A Practical Method for Analytical Evaluation of Approximate Solutions of Fisher ’ s Equations

In this article, a framework is developed to get more approximate solutions to nonlinear partial differential equations by applying perturbation iteration technique. This technique is modified and improved to solve nonlinear diffusion equations of the Fisher type. Some problems are investigated to illustrate the efficiency of the method. Comparisons between the new results and the solutions obtained by other techniques prove that this technique is highly effective and accurate in solving nonlinear problems. Convergence analysis and error estimate are also provided by using some related theorems. The basic ideas indicated in this work are anticipated to be further developed to handle nonlinear models.


Introduction
In the past few decades many analytical and numerical methods have been suggested to examine the solutions of partial differential equations (PDEs) such as Adomian decomposition method [1,2], variational iteration method [3,9], homotopy perturbation method [10][11][12], homotopy analysis method [13][14][15][16] and others [4][5][6][7][8].However, even these methods may not always be sufficient to reach satisfactory results.For this reason, much attention has been recently devoted to the newly developed techniques for obtaining more accurate and efficient solutions.Two of these analytical methods are the optimal homotopy asymptotic method and optimal perturbation iteration method which are recently developed by Marinca et al and Deniz et al respectively.Many authors have applied these methods to solve nonlinear differential equations arising in scientific problems [17][18][19][20][21][22].
In this study, we consider the generalized Fisher's equation where a,b and α are real numbers.It is used as a model equation for the evolution of a neutron population in a nuclear reactor and a prototype model for a propagating flame.It also represents the propagation of a viral mutant in an infinitely long habitat [23,24].A traveling wave solution u (x, t) = u (ξ = x − ct), spreading with a speed c, must be positive and bounded [25].Therefore the boundary conditions for the traveling wave solution are usually or u (x, 0) = v (x) , lim |x|→1 u (x, t) = 0, |x| < 1, t > 0 (3) where |x| < 1, t > 0 and v (x) is a given function.
In this study, we modify perturbation iteration method (PIM), which is recently constructed by Pakdemirli et al. [26,27], to apply it to solve PDEs, particularly for the Eq. ( 1).We also provide the convergence analysis and error estimate of PIM.

Method of Solution
To give the basic concepts of PIM, let us consider the Fisher's equation in closed form: where u = u(x, t).Introducing the perturbation parameter ε = 1 with nonlinear term yields F can also be divided as where is the linear simpler part, which is easy to deal with and is the remaining part.One correction term (u c ) is taken from the perturbation expansion as: where n ∈ N ∪ {0}.Perturbation iteration algorithms will be classified with respect to the degrees of derivatives in the Taylor expansions (m).Briefly, this process will be represented as PIA-m.
PIA-1: Substituting (7) into the remaining part R in (6), then expanding in a Taylor series only with first derivative gives PIA-1: where Evaluating all derivatives and functions at ε = 0 yields We can easily obtain first correction term (u c ) 0 from ( 9) by determining a trial function u 0 which satisfies initial condition(s).Then, u 1 is determined by using (u c ) 0 , (7) and initial condition(s).One can obtain satisfactory results by repeating iteration process with the aid of ( 7) and (9).

PIA-2:
Following the same procedure as in the previous algorithm with second derivative gives PIA-2: By computing all derivatives and functions at ε = 0, we have for the Fisher's equation (5).By means of ( 7) and ( 11), iterative scheme is developed.
After obtaining first correction term (u c ) 0 , first approximate solutions are defined by using prescribed initial conditions.Suppose now that Then we have Consequently, one can represent the solution as

Convergence Analysis and Error Estimate
In this section, we investigate the convergence of the PIM by giving some theorems.
Theorem 1:(Banach's fixed point theorem).Suppose that B be a Banach space and is a nonlinear mapping, and assume that for some constant β < 1 .Then A has a unique fixed point.Also, the sequence with an arbitrary choice of u 0 ∈ B , converges to the fixed point of A and The following theorem, which is needed for our analysis, can be deduced from the Banach fixed point theorem [28].
Theorem 2. Let B be a Banach space denoted with a suitable norm .over which the series ∞ i=0 C i is defined and assume that the initial guess u 0 = C 0 remains inside the ball of the solution u.The series solution ∞ i=0 C i converges if there exists β such that Proof: We first define a sequence as: Now, we must show that {A n } ∞ n=0 is a Cauchy sequence in B. To do this, consider that For every n, k ∈ N,n ≥ k : Since we also have 0 < β < 1 , we can obtain from ( 22) Thus, {A n } ∞ n=0 is a Cauchy sequence in B and this implies that the series solution ( 14) is convergent.
Theorem 3. If the initial guess u 0 = C 0 remains inside the ball of the solution u then A n = n i=0 C i also remains inside the ball of the solution.
is the ball of the solution u.By hypothesis u = lim n→∞ A n = ∞ i=0 C i and from Theorem 2, we have where β ∈ (0, 1)and n ∈ N.This completes the proof.
Theorem 4. Assume that the obtained solution ∞ i=0 C i is convergent to the solution u.If the truncated series k i=0 C i is used as an approximation to the solution u of problem (4), then the maximum error is given as, Proof: From the Eq. ( 22), we have for n ≥ k.By knowing it is obvious that and also we can write Here β is selected as β = max {β i , i = 0, 1, . . ., n} where

Applications
In this section, we will take a few examples with a known exact and approximate solutions obtained via other methods in order to show the effectiveness of both perturbation iteration algorithms.

PIA-1 solution:
For the Eq. ( 35), we have By substituting (36) into Eq.( 9), PIA-1 turns into One may choose the initial function as: which satisfies the initial condition.Using the Eqs.( 13), (37), (38) with initial condition, the iterations are reached as: and so on.In [30] , the exact solution is given by It is worth mentioning that the coefficients of higher order terms of PIA-1 solutions deviate from the Taylor expansion of (42) after the second term u 1 .

PIA-2 solution:
Proceeding as in PIA-1, Eq. ( 11) reduces to for the Eq. ( 35).Using the same starting function u 0 = C 0 = λ and applying the initial condition gives For the other iterations, the term u n in the left hand side of equation ( 43) can be approximated as u 0 = λ for simplicity.By considering this, we have and so on.One can proceed to reach higher iterations by using a symbolic computer program.This problem has been studied by several authors via different methods.Table 1 and Table 2 display the 3-terms approximation obtained by Adomian decomposition method (ADM) [29], variational iteration method (VIM) [31] , homotopy perturbation method (HPM) [32], PIA-1 with 2-terms and 3-terms approximation of PIA-2 for different λ's.It is clear that PIA-2 gives better results than other techniques even for n=2.and different values of λ.

PIA-2 solution:
In the light of previous example, one can construct PIA-2 as: For the sake of simplicity and easier calculating, the term u n in the left hand side of equation ( 56) is taken as 0.25 which comes from Thus, we have for (49).Using the equations ( 13), ( 52) and (58) with initial condition yields 3  (59) and so on.The solution in a closed form by Wazwaz [29] given by The problem is already solved with ADM [29], HPM [32].However, table (3) reveals that PIA-2 solutions gives more effective results than ADM [29], HPM [32] and PIA-1 solutions for n=3. Figure 1 also shows the accuracy of the fourth order PIA-2 solutions.

PIA-2 solution:
Proceeding as in the other examples, PIA-2 can be formed as: The term u n in the left hand side of (71) is taken as 1 2 1/3 due to Therefore, (71) turns into simplified form: After making the relevant calculations, the following results are obtained: (75) and so on.Table 4 shows a comparison between the four terms approximate solutions obtained by ADM [29], PIA-1 and PIA-2.Figure 2 represents the absolute errors of five terms PIM solutions.

Discussion
In this study, perturbation iteration method (PIM) is improved and new iteration algorithms are constructed to solve nonlinear partial differential equations (PDEs).These algorithms are employed specifically to nonlinear generalized Fisher's equations to get more effective and approximate solutions.The obtained solutions are compared with exact solutions as well as Adomian decomposition method (ADM), variational iteration method (VIM) and homotopy perturbation method (HPM).Approximate results have been also displayed in the tables and figures to justify the efficiency and accuracy of the proposed algorithms.Finally, we can say that this paper presents a detailed study of the newly developed perturbation iteration technique for PDEs and it will be encouraging for further studies.

Figure 1 :
Figure 1: Absolute errors obtained by PIM for Example 2.

Figure 2 :
Figure 2: Absolute errors obtained by PIM for Example 3.

Table 1 :
The approximate solutions of Fisher equation (33) using different methods for λ = 2.

Table 2 :
The approximate solutions of Fisher equation (33) using different methods for λ = 4.

Table 3 :
The four terms approximate solutions of Fisher equation (47) using different methods.

Table 4 :
The three terms approximate solutions of Fisher equation (63) by ADM and PIM.