FINDING THE APPROXIMATE ANALYTICAL SOLUTIONS OF 2 n ( n ε R ) ORDER DIFFERENTIAL EQUATION WITH BOUNDARY VALUE PROBLEM USING VARIOUS TECHNIQUES

This paper judge against the error estimated by Approximate analytical solutions are obtained using homotopy perturbation method (HPM), and Modified power series method. HPM is a combination of traditional perturbation method and the homotopy method. A numerical example has been considered to demonstrate the effectiveness, exactness and implementation of the method and the results of errors are compared. To attain sufficiently exact results with HPM, it is generally required to calculate at least two statements of the S -terms. However, it was exposed in the numerical examples that highly accurate results were obtained by calculating only one S-term of the series, revealing the effectiveness of the HPM solution. It is concluded that HPM is a powerful tool for solving high-order boundary value problem as it shows less error than MPSAM.


INTRODUCTION
The homotopy perturbation method applies to solve a system of differential equation.We consider a boundary value problem  Where a, b ,c ,d ,e ,f , g , h ,I ,j ,k , l are finite real constant and the functions f (t) and g (t) are continuous on [  ,m].Liao [3] proposed a powerful analytic method for nonlinear problem, namely the homotopy perturbation method.The solution of the boundary value problems has been obtained in term of convergent series with easily computable mechanism.The homotopy perturbation method (HPM) was introduced by He [1], [2] by J. H.In (HPM) method the solution is well thought-out as the summation of an infinite series which frequently converges swiftly to the exact solution.This trouble-free method has been applied to solve linear and nonlinear equations.Since He's apply this method for solving Blasius equation [4].Obtained solutions in contrast with earlier HPM results provide the higher accuracy.
The modified PSAM takes a slight similarity of the Taylor series method (TSM).However, the methodology differs.In the case of TSM, we determine the successive derivative of the boundary value problem and estimate each derivative using the boundary conditions. in conclusion, to fulfill the boundary conditions, we assess the solution at such a point, and then the resultant arrangement of equations is solved to obtain the unknowns.While in the modified PSAM, the BVP and its boundary conditions are first transformed into systems of ODEs.The general solution is then given in power series in t where the constant equals !i i  and  are the boundaries correspondingly.To fulfill the boundary conditions, we estimate the general solution at such a point, say, t = 1; then the resultant classification of equations is solved to obtain the unknowns.Also, ZP and ZC are termed the particular and complementary solution are used for more compact generalized series solution.Numerical examples are well thought-out to show the rate of convergence of the modified PSAM as compared with the HPM and analytic solution obtainable in the text.

HOMOTOPY PERTURBATION METHOD.
We consider the following nonlinear differential equation () -f () = 0, q ∈ Ω (1.1) With the boundary condition Where A, B is a common differential operator and boundary operator respectively, u is known analytical function, and Γ is the boundary of the domain Ω.The operator A can be separated into two parts L and N, where L is linear, while N is nonlinear.So (1.1) can be rewritten as (1.3) By Liao [3], we can construct a homotopy Where q ∈ Γ and S∈ [0, 1] is an embedding parameter, u0 is an initial approximation of (1.1), which satisfies the boundary conditions.Perceptibly from Equations (1.4) and (1.5) we will have: Altering process of S from zero to unity is just that of H( ,S) from L ( ) -L(p0) to A( )f (q).In topology, this is called deformation, L ( ) -L (p0) and A( )f (q) is called homotopic.The embedding parameter S is introduced a great deal more logically, unaltered by mock factors.Additionally, it can be considered as a small parameter for 0 < S ≤ 1.So, it is very normal to suppose that the solution of (1.4), (1. 5) can be written as The grouping of the perturbation method and the homotopy method is called the HPM, which eliminates the drawbacks of the traditional perturbation methods while observance of all its reward.The series (1.8) is convergent for nearly all cases.We take the common higher order boundary value problems of the type (1.9) Boundary condition


, k = 0, 1, 2… (u -1) are real finite constants, The constants 2k  ,k = 0, 1,2, 3,…, (u -1) illustrate the even order derivatives at the boundary Convergence of the above series for the application of (HPM) to (1.15) By equating the terms with identical powers of S, like S 0 ., S 1 ., S 2 .…., S 2k .Combining all the terms of Equations give the solution of the problem, by using the boundary conditions (1.10) and (1.11) we can obtain all parameters

MODIFIED POWER SERIES APPROXIMATION METHOd (MPSAM)
We will look of power series app.method.
, '''(1) ,........... ( 1) , (2.5) By modified PSAM app.Solution is uniquely given as Where ai, i= 0(1) N are unknown constants to be determined and If the definitions and propositions of equations (2.1) -(2.6) are sustained, then a more compact generalized series solution at the primary boundary, t = 0 can be written as where Z and ZC are termed the particular and complementary solutions, and are defined as are the unknowns to be estimated, and  is the order of the boundary value problem.To prove the above result.We assume a series approximation of a Power series of the form Continuing in this way we will get Similarly we will solve for ZC at t = 0, here m= 1, as  = 0(1)( − 1).Substituting ZC in (2.3) as we solved for of Z, we get 16) are estimated at secondary boundary x=1 and following relation where pochhammer is notation with negative index After simplification some recurrence equations are as m=0,1,2,3…..

MAIN RESULTS
In this section linear and nonlinear BVPs will be tested by the Homotopy perturbation method and modified power series method of the twelfth and tenth order.
Example1: Consider the following linear twelfth-order problem solve first using HPM and by Modified PSAM ,and then by DTM Proceeding in this way we get S 12 then adding above coefficient from S 0 to S 12  Applying the operation of DTM, the following recurrence relation is obtained ( ) By using Eq, ( 220)]the following transformed boundary conditions at x=0 can be obtained Where According to Eq (2.1)in [14] we have The constant , , ,  then using inverse transformation rule in (2.21) we will get solution up to N=16   The exact solution of the problem is ( ) Applying the convex homotopy method  The series solution is given as: 2) exhibits the solution and the series solution along with the error obtained by using the homotopy perturbation method.It is obvious that the error can be reduced further, and higher accuracy can be obtained by evaluating more components of y(x).Now we will solve same problem using Modified PSAM with the boundary condition The all ci ,i= 5(6)9 are estimated at the boundary x=1, hence using (2.17After solving above equation, we obtain C5=1.000496,C6=0.990240,C7=1.087700,C8=0.561900and C9 =2.156400 so put all these value in expansion Applying the operation of DTM, the following recurrence relation is obtained ( )

4 .
DIFFERENTIAL TRANSFORMATION FOR BOUNDARY VALUE PROBLEMSuppose differential transformation of deflection function Y(x)[16] y(x) in the term of Differential transformation; t the boundary condition.Given in equation (2.21) for x=1 by taking Now let's solve same problem using differential transformation method subject to boundary conditions[ )-(2.19) we have

Table 1 .
Consider the following nonlinear boundary value problem of tenth order solve using HPM, Modified power series method and DTM Table(1.1): Shows the comparison of results obtained by the HPM and modified PSAM using the error estimates and DTM using Eq,( 2.20)the following transformed boundary conditions at x=0 can be obtained

Table ( 1.2) Shows the comparison of result obtained by modified PSAM and HPM for Example :2 using error estimates
Then by using invers transformation rule(2.22)we get following sesies solution is evaluated up