordinar differential equation ( ) + a( ) ( ) + b( ) ( ) + c( ) ( ) + d( ) ( ) = g( ) () For brevit the smbol will be used to denote ( ) to denote ( ) and will mean the first derivatives of at point and respectivel Then equation () can be simpl written as + a( ) + b( ) + c( ) + d( ) = g( ) () Here it is assumed that b + c 0 Application of this equation can be found in biolog phsics and engineering where it is used to model natural phenomena Lie Group of Transformations In [6] a definition of admitted Lie group of transformations for dela differential equations was developed A theor of equivalence Lie groups can be considered similar to that of admitted Lie groups Here this analsis is developed Let ϕ :Ω Δ Ω be a transformation where Ω is a set of variables ( ) = ( abcd g) and Δ R is a smmetric interval with respect to zero The variable ε Δ is considered as a parameter of the transformation ϕ This transform maps the variables ( ) to variables ( ) Let ϕ( ε ; ) be denoted b ϕε ( ) The set of functions ϕ ε forms a one-parameter transformation Lie group of the space Ω if it contains the identit transformation as well as inverse of its elements and their composition [456] Alternativel the notation = ϕ ( ε ; ) = ϕ ( ε ; ) = ϕ ( ε) ; is used instead of ϕε = ( ) The transformed variable with dela term it s derivatives and it s derivatives with dela term d d are defined b = ( ) = and = ( ) respectivel d d Let us consider a dela differential equation F( ) = (3) 0 33

2 A Lie group of transformations is called admitted if each transformation maps a solution of the differential equation to a solution of the same equation Such transformation are called smmetries For this reason equations for defining smmetries were constructed under the assumption that a Lie group of transformations maps a solution of a dela differential equation into a solution of the same equation This assumption leads to F( ) ε= 0= XF ( )!) =0 (4) Here the operator X is defined b where X ( ) ( ) = η ξ + η ξ + η + η + η + ( ζ ξ) + ζ ξ( ) = ( ; 0) η( ) = ( ; 0) ζ ( ) = ( ; 0) ξ( ) = ξ( ) η = D ( η ξ) η = D ( η ) η ( ) = η( ) η = D( η ξ ) ζ = D( ζ ξ) D = ( + ) + ( + ) D = a The operator X is called a canonical Lie-B cklund infinitesimal operator of a smmetr Lie s theor [456] shows that there is a one-to-one correspondence between generators and smmetries This operator is also equivalent to an infinitesimal generator [6] X = ξ + η + ζ Equation (4) gives a definition of an action of the infinitesimal generator of a Lie group onto a dela differential equation Determining Equations Definition A Lie group of transformations is called admitted if it satisfies the equation XF ( ) = 0 () (5) for an solution of (3) Equation (5) is called a determining equation For solving the determining equation one can use the theor of eistence of a solution of an initial value problem for dela equation () [] This problem is formulated as follows Let a function χ( ) ( 0 0) be given Find a solution ( ) [ ε ) which satisfies the condition ( ) = χ( ) ( ) 0 0 Because the initial values are arbitrar the variables and their derivatives can be considered as arbitrar elements Thus if the determining equation (5) is written as a polnomial of of variables and their derivatives the coefficients of these variables in the equations must vanish The method for obtaining the overdetermined sstem of equations is called splitting the determining equation This gives an overdetermined sstem of partial differential equations for the coefficients of the infinitesimal generator The unknown functions ξ η and ζ can be found b solving this sstem 34

3 Equivalence Problem The problem of finding all equations which are equivalent to a given equation is called an equivalence problem If the given equation is a linear equation then the equivalence problem is called a linearization problem In this section the importance of an equivalence Lie group of transformations is shown onsider a linear second order ordinar differential equation + a( ) + c( ) = g( ) (6) A Lie group of transformations of the independent variables dependent variables and coefficients which conserves the differential structure of the equation is called an equivalent Lie group This group allows simplifing the coefficients of equations For eample S Lie showed that an linear second order ordinar differential equation (6) is equivalent to the equation = 0 (7) Equation (7) admits the eight-dimensional Lie algebra spanned b the generators X = X = X = X = X = X = X = + X = If one tries to find an admitted Lie group for equation (6) then the sstem of determining equations consists of four second-order ordinar differential equations In general this sstem cannot be solved The purpose of this manuscript is to do group classification of equation () 3 Equivalence Smmetries of () Let p be a particular solution of equation () onsidering the change = = p equation () is reduced to the equation + a( ) + b( ) + c( ) + d( ) = 0 Similar to a second-order ordinar differential equation the coefficient a ( ) can be reduced b change = v( ) q( ) with q satisfing the equation q + aq = 0 We will consider equivalence smmetries of equation + b( ) + c( ) + d( ) = 0 (8) instead of () Letting F = + b( ) + c( ) + d( ) then equation (5) becomes ( ) (3) X b ( ) c ( ) d ( ) = 0 (9) Splitting this equation with respect to b c d b c d and later with respect to one finds b ξ = ξ = α η = β+ γ ζ = b ( α + β β ) c d ζ = β c α ζ = b β d α + dα dα where α( ) β ( ) are arbitrar periodic functions with period and γ ( ) is an arbitrar solution of (3) Thus the equivalence smmetr is = α( ) = β( ) + γ( ) (0) 35

4 4 Group lassification of Equation (3) A Lie group of transformations admitted b equation (3) has to satisf the following determining equation [3] Y ( + b ( ) + c ( ) + d ( ) ) = 0 (3) () Here the operator Y is defined b where ( ) ( ) Y = η - ' ξ + η - ξ + η + η + η ' ' ' " ' ' " 4 ξ( ) = ( ; 0 ) η( ) = ( ; 0 ) ξ = ξ - η = η - ( ) ( ) ( ) ( ) ( ) ( ) ( ) η η ξ η η ξ η η = D - ' = D - = D D = + ' ' ' Splitting this equation with respect to and later with respect to one finds ξ = ξ η = β + γ () ' ξ = β β" = -c' ξ - c ξ (3) γ " = -b γ - c γ d γ (4) ( ) ( β β ) ξ β ξ b β - β = b ξ + ξ b (5) d - = d + b + d (6) B integrating (3) one finds β = ξ / + where is an arbitrar constant Since ξ = ξ it implies β = β Hence integrating equation (5) one has b ξ = (7) is an arbitrar constant Equation (6) is written as ase b 0 d 0 b d ξ + ξ d = - ξ (8) Substituting β into the second equation (3) one gets 3 is an arbitrar constant If 0 ξ ξξ + cξ = 3 (9) then from equations (7) (8) and (9) one obtains ξ = η = + + γ b b 3 b b b c = 5b ( ) d 4b + = + b b where 4 is an arbitrar constants 5 = 3/ and γ ( ) is an arbitrar solution of (3) Since ξ = ξ the the coefficient b has to satisf the same propert b = b The infinitesimal generator obtained is 36

5 X = γ (0) b b If = 0 then ξ = 0 η = +γ and all coefficients are arbitrar The infinitesimal generator is X = ( + γ ) ase b 0 d = 0 () Solving equations (6) (7) (8) and the second equation of (3) one obtains β = bξ = ξ = + cξ = where are arbitrar constants Since ξ = ξ then If then = c = b = () The infinitesimal generator of the admitted Lie group is X = + ( + γ ) (3) 8 6 If 8 = 0 then ξ = 0 η = 6 + γ b and c are arbitrar γ ( ) is an arbitrar solution of (8) The infinitesimal generator is X = ( 6 + γ ) ase b= 0 d 0 From equation (8) one finds dξ = 0 where 0 is an arbitrar constant Hence / / 4 d ξ d = d η = + + γ (4) If 0 0 then from equation (9) one finds The infinitesimal generator obtained is c = + d + d (5) 3 ( d ) 0 d 8 d X / 0 0 d = + 3 γ / + + / (6) d d If 0 = 0 then ξ = 0 β = η = +γ and the coefficients c and are arbitrar functions Hence the infinitesimal generator is X = ( + γ ) (7) 5 onclusion The linear second-order dela ordinar differential equation is classified into three cases as the followings: b 0 and d 0 The infinitesimal generator admitted b the equation of this case is (0) b 0 and d = 0 The infinitesimal generator admitted b the equation of this case is (3) b = 0 and d 0 The infinitesimal generator admitted b the equation of this case is (6) Acknowledgements This work is supported b Ministr of Universit Affairs of Thailand (MUA) and Nakhon Phanom Universit Author is deepl indebted to S V Meleshko for his frequent help d 37

6 References [] R D Driver Ordinar and Dela Differential Equations New York: Springer-Verlag 977 [] J Thanthanuch Application of group analsis to functional differential equations PhD Thesis Nakhonratchasima Thailand: Suranaree Universit of Technolog 003 [3] J Thanthanuch and S V Meleshko On definition of an admitted Lie group for functional differential equations ommunication in Nonlinear Science and Numerical Simulation 9(004) 7 5 [4] N H Ibragimov Elementrar Lie Group Analsis and Ordinar Differential Equations London: John Wile & Sons Ltd 999 [5] N H Ibragimov (Ed) Lie Group Analsis of Differential Equations Vol -3 Florida: R Press 994 [6] L V Ovsiannikov Group Analsis of Differential Equations New York: Academic Press 98 38