The elliptic-hyperbolic classic Tricomi equation on a region Ω in R2 may be written in the form ,
which is usually used to study characteristics of the solution between subsonic and supersonic flows of the compressible gas, namely transonic flow. Compared to the Chaplygin equation , neither the velocity of the sound nor the nonlinear boundary conditions affects this equation. If x → xm, the principal part of Eq. (1), i.e. wxx – xmwyy = 0, is called the Keldysh equation. Herein, the most pioneering researches undertaken on the Tricomi equation are reviewed.
Singh et al.  employed the homotopy perturbation Sumudu transform method (HPSTM) to solve the local fractional Tricomi equation in the fractal transonic flow. They reported that their proposed model provides the results without any transformation of the equations into the discrete counterparts and is completely free of the round-off errors. Yagdjian  analyzed Eq. (1) by using the integral transform approach which was introduced in Ref.  for the hyperbolic domain. Nazipov  presented solution of the spatial Tricomi problem for a single mixed-type equation in which the study was carried out in a bound spatial domain. In this regard, Sabitov  presented a similar work for a mixed parabolic-hyperbolic equation. Quintanilla  analyzed the spatial behavior of the Tricomi equation by reaching the decay for an equation which may be elliptic, hyperbolic and parabolic depending on the different points of the region. Zhang  converted the linear Tricomi equation into a confluent hypergeometric equation using partial Fourier-transformation. He could also capture an explicit solution of the initial value problem in terms of two integral operators.
Recently, the analytic approximation of the partial differential equations has been extensively studied through some nonperturbative techniques: Adomian decomposition method (ADM) [10, 11], VIM [12, 13], inverse scattering method , δ-expansion method  etc. The HAM [16, 17, 18, 19, 20] is one of the most effective methods for finding the analytic solution of the partial differential equations. It is required to set ℏ ≠ 0 as an auxiliary parameter for the case in which the variation of this parameter may lead to a uniformly continuous family of functions. In this regard, Sajid and Hayat  presented an analytic solution to the heat conduction/convection equation when the effects of nonlinearity had been taken into account. They also compared the HAM with the homotopy perturbation method (HPM) and drew a conclusion that if ℏ = –1, the HPM is a special case of the HAM. Odibat  captured the Legendre polynomials to solve the nonlinear fractional differential equations by approximating the non-homogeneous and nonlinear terms within the HAM and VIM. Liao  introduced a relationship between the HAM and Euler’s transformation via a nonlinear ordinary differential equation. He concluded that the Euler’s transformation is a special case of the HAM for some special choices of the initial approximation and auxiliary parameter. Shukla et al.  investigated the HAM by adding a non-homogenous term to the auxiliary linear operator L and concluded that this term plays a significant role in reducing the square residual error. Mastroberardino  studied a nonlinear boundary value problem for the electrohydrodynamic flow of a fluid in a circular cylindrical conduit using the HAM. He also showed that the HPM solution yields divergent in all cases. Motsa et al.  studied a fully-developed parallel flow through a horizontal channel using spectral homotopy analysis method (SHAM). They showed that the SHAM can utilize any form of the initial approximation and concluded that the 4th-order approximation of SHAM is in good agreement with the numerical findings.
The main purpose of the present paper is to provide an analytic solution of the Tricomi equation. To this end, the HAM is optimized to accelerate convergence of the series solution. Due to the fact that the VIM is a special case of the HAM , the HAM results are compared with those results obtained through this method as well as those obtained through a closed-form equation. To the best of the authors’ knowledge, no previous work has been conducted in open literature.
2 Series solution to the Tricomi equation using HAM
Let’s consider the following nonlinear algebraic equation,
where N and w (x, y) are a nonlinear differential operator and an unknown function, respectively. Using q ϵ [0, 1] as an embedding parameter in topology, the following zero-order deformation equation is constructed,
where ψ (x, y; q) and w0(x, y) are an unknown function and an initial approximation of w(x, y), respectively. It is to be noted that by setting q = 0 and q = 1, it holds,
respectively. As q varies from 0 to 1, ψ (x, y; q) varies from the initial approximation w0(x, y) to the solution w(x, y). By expanding ψ (x, y; q) in a Taylor’s series with respect to q, the following homotopy-series is constructed,
If Eq. (5) converges at q = 1, one has,
Differentiating the zero-order deformation Eq. (3) i-times with respect to q, dividing by i! and setting q = 1, the ith-order deformation equation will be constructed,
To apply the HAM on the present problem, consider Eq. (1) and its corresponding boundary conditions given,
It should be noted that the HAM enables us to choose an auxiliary linear operator . To this end, the auxiliary linear operator L will be defined,
which has the property of,
where b0 and b1 are integration constants to be determined by the corresponding boundary conditions. Furthermore, the nonlinear differential operator N can be chosen in terms of Eq. (1),
By solving the ith-order deformation Eq. (8), the corresponding ith-order approximated solution can be obtained,
Therefore, the ith-order approximated solution of w(x, y) will be obtained,
In theory, at the pth-order of approximation, the square residual error can be defined ,
where ξ = ξ (x, y). It should be noted that by decreasing the values of Δp, the convergence for corresponding series solution would be faster .
3 VIM based Lagrange multiplier method
Generating the correction functional for the VIM to solve the partial differential equations is one of the procedures which can ensure a rapid convergence of the series solution. It is to be noted that neither the small parameter nor the very large one affects the solution .
By introducing the differential equation as a combination of linear and nonlinear operators in the form of L[w (x, y)]+N[w (x, y)] = f (x, y), where f (x, y) is a known analytic function, the correction functional can be written,
where λ is a general Lagrange multiplier which can be calculated by variational theory, wi(ξ) is considered as a restricted variation which means δwi(ξ) = 0. It is noted that the exact solution can be determined by w(x, y) = limi→∞ wi(x, y). The application of the restricted variation in Eq. (19) is to simplify the determination of the multiplier .
Calculating the variation with respect to wi(x, y) and δwi+1(x, y) = 0 yields,
where k is the highest-order derivative. By setting k = 2, the Lagrange multiplier for the present problem can be determined,
After finding Eq. (23), the iteration formula can be given,
4 Results and discussion
To validate the present analytic solutions (i.e. HAM and VIM), present findings are compared with the separable solution of the generalized Tricomi equation, wxx – f(x)wyy = 0, which can be determined ,
where c1, c2, and λ are arbitrary constants, and the function H = H(x) is calculated by the ordinary differential equation + λ2f(x)H = 0 . In Table 1, results of the present analytic solutions are compared with those obtained through Eq. (25) for solving the Tricomi Eq. (1). The initial approximation in this case is taken as w0(x, y) = y2. Based on the results of Table 1, by increasing in the values of x, w (x, y) will be decreased in all cases. It is seen that although the convergence is occurred at the 11th-order approximation of the HAM, the results of the 9th- and 7th-order approximated solution only suffer from a relative error of at most 1.127% and 1.737%, respectively, compared with the results of Eq. (25).
To determine the valid values of the auxiliary parameter ℏ, one needs to examine the properties of the series solution. It should be noted that since the auxiliary parameter ℏ significantly influences on the convergence of the series solution , it is straightforward to use ℏ = 1 which is seen in Table 1. Furthermore, if the auxiliary parameter ℏ is properly chosen, the series solution may converge fast. However, for cases subjected to obtain higher convergence, the HAM may be optimized.
To investigate the accuracy of the VIM findings from Table 1, the relative error between VIM and Eq. (25) does not exceed 0.985%. It should be noted that although the present VIM findings suffer from a large error compared with the 11th-order approximation of the HAM ( = 0.551%), using this technique for analyzing the partial differential equations is still recommended. It is due to the fact that this scheme provides high convergence with more iteration by introducing a Lagrange multiplier λ  which is given in Eq. (23) for this problem. Moreover, solving Eq. (24) in terms of x and y and using the fact that ∫ ykdξ = , may lead to the threshold of λ. In this way, one would replace above series with Eq. (23), and vary the integral part of Eq. (24) to find its more convergent values in which the Lagrange multiplier takes a value in the range 0 ≤ λ ≤ This fact can be considered as verification of the VIM given in Eq. (24) for solving the Tricomi Eq. (1).
The optimal values of ℏ can be found by minimizing the square residual error Δp, which is given in Eq. (18). In this case, the variation of Δp at each order of the HAM solution should be plotted versus the variation of ℏ numerically. Hence, the optimal values of ℏ can be obtained at the minimum of the resulting graph which is illustrated in Figure 1 for 11th-order approximation of the HAM. As Figure 1 depicts, Δp indicates descending linear behaviour for ℏ ≤ 1.017 while ℏ > 1.017 suggests ascending linear behaviour of this curve.
Table 2 investigates the optimal values of ℏ as well as the minimum value of the corresponded Δp for the Tricomi Eq. (1) at 7th-, 9th-, and 11th-order of the HAM solution. From this table, Δp decreases with an increase of the order of the approximation in series solution which indicates the convergence of the present HAM solution. In other words, using the optimal values of ℏ accelerates the rate of convergence of the series solution. It is seen that by increasing order of the approximation, the auxiliary parameter ℏ becomes more and more close to unity.
A comparison between the values of w(x, y) obtained by different orders of the HAM with and without utilizing the optimal values of ℏ is presented in Table 3. This table also compares the optimized HAM findings with the results obtained through Eq. (25). According to the results of this table, the relative error between the optimized 11th-order approximation of the HAM and Eq. (25) becomes only 0.052%. This is due to the fact that the optimized 11th-order approximation of the HAM can provide more accurate results than the non-optimized one. Therefore, one can conclude that this exhibits the significant effect of the optimal values of ℏ on the series solution.
Utilizing the HAM and VIM to solve the Tricomi equation, which is an abstraction of the Euler equation on a 2D fluid motion near the sonic condition, was the main objective of the present study. Due to the fact that the HAM contains the auxiliary parameter ℏ ≠ 0, the series solution was optimized by minimizing its square residual error at any order of the analytic approximation. It was found that the optimized HAM can accelerate convergence of the series solution. Furthermore, it was shown that the HAM can provide more accurate results than the VIM for solving the Tricomi equation. Comparison between the results found by the present analytic solutions and those prepared by a closed-form equation showed that such techniques can be considered as a promising tool for analyzing partial differential equations. This fact is due to the existing small differences between the results.
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About the article
Published Online: 2018-04-18
Citation Information: Open Physics, Volume 16, Issue 1, Pages 143–148, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0022.
© 2018 E. Khoshrouye Ghiasi and R. Saleh, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0