Basic ideasof the homotopy analysis methodShijun LIAODeptofMath.,UCF,April2007
Basic ideas of the homotopy analysis method Shijun LIAO Dept of Math., UCF , April 2007
Consider the nonlinear differential equationf"+f f"-β fi2 = 0f(0) = 0, f'(0) = 1, f(+) = 0Using the transform ationg() = f(n),， =n,the problem becomesg"+gg"-βg'2=0,g(0)= 0, g'(0) = 1, g'(+0) = 0where the constant > 0 is unknown
Consider the nonlinear differential equation: where the constant 0 is unknown (0) 0, '(0) 1, '( ) 0 ' 0, the problem becomes ( ) ( ), , Using the transform ation (0) 0, '(0) 1, '( ) 0 ''' '' ' 0 2 2 = = + = + − = = = = = + = + − = g g g λ g''' g g'' g g f f f f f f f f
High-order deformation equationSubstituti ngG(5;q)=Zgr(5)qk, A(g)=ghk=0k=0into the zeroth -order deformatio n equation, and equating the coefficien tof the like - power of q, we have the high - order deformatio n equationL[gm(5)- Xm-18m-1())=hRm(5)subjectto theboundaryconditionsgm(0) = 0, gm(+0) = 0,Z2[gm(0) - Xm-1g'm-1 (0)]= ngk (0) -(1- Xmm-11k=0where[0,m≤1Xm[1,m≥2
High-order deformation equation = − = − − = + = − = = = − = − − − − − − + = + = 1, 2 0, 1 where '(0) ' (0) ' (0) (1 ) (0) 0, '( ) 0, subject to the boundary conditions ( ) ( ) ( ) of the like - power of , we have the high - order deformatio n equation into the zeroth - order deformatio n equation, and equating the coefficien t ( ; ) ( ) , ( ) Substituti ng 1 0 0 1 1 1 1 1 0 0 m m g g g g g L g g R q G q g q q q m m k m m m k m k m m m m m m m k k k k k k
High-order deformation equationwherem-Rm =Z(ag"m-1-k+gkg"m--k-βg' g'm-1-k k=0For details, please refer to :Liao, S.J. and Tan, Y., " A general approach to giveseries solution of nonlinear differenti al equations"(Appendix A and B), J. of Applied Mathematic s,in press
High-order deformation equation ( ) in press ( Appendix A and B), J. of Applied Mathematic s, series solution of nonlinear differenti al equations" , Liao, S.J. and Tan, Y., "A general approach to give For details, please refer to : ''' '' ' ' where 1 0 1 1 1 − = = − − + − − − − − m k m k m k k m k k m k R g g g g g
Choice of auxiliary linear operatorLet gm denote a special solution of the high - orderdeformatio n equation, ie.L[gm()] = hR,and w,(), w2(), w,()are three solutions of Lu = 0, ieL[wi()] = 0, L[w2(5)] = 0, L[w3()] = 0Then, the common solution of the high -orderdeformatio n equation readsgm(E) = Xm-1gm-1()+gm()+C,W)()+C,W2()+C,W(5)where Ci,C2,C3 are integral constants
Choice of auxiliary linear operator where , , are integral constants ( ) ( ) ( ) ( ) ( ) ( ) deformatio n equation reads Then, the common solution of the high - order [ ( )] 0, [ ( )] 0, [ ( )] 0 and ( ), ( ), ( ) are three solutions of 0, i.e. [ ( )] , deformatio n equation, i.e. Let denote a special solution of the high - order 1 2 3 1 1 2 2 3 3 * 1 1 1 2 3 1 2 3 * * C C C g g g C w C w C w L w L w L w w w w Lu L g R g m m m m m m m = + + + + = = = = = − −