Research Article | Open Access
A General Solution for Troesch's Problem
The homotopy perturbation method (HPM) is employed to obtain an approximate solution for the nonlinear differential equation which describes Troesch’s problem. In contrast to other reported solutions obtained by using variational iteration method, decomposition method approximation, homotopy analysis method, Laplace transform decomposition method, and HPM method, the proposed solution shows the highest degree of accuracy in the results for a remarkable wide range of values of Troesch’s parameter.
Troesch’s equation is a boundary value problem (BVP) expressed as where prime denotes differentiation with respect to and is known as Troesch’s parameter.
Equation (1.1) arises in the investigation of confinement of a plasma column by a radiation pressure  and also in the theory of gas porous electrodes [2, 3]. This BVP problem has a pole  approximately located at which makes the solution of (1.1) a difficult task for numerical methods.
In order to overcome such difficulties, there are several reported numerical solutions for Troesch’s problem [4–10]. Recently, after some decades using wrong numerical results, in [8, 9] were reported the most accurate solutions for and . Besides, there are approximated analytical solutions obtained by using different methods like homotopy perturbation method (HPM) [11, 12], decomposition method approximation (DMA) [12, 13], homotopy analysis method (HAM) , variational iteration method (VIM) , and Laplace transform decomposition method (LTDM) . The main disadvantage of aforementioned approximated solutions is that they are obtained for specific values of , like , or . In contrast, we propose a general approximate solution for Troesch’s problem, useful for , by using HPM method [17–27].
This paper is organized as follows. In Section 2, we provide a brief review of HPM method. In Section 3, we obtain the solution of Troesch’s problem employing HPM. Section 4 shows numerical simulations and discuss our findings. Finally, a brief conclusion is given in Section 5.
2. Basic Idea of HPM Method
Basically, the HPM method [17–26, 31–53] introduces a homotopy parameter , which takes values ranging from up to . When parameter , the equation usually reduces to a simple, or trivial, equation to solve. Then, is gradually increased to 1, producing a sequence of deformations. Eventually, at , the homotopy equation takes the original form of the equation to solve, and the final stage of deformation provides the desired solution. Usually, few iterations are required to obtain good results [17–19].
In the HPM method, it is considered that a nonlinear differential equation can be expressed as with the boundary condition where is a general differential operator, is a known analytic function, is a linear initial/boundary operator, is the boundary of domain , and denotes differentiation along the normal drawn outwards from . The operator, generally, can be divided into two operators, and , where is the linear operator, and is the nonlinear operator. Hence, (2.1) can be rewritten as
Now, a possible homotopy formulation is where is the initial approximation for the solution of (2.3) which satisfies the boundary conditions, and is known as the perturbation homotopy parameter.
We assume that the solution of (2.4) can be written as a power series of as
When , results that the approximate solution for (2.1) is
3. Solution of Troesch’s Problem by Using HPM Method
Straight forward application of HPM to solve (1.1) is not possible due to the hyperbolic term of dependent variable. However, in [11, 12] were reported HPM solutions, obtained by using a power series expansion of the term of (1.1), which are limited to , due to the truncate power series. Nevertheless, the polynomial type nonlinearities are easier to handle by the HPM method. Therefore, in order to apply HPM successfully for a wide range of Troesch’s parameter (), we convert the hyperbolic-type nonlinearity in Troesch’s problem into a polynomial type nonlinearity, using the variable transformation reported in . First, we consider that from which we find where prime denotes differentiation with respect to .
Then, equating (3.3) and (3.4), we achieve to the following transformed problem: where conditions are obtained by using variable transformation (see (3.1)) and substituting original boundary conditions and into above equation, results
We solve (3.9) by using Maple software, resulting where is and is
Next, calculating the limit when , we obtain the second-order approximated solution of (3.5)
3.1. Interval of Solution
The real branch of is restricted to the range . Therefore, (3.14) requires where is delimited by the boundary conditions as
In order to show that (3.14) fullfills the conditions (3.15) and (3.16), we plot (3.13) in Figure 1. From such figure, we can observe that is valid in the intervals of and . Now, from (3.13) and (3.14), we calculate the following limits: From (3.17), (3.18), (3.19), and Figure 1, we can conclude that the maximum value of (3.13) is 1 in the range of . Therefore, (3.13) fullfills (3.15) in the range given by (3.16). Additionally, limit (3.19) shows that for , the presented solution (3.14) becomes the exact/trivial solution for (1.1).
4. Numerical Simulation and Discussion
In the case of (see Table 1), we can observe that the lowest average absolute relative error (A.A.R.E.) is for LDTM , followed closely by the proposed solution (3.14). A possible reason can lie in the fact that (3.14) is a second-order approximation, while LDTM is of third order. For (see Table 2), there is a change now the lowest A.A.R.E. is for the proposed solution (3.14), followed by LDTM solution. For both cases, the other approximations ADM , HPM , HPM , and HAM  have lower accuracy than (3.14). Equation (3.14) did not require an adjustment parameter, unlike LDTM solution, which required a specific adjustment parameter calculated for each value of Troesch’s parameter . Therefore, the proposed solution is easier to use than LDTM solution.
In Table 3, we can observe a comparison of (3.14) with numerical solution , and other solutions obtained by HAM , VIM , and HPM  for . Approximation (3.14) has the lowest A.A.R.E. from all above solutions, followed by VIM approximation. Besides, HAM  has an relatively poor value of A.A.R.E., despite the fact that it is a sixth-order approximation. Furthermore, HPM  shows divergence from the numerical solution. In addition, (3.14) do not require an adjustment parameter, nevertheless, VIM solution required a specific adjustment parameter calculated for each value of Troesch’s parameter . Therefore, the proposed solution is easier to use than VIM solution.
In Table 4, is presented a comparison of initial slope and the results reported in [4, 5, 15, 28, 29] for the range of ; resulting that the proposed solution is the only one reported in literature with high accuracy in the complete aforementioned range. Moreover, in order to compare the derivative of (3.14) for , we use the approximate reported in  resulting in a remarkable accuracy for the range .