Research Article | Open Access
A new tool for the solution of nonlinear differential equations is presented. The Fixed-Term Homotopy (FTH) delivers a high precision representation of the nonlinear differential equation using only a few linear algebraic terms. In addition to this tool, a procedure based on Laplace-Padé to deal with the truncate power series resulting from the FTH method is also proposed. In order to assess the benefits of this proposal, two nonlinear problems are solved and compared against other semianalytic methods. The obtained results show that FTH is a power tool capable of generating highly accurate solutions compared with other methods of literature.
Many physical phenomena are commonly modelled using nonlinear differential equations, which is a straightforward way to describe the behaviour of their dynamics. Among these methods, the most commonly used is the Homotopy Perturbation Method (HPM) [1–49]. This method is based in the use of a power series of the homotopy parameter, which transforms the original nonlinear differential equation into a series of linear differential equations. In this paper, a generalization of this concept using a product of two power series of the homotopy parameter called Fixed Term Homotopy (FTH) method is proposed. FTH method transforms the nonlinear differential equation into a series of linear differential equations, generating high precision expressions with fewer algebraic terms, reducing the computing cost. Furthermore, in order to deal with the truncate power series obtained with FTH method, the use of Laplace-Pade after-treatment is also proposed. To assess the potential of the proposed methodology, two nonlinear problems, Van Der Pol Oscillator [6, 50] and Troesch’s equation [51–57], will be solved and compared using similar methodologies.
This paper is organized as follows. In Section 2, the fundamental idea of the FTH method is described. Section 3 presents a study of convergence for the proposed method. Section 4 introduces the Laplace-Padé after-treatment. In Sections 5 and 6, the solution procedure of two nonlinear problems is presented. Additionally, a discussion of the obtained results and the finds of this work are summarized in Section 7. Finally, the conclusions are presented in Section 8.
2. Basic Concept of FTH Method
The FTH and HPM methods share common foundations. Both methods consider that a nonlinear differential equation can be expressed as which has as boundary condition where is a general differential operator, is a known analytic function, is a boundary operator, is the boundary of domain , and denotes differentation along the normal drawn outwards from . In general, the operator can be divided into two operators and , which are the corresponding linear and nonlinear operators, respectively. Hence, (1) can be rewritten as
Now, a possible homotopy formulation is given by the expression where is the trial function (initial approximation) for (3) which satises the boundary conditions, and is known as the perturbation homotopy parameter. From analyzing (4), it can be concluded that
For , the homotopy map (4) is reduced to the problem (5) that possesses a trivial solution . Moreover, for , the homotopy map (4) is transformed into the original nonlinear problem (6) that possesses the sought solution.
Considering that , it results that the approximate solution for (1) is
In , a homotopy which uses the auxiliary term was reported. Then, modifying that version, it results in the following proposed homotopy: where is an arbitrary function.
When or , the auxiliary term is set to zero. Then, does not affect or change the initial solution when or the sought solution at . Moreover, a properly selection of can be useful to improve convergence of the homotopy. Now, for the FTH method, (7) can be rewritten as the product of two power series, such that where are unknown functions to be determined by the FTH method, and are arbitrarily chosen fixed terms.
After substituting (10) into (9), and equating terms with the same order of , we obtain a set of linear equations that lead us to calculate . The limit of (10), when , provides an approximate solution for (3) in the form of
The upper limit exists in the event that both limits exists so and fixed term exist.
3. Convergence of FTH Method
To analyze the convergence of FTH, (9) is rewritten as
Applying the inverse operator to both sides of (14), we obtain
For the convergence analysis of the FTH method, we used the Banach Theorem as reported in [1, 2, 5, 60]. Such theorem relates the solution of (3) and the fixed point problem of the nonlinear operator .
Theorem 1 (Sufficient Condition of Convergence). Suppose that and are Banach spaces and is a contractive nonlinear mapping, then
According to Banach Fixed Point Theorem, has a unique fixed point such that . Assume that the sequence generated by the FTH method can be written as
If one assumes that , where . Then, under these conditions:(i),(ii).
Proof. (i) By inductive approach, we have for
Assuming as induction hypothesis that , then
Using (i), we have
(ii) Because of and , , that is,
4. Laplace-Padé after Treatment for FTH Series
The coupling of Laplace transform and Padé approximant  is used in order to recover part of the lost information due to the truncated power series [60, 62–70]. The process can be recast as follows.(1)First, Laplace transformation is applied to power series obtained by FTH method. (2)Next, is substituted by in the resulting equation. (3)After that, we convert the transformed series into a meromorphic function by forming its Padé approximant of order . and are arbitrarily chosen, but they should be of smaller value than the order of the power series. In this step, the Padé approximant extends the domain of the truncated series solution to obtain better accuracy and convergence. (4)Then, is substituted by .(5)Finally, by using the inverse Laplace transformation, we obtain the modified approximate solution.
We will denominate to this process as the Laplace-Padé fixed term homotopy (LPFTH) method.
5. Van Der Pol Oscillator
In order to find an approximate solution for (25) by means of LPFTH, we obtain the Taylor series of the trigonometric terms, resulting is where the expansion order is 7.
From (9), we establish the following homotopy equation where is an adjustment parameter due to the auxiliary term .
By solving (30), we obtain
Then, we select the adjustment parameters as: , ; where the parameters are calculated using the NonlinearFit command from Maple Release 13 [5, 32–34]. Moreover, NonlinearFit command finds values of the approximate model parameters such that the sum of the squared -residuals is minimized.
In order to guarantee the validity of the approximate solution (32) for large , the series solution is transformed by the Laplace-Padé after-treatment. First, Laplace transformation is applied to (32) and then is written in place of in the equation. Then, the Padé approximant is applied and is written in place of . Finally, by using the inverse Laplace transformation, we obtain the modified approximate solution
6. Troesch’s Problem
The Troesch’s equation is a boundary value problem (BVP) that arises in the investigation of confinement of a plasma column by a radiation pressure  and also in the theory of gas porous electrodes [72, 73]. The problem is expressed as where prime denotes differentiation with respect to and is known as Troesch’s parameter.
Straightforward application of FTH to solve (34) is not possible due to the hyperbolic term of dependent variable. Nevertheless, the polynomial type nonlinearities are easier to handle by the FTH method. Therefore, in order to apply FTH successfully, we convert the hyperbolic-type nonlinearity in Troesch’s problem into a polynomial type nonlinearity, using the variable transformation reported in [51, 52]
Substituting original boundary conditions and into (35) results in
We solve (40) by using Maple software, resulting in
If we consider , then we choose the adjustment parameter as: , , and , by using the procedure explained for the Van der Pol Oscillator problem.
7. Numerical Simulation and Discussion
A comparison of the exact solution (26) of the Van der Pol Oscillator against the approximated solutions obtained by LPFTH (33), HPM , and VIM  is shown in Table 1 and Figure 1. In the comparison can be seen that the LPFTH method exhibited the higher accuracy for a large period of time reproducing successfully the oscillatory behaviour of the exact solution (26), therefore, being a suitable alternative to solve nonlinear oscillators.
The Troesch’s BVP problem is a benchmark equation for numerical [74, 75] and semi-analytical methods [51–57] which has been solved by FTH to obtain the approximated solution (43). Table 2 shows a comparison of the results obtained with other semi-analytical methods as: homotopy perturbation method (HPM) [52, 54, 55], decomposition method approximation (DMA) [53, 54], homotopy analysis method (HAM) , and Laplace transform decomposition method (LTDM) . The comparison shows that the average absolute relative error (A.A.R.E.) of (43) is lower than most of the reported results and similar to LDTM .
For both cases of study, polynomial functions were used for fixed and auxiliary terms as they generate the best results. Nonetheless, functions as exponential, trigonometric or hyperbolic may enlarge the domain of convergence. Thereby, a methodology that allows the selection of fixed and auxiliary terms like these to obtain more accurate solutions is a window of opportunity to be exploited.
Since FTH does not rely in the linearization of the input equations, a perturbation parameter nor assumption of weak nonlinearity, the solution generated may be general and more realistic than the method of simplifying the physical problems.
In this work, the Fixed Term Homotopy and the Laplace-Padé Fixed Term Homotopy methods are presented as novel tools to solve nonlinear ordinary differential equations. The proposed methods were tested using two nonlinear problems: a second order nonlinear oscillator and a high nonlinear boundary value problem. From a comparison against several semi-analytical methods from the literature, FTH and LPFTH probed to be power tools that generate highly accurate easy manageable expressions. Furthermore, the homotopy given in (9) may be replaced or modified by a formulation inspired by some other homotopy method proposed in literature [76–96], which can lead to improvement of convergence of proposed methods. Additionally, being FTH a modified version of HPM can be assumed that the differential equations solved by HPM should be also solvable by FTH. In that fashion, further research can be focused on applying FTH method in the solution of nonlinear partial differential equations, nonlinear fractional differential equations, among others.
The authors gratefully acknowledge the financial support of the National Council for Science and Technology of Mexico (CONACyT) through Grant CB-2010-01 no. 157024. The authors would like to thank Roberto Castaneda-Sheissa, Rogelio-Alejandro Callejas-Molina, and Roberto Ruiz-Gomez for their contribution to this project.
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