A General Solution for Troesch's Problem

Abstract

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.

1. Introduction

Troesch’s equation is a boundary value problem (BVP) expressed as 𝑦^′′=𝑛sinh𝑛𝑦,𝑦(0)=0,𝑦(1)=1, (1.1) where prime denotes differentiation with respect to 𝑥 and 𝑛 is known as Troesch’s parameter.

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 𝑥_∞=1𝑛8ln𝑦^′(0), (1.2) 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.

2. 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 𝐴(𝑢)−𝑓(𝑟)=0,𝑟∈Ω, (2.1) with the boundary condition 𝐵𝑢,𝜕𝑢𝜕𝜂=0,𝑟∈Γ, (2.2) 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 𝐿(𝑢)+𝑁(𝑢)−𝑓(𝑟)=0. (2.3)

Now, a possible homotopy formulation is 𝐻𝑣,𝑝=1−𝑝𝐿(𝑣)−𝐿𝑢₀+𝑝𝐿(𝑣)+𝑁(𝑣)−𝑓(𝑟)=0,𝑝∈[0,1], (2.4) where 𝑢₀ 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 𝑣=𝑝⁰𝑣₀+𝑝¹𝑣₁+𝑝²𝑣₂+⋯. (2.5)

When 𝑝→1, results that the approximate solution for (2.1) is 𝑢=lim_𝑝→1𝑣=𝑣₀+𝑣₁+𝑣₂+⋯. (2.6)

The series (2.6) is convergent on most cases, nevertheless, the convergence depends on the nonlinear operator 𝐴(𝑣) [19, 21, 35, 36, 52].

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 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)=4𝑛tanh⁻¹(𝑢(𝑥)), (3.1) from which we find 𝑦^′=4𝑛1−𝑢²𝑢^′,𝑦^′′=4𝑛1−𝑢²𝑢^′′+8𝑢𝑛1−𝑢²²𝑢^′2, (3.2)(3.3) where prime denotes differentiation with respect to 𝑥.

Now, by substituting (3.1) into the sinh term of (1.1), we obtain 𝑦^′′=𝑛sinh4tanh⁻¹(𝑢(𝑥=𝑢))4𝑛𝑢²+1(𝑢+1)²(𝑢−1)². (3.4)

Then, equating (3.3) and (3.4), we achieve to the following transformed problem: 1−𝑢²𝑢^′′+2𝑢𝑢^′2−𝑛²𝑢1+𝑢²=0, (3.5) where conditions are obtained by using variable transformation (see (3.1)) 𝑢(𝑥)𝑛=tanh4𝑦(𝑥), (3.6) and substituting original boundary conditions 𝑦(0)=0 and 𝑦(1)=1 into above equation, results 𝑢(0)(1)𝑛=0,𝑢=tanh4. (3.7)

From (2.4) and (3.5), we can formulate the following homotopy [17–19]: 𝐻𝑣,𝑝=1−𝑝𝑣^′′−𝑛²𝑣+𝑝1−𝑣²𝑣^′′+2𝑣𝑣^′2−𝑛²𝑣1+𝑣²=0, (3.8) where 𝑝 is the homotopy parameter.

Substituting (2.5) into (3.8) and equating identical powers of 𝑝 terms, we obtain 𝑝⁰∶𝑣₀^′′−𝑛²𝑣₀=0,𝑣₀(0)=0,𝑣₀(1)𝑛=tanh4,𝑝¹∶𝑣₁^′′−𝑛²𝑣₁−𝑛²𝑣³₀−𝑣²₀𝑣₀^′′+2𝑣₀𝑢^′₀²=0,𝑣₁(0)=0,𝑣₁(1)𝑝=0,₂∶𝑣₂^′′−𝑛²𝑣₂−3𝑛²𝑣²₀𝑣₁+4𝑣₀𝑣^′₀𝑣^′₁−2𝑣₀𝑣₁𝑣₀^′′−𝑣²₀𝑣₁^′′+2𝑣₁𝑣^′₀²=0,𝑣₂(0)=0,𝑣₂(1)=0. (3.9)

We solve (3.9) by using Maple software, resulting 𝑣₀()()=𝑤sinh𝑛𝑥,𝑤=tanh𝑛/4(𝑛),𝑣sinh₁𝑤=−³𝑛exp(−𝑛𝑥)())((+()()(),𝑣2exp2𝑛−2(𝑥−1exp2𝑛𝑥+1))−1𝑥+1exp2𝑛−exp2𝑛𝑥₂()=𝑚+𝑞exp−3𝑛𝑥×32𝑛4+−𝑥²+6𝑥𝑛²+−32𝑛)𝑛)−1𝑥−3exp((4𝑥+28)𝑛)+−3exp((6𝑥+24+6𝑥+𝑥²𝑛²+−32𝑛((+3𝑥+3exp2𝑛𝑥+1))4+−𝑥²+6𝑥𝑛²+33+2𝑥𝑛()+−3exp2𝑛𝑥4+6𝑥+𝑥²𝑛²+32𝑛()−1𝑥−3exp4𝑛𝑥8+18()+1exp6𝑛𝑥8exp(2𝑛), (3.10) where 𝑚 is 𝑞()()()𝑚=exp−𝑛𝑥−exp𝑛𝑥exp−3𝑛(𝑛)()×()16𝑛exp−exp−𝑛7exp6𝑛+40𝑛²()()()()exp6𝑛−36𝑛exp6𝑛−exp8𝑛−12exp4𝑛+112𝑛²exp(4𝑛)+7exp(2𝑛)+40𝑛²exp(2𝑛)+36𝑛exp(2𝑛)−1, (3.11) and 𝑞 is 𝑤𝑞=−⁵𝑛().2exp2𝑛−2 (3.12)

Next, calculating the limit when 𝑝→1, we obtain the second-order approximated solution of (3.5) 𝑢₂(𝑥)=lim_𝑝→1𝑣=𝑣₀+𝑣₁+𝑣₂. (3.13)

Finally, from (3.1) and (3.13), the proposed solution of Troesch’s problem is 𝑦(𝑥)=4𝑛tanh⁻¹𝑢₂(𝑥),0≤𝑥≤1,𝑛≥0. (3.14)

3.1. Interval of Solution

The real branch of tanh⁻¹(𝑧) is restricted to the range −1≤𝑧≤1. Therefore, (3.14) requires −1≤𝑢₂(𝑥)≤1, (3.15) 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<𝑢₂(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: lim_𝑛→∞𝑢₂(1)=1,lim_𝑥→0𝑢₂(𝑥)=0,lim_𝑛→0𝑦(𝑥)=𝑥. (3.17)(3.18)(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).

Figure 1 

Plot for (3.13) in the range of 𝑥∈[0,1] and 𝑛∈[0,30].

4. 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.

Table 1 

Comparison between (3.14), exact solution [8, 9], and other reported approximate solutions, using 𝑛=0.5.

Table 2 

Comparison between (3.14), exact solution [8, 9], and other reported approximate solutions, using 𝑛=1.

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 sixth-order 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.

Table 3 

Comparison between (3.14), numerical solution [15], and other reported approximate solutions, using 𝑛=10.

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 order to compare the derivative of (3.14) for 𝑛>20, we use the approximate 𝑦^′(0) reported in [29] 𝑦^′(0)=10⁻¹²exp(29.71−𝑛),𝑦^′(0)<10⁻⁴, (4.1) resulting in a remarkable accuracy for the range 𝑛∈[22,1𝑒(+12)].

Table 4 

Initial slope 𝑦^′(0) of Troesch's problem.

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].

Table 5 

Slope 𝑦^′(1) of Troesch's problem.

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.

Figure 2 

Numerical solution (different symbols) of (1.1) and proposed solution (3.14) (solid line) for different values of 𝑛.

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.

5. 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.

Acknowledgments

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 Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution to this project.