Homotopy Perturbation Method to Obtain Positive Solutions of Nonlinear Boundary Value Problems of Fractional Order

and Applied Analysis 3

In this paper, we employ the HPM to obtain positive solutions of (1) and then we compare the obtained result with those obtained by the Adomian decomposition method.

Homotopy Perturbation Method
In this section we employ the HPM to obtain positive solution of nonlinear fractional order BVP (1).We construct the following homotopy for (1): The embedding parameter  monotonically increases from 0 to 1 as the trivial problem   () = 0 is continuously transformed to the original problem   () + (, ()) = 0.If  = 0, then (5) becomes a linear equation and when  = 1, then ( 5) turns out to be the original equation (1).
The HPM uses the embedding parameter  as a "small parameter, " and writes the solution of (5) as a power series of ; that is, where Substituting ( 7) and ( 8) into (5) leads to Now equating the terms with identical powers of , we can obtain a series of equations of the following form: It is obvious that the system of nonlinear equations in (11) is easy to solve and the components V  ,  ≥ 0, of the homotopy perturbation method can be completely determined and the series solutions are thus entirely determined.
Setting  = 1 results in the approximate solution of ( 7): We can approximate the solution  by accelerating  = 1 and the truncated series: where  = V  (0) will be determined by applying suitable boundary conditions of (1).

Illustrative Examples
In this section, to give a clear overview of the HPM for fractional nonlinear BVP, we present the following examples.We apply the HPM and compare the results with the ADM.
Example 1.Consider the following nonlinear boundary value problem [11]: According to the modified HPM [17], we construct the following homotopy: Substitution of ( 7) and ( 8) into (15) and then equating the terms with same powers of  yield the following series of equations: where   are defined as In view of ( 4) and (17), by applying the inverse operator   on both sides of ( 16) and solving corresponding integrals we get Other components are determined similarly.Further we compute () for various values of .
It should be remarked that the graphs drawn here using the HPM are in excellent agreement with those drawn using the ADM [11].
In view of (5) we construct the following homotopy: Substituting ( 7) and ( 8) into (25) and proceeding as before we have where the first few Adomian polynomials   that represent the nonlinear term  V() are defined as In Figure 2 we plot   ,  = 1, 2, 3, 4 where   are as defined in Example 2 and  = 1.
Remark 5.The graphs drawn in Figure 2 are in excellent agreement with those drawn in [11] using the ADM.
Applying the inverse operator   on both sides of (30), we have We know   =   .Substituting (31) in ( 7) leads us to we observe that the power series V 0 + V 1  + V 2  2 + ⋅ ⋅ ⋅ corresponds to the solution of the equation H(V; ) =   V + (, V()) = 0 and becomes the approximate solution of (1) if  → 1.This shows that the homotopy perturbation method is Adomian's decomposition method with the homotopy H(V; ) given by (34).The proof of Theorem 6 is completed.