A General Solution for Troesch ’ s Problem

1 Electronic Instrumentation and Atmospheric Sciences School, University of Veracruz, Xalapa, VER 91000, Mexico 2 Department of Mathematics, Zhejiang University, Hangzhou 310027, China 3 Departamento de Fı́sica y Matemáticas, Universidad Iberoamericana, Prolongación Paseo de la Reforma 880, 01219 Mexico DF, Mexico 4 Micro and Nanotechnology Research Center, University of Veracruz, Calzada Ruiz Cortines 455, 94292 Boca del Rio, VER, Mexico 5 National Institute for Astrophysics, Optics and Electronics Luis Enrique Erro No.1, 72840 Santa Marı́a Tonantzintla, PUE, Mexico


Introduction
where prime denotes differentiation with respect to x and n is known as Troesch's parameter.

Mathematical Problems in Engineering
Equation 1.1 arises in the investigation of confinement of a plasma column by a radiation pressure 1 and also in the theory of gas porous electrodes 2, 3 .This BVP problem has a pole 4 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 n 0.5 and n 1. 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 14 , variational iteration method VIM 15 , and Laplace transform decomposition method LTDM 16 .The main disadvantage of aforementioned approximated solutions is that they are obtained for specific values of n, like n 0.5, 1 or 10.In contrast, we propose a general approximate solution for Troesch's problem, useful for n ≥ 0, 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.

Basic Idea of HPM Method
Basically, the HPM method 17-26, 31-53 introduces a homotopy parameter p, which takes values ranging from 0 up to 1.When parameter p 0, the equation usually reduces to a simple, or trivial, equation to solve.Then, p is gradually increased to 1, producing a sequence of deformations.Eventually, at p 1, 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 A is a general differential operator, f r is a known analytic function, B is a linear initial/boundary operator, Γ is the boundary of domain Ω, and ∂u/∂η denotes differentiation along the normal drawn outwards from Ω.The A operator, generally, can be divided into two operators, L and N, where L is the linear operator, and N is the nonlinear operator.Hence, 2.1 can be rewritten as Now, a possible homotopy formulation is where u 0 is the initial approximation for the solution of 2.3 which satisfies the boundary conditions, and p is known as the perturbation homotopy parameter.We assume that the solution of 2.4 can be written as a power series of p as When p → 1, results that the approximate solution for 2.1 is The series 2.6 is convergent on most cases, nevertheless, the convergence depends on the nonlinear operator A v 19, 21, 35, 36, 52 .

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 sin term of dependent variable.However, in 11, 12 were reported HPM solutions, obtained by using a power series expansion of the sinh term of 1.1 , which are limited to n ≤ 1, 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 n , we convert the hyperbolic-type nonlinearity in Troesch's problem into a polynomial type nonlinearity, using the variable transformation reported in 15 .First, we consider that y x 4 n tanh −1 u x , 3.1 from which we find where prime denotes differentiation with respect to x.Now, by substituting 3.1 into the sinh term of 1.1 , we obtain 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 y 0 0 and y 1 1 into above equation, results From 2.4 and 3.5 , we can formulate the following homotopy 17-19 : where p is the homotopy parameter.Substituting 2.5 into 3.8 and equating identical powers of p terms, we obtain We solve 3.9 by using Maple software, resulting and q is q − w 5 n 2 exp 2n − 2 .

3.12
Next, calculating the limit when p → 1, we obtain the second-order approximated solution of 3.5

3.13
Finally, from 3.1 and 3.13 , the proposed solution of Troesch's problem is 3.14

Interval of Solution
The real branch of tanh −1 z is restricted to the range −1 ≤ z ≤ 1.Therefore, 3.14 requires where x is delimited by the boundary conditions as 0 ≤ x ≤ 1.

3.16
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 0 < u 2 x < 1 is valid in the intervals of x ∈ 0, 1 and n ∈ 0, 30 .Now, from 3.13 and 3.14 , we calculate the following limits:

3.19
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 0 ≤ n ≤ ∞.Therefore, 3.13 fullfills 3.15 in the range given by 3.16 .Additionally, limit 3.19 shows that for n 0, the presented solution 3.14 becomes the exact/trivial solution for 1.1 .

Numerical Simulation and Discussion
In the case of n 0.5 see Table 1 , we can observe that the lowest average absolute relative error A.A.R.E. is for LDTM 16 , 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 n 1 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 13 , HPM 12 , HPM 11 , and HAM 14 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 n.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 15 , and other solutions obtained by HAM 14 , VIM 15 , and HPM 12 for n 10.Approximation 3.14 has the lowest A.A.R.E.from all above solutions, followed by VIM approximation.Besides, HAM 14 has an relatively poor value of A.A.R.E., despite the fact that it is a sixthorder approximation.Furthermore, HPM 12 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 n.Therefore, the proposed solution is easier to use than VIM solution.
In Table 4, is presented a comparison of initial slope y 0 and the results reported in 4, 5, 15, 28, 29 for the range of 1 ≤ n ≤ 20; resulting that the proposed solution is the only one reported in literature with high accuracy in the complete aforementioned range.Moreover, in In the same fashion, in Table 5, we compare y 1 for proposed solution and other reported numerical solutions 4, 5, 28, 30 .The results shows that 3.14 has a good accuracy for y 1 at least in the range n ∈ 1, 20 .
Figure 2 results from Tables 1, 2, and 3 and Maple routines of numerical solution of differential equations.We can observe the overlap of numerical solution and 3.14 , for different values of n ∈ 0.5, 10 .Therefore, 3.14 is a high accurate approximate solution, valid for a wide range of values of Troesch's parameter.
Further research should be performed in order to verify the accuracy of 3.14 for large values of n.Nevertheless, as we know from above discussion y 0 and y 1 are valid and accurate in the ranges 1 ≤ n ≤ 1e 12 and 1 ≤ n ≤ 20, respectively, therefore, we can expect a high accuracy of 3.14 in the same ranges.

Concluding Remarks
In this work, we obtained an approximate solution for Troesch's problem.Besides, we presented a comparison between the numerical solution, the proposed solution, and other approximations reported in the literature.The numerical and graphical results show that the proposed solution is the most accurate one, for a wide range of values of Troesch's parameter n .Moreover, Troesch's problem approximation exhibit a remarkable accuracy for y 0 at least in the range 1 ≤ n ≤ 1e 12 .In the same fashion, y 1 exhibits a good accuracy at least in the range 1 ≤ n ≤ 20.Accordingly, we can expect a good accuracy of proposed solution for the range 0 ≤ n ≤ 1e 12 and maybe for even more larger values of n.Additionally, the     proposed approximated solution does not require any adjustment parameter as reported for solutions obtained by using LDTM and VIM methods, which makes our proposed solution easier to use than those approximations.Finally, further research is necessary in order to verify the accuracy of our proposed approximation for large values of n.

Figure 2 :
Figure 2: Numerical solution different symbols of 1.1 and proposed solution 3.14 solid line for different values of n.

Table 3 :
Comparison between 3.14 , numerical solution 15 , and other reported approximate solutions, using n 10.