A numerical approach for solving the high-order nonlinear singular Emden–Fowler type equations
- Atta Dezhbord^{1},
- Taher Lotfi^{1}Email authorView ORCID ID profile and
- Katayoun Mahdiani^{1}
https://doi.org/10.1186/s13662-018-1529-7
© The Author(s) 2018
Received: 20 September 2017
Accepted: 15 February 2018
Published: 3 May 2018
Abstract
Reproducing kernel Hilbert space method (RKHSM) is an analytical technique, which can overcome the difficulty at the singular point of non-homogeneous, linear singular initial value problems; especially when the singularity appears on the right-hand side of this type of equations, so it can solve powerfully linear singular initial value problems. Therefore, using advantages of this method, in this paper the high-order nonlinear singular Emden–Fowler type equations are investigated by RKHSM. Then we present five numerical examples to show that the proposed scheme is accurate and reliable.
Keywords
1 Introduction
For \(f(x)=1 \), \(g(u)=u^{m} \), Eq. (1) becomes the standard Lane–Emden equation of the first order and index m, and for \(f(x)=1 \), \(g(u)=\exp(u) \) becomes the second order.
The Emden–Fowler equation was studied by Fowler [4] to describe a variety of phenomena in fluid mechanics and relativistic mechanics among others. The singular behavior that occurs at \(x = 0\) is the main difficulty of Eq. (1).
During the last few decades, many analytic and numeric methods were developed to study and to obtain approximate solutions for different types of Lane–Emden equations and Emden–Fowler equation. The Adomian decomposition method (ADM) was presented by Wazwaz [5–7]; the variational iteration method (VIM) was investigated in [8–10]. The authors of [11] solved singular IVPs of Lane–Emden type by the homotopy perturbation method (HPM). Parand et al. investigated nonlinear differential equations of Lane–Emden type by the rational Legendre pseudospectral approach [12].
Naturally, solving high-order models with usual methods is difficult, so providing appropriate methods to solve these types of equations is useful. The theory of reproducing kernels [13] was used for the first time at the beginning of the twentieth century by Zaremba in his work on boundary value problems for harmonic and biharmonic functions.
This theory has been successfully applied to integral equations [14, 15], partial differential equations [16], boundary value problems [17–22], fractional differential equations [23], and so on [24–29].
In this paper, we generalize the idea of the RKHSM to provide a numerical solution for Eqs. (4)–(8). The main idea is to construct the reproducing kernel space satisfying the conditions for determining solution of the new type of Emden–Fowler equations stated in the third and forth order. The analytical solution is represented in the form of series through the function value at the right-hand side of the equation. To demonstrate the effectiveness of the RKHSM algorithm, several numerical experiments of Eqs. (4)–(8) are presented.
The outline of the paper is as follows: several reproducing kernel spaces are described in the next section. In Section 3, linear operators, a complete normal orthogonal system, and some essential results are introduced. Also, a method for the existence of solutions for Eqs. (4)–(8) based on a reproducing kernel space is described. Various numerical examples are presented in Section 4. Section 5 ends this work with a brief conclusion.
2 Several reproducing kernel spaces
In this section, several reproducing kernels needed are constructed in order to solve Eqs. (4)–(8) using the reproducing kernel spaces method.
1. The reproducing kernel space \(W_{2}^{4,0}[0,1] \) for Eqs. ( 4 )–( 5 )
Theorem 1
Proof
The proof of the completeness and reproducing property of \(W_{2}^{4,0} \) is similar to the proof of Theorem 1.3.2 in [29].
This completes the proof. □
2. The reproducing kernel space \(W_{2}^{5,0}[0,1] \) for ( 6 )–( 8 )
Theorem 2
Proof
The proof of this theorem is similar to that of Theorem 1. Therefore the proof is omitted. □
3. The reproducing kernel space \(W_{2}^{1}[0,1] \)
Theorem 3
([29])
3 Solving Eqs. (4)–(8) in reproducing kernel spaces
Equations (4)–(8) cannot be solved directly using the reproducing kernel method, since it is impossible to obtain a reproducing kernel satisfying the initial conditions of Eqs. (4)–(8). So, we need homogenize the conditions of Eqs. (4)–(8).
In Eqs. (27)–(31), since \(\overline{u}(x)\) is sufficiently smooth, we see that \(L_{1}^{T}, L_{2}^{T}, L_{1}^{F}, L_{2}^{F} \), and \(L_{3}^{F} \) are bounded linear operators.
Theorem 4
The linear operators \(L_{1}^{T}, L_{2}^{T}, L_{1}^{F}, L_{2}^{F} \), and \(L_{3}^{F} \) defined by Eqs. (27)–(31) are bounded linear operators.
Proof
This completes the proof. □
Lemma
- 1.
- 2.
- 3.
- 4.
Proof
By Theorem 3, one can get the proof of parts 1 and 2 immediately.
Note that \(\{x_{i}\}_{i=1}^{\infty}\) is dense on \([0, 1]\), and hence \((L_{1}^{T}u)(x) = 0\). It follows that \(u \equiv 0\) from the existence of \((L_{1}^{T})^{-1}\). So \(\{\psi_{i}(x)\}_{i=1}^{\infty}\) is complete in \(W_{2}^{4,0}\).
4. The proof of this part is similar to that of part 3. □
Theorem 5
Proof
Remark
If Eqs. (27)–(31) are linear, that is, \(H( x,\overline{u}(x)+u_{0})=H( x) \), then the solution of equations can be obtained directly from Eq. (36).
For the proof of convergence of the iterative formula (Eqs. (38)), see [22].
Remark
In the iteration process of Eq. (38), we can guarantee that the approximation \(\overline{u}_{n}(x) \) satisfies the initial conditions of Eqs. (27)–(31).
In the following algorithm, we summarize how the method works.
Algorithm
- 1.
Set \(m=4 \) or \(m=5 \) in \(W_{2}^{m,0}\).
- 2.
Choose N collocation points in the domain set \([0, 1]\).
- 3.For \(i=1,2,\ldots, N\), set$$ \psi_{i}(x)= \textstyle\begin{cases} \frac{\partial^{3} R_{x}(x_{i})}{\partial x^{3}}+\frac{2\beta}{x}\frac{\partial^{2} R_{x}(x_{i})}{\partial x^{2}}+\frac{\beta(\beta-1)}{x^{2}}\frac{\partial R_{x}(x_{i})}{\partial x}&\text{ for (27)},\\ \frac{\partial^{3} R_{x}(x_{i})}{\partial x^{3}}+\frac{\beta}{x}\frac{\partial^{2} R_{x}(x_{i})}{\partial x^{2}}&\text{ for (28)},\\ \frac{\partial^{4} R_{x}(x_{i})}{\partial x^{4}}+\frac{3\beta}{x}\frac{\partial^{3} R_{x}(x_{i})}{\partial x^{3}}+\frac{3\beta(\beta-1)}{x^{2}}\frac{\partial^{2} R_{x}(x_{i})}{\partial x^{2}}+\frac{\beta(\beta-1)(\beta-2)}{x^{3}}\frac{\partial R_{x}(x_{i})}{\partial x}&\text{ for (29)},\\ \frac{\partial^{4} R_{x}(x_{i})}{\partial x^{4}}+\frac{2\beta}{x}\frac{\partial^{3} R_{x}(x_{i})}{\partial x^{3}}+\frac{\beta(\beta-1)}{x^{2}}\frac{\partial^{2} R_{x}(x_{i})}{\partial x^{2}}&\text{ for (30)},\\ \frac{\partial^{4} R_{x}(x_{i})}{\partial x^{4}}+\frac{\beta}{x}\frac{\partial^{3} R_{x}(x_{i})}{\partial x^{3}}&\text{ for (31)}. \end{cases} $$
- 4.
Set \(\bar{\psi}_{i}(x) =\sum_{k=1}^{i} \beta_{ik}\psi_{k}(x)\), \(i=1,2,\ldots,N\).
- 5.
Choose an initial function \(u_{0}(x)\).
- 6.
Set \(n=1\).
- 7.
Set \(B_{n}=\sum_{k = 1}^{n}\beta_{nk}H( x_{k},\overline{u}_{n-1}(x_{k})+u_{0})\).
- 8.
Set \(\overline{u}_{n}^{N}(x)=\sum_{i = 1}^{n}B_{i}\bar{\psi}_{i}(x)\).
- 9.
If \(n < N\), then set \(n = n + 1\) and go to step 7, else stop.
4 Numerical experiments
In this section, we present and discuss the numerical results by employing the reproducing kernel space for two examples of third order and three examples of forth order Emden–Fowler type equations in spaces \(W_{2}^{4,0}\) and \(W_{2}^{5,0}\), respectively. For each example, we demonstrate a figure of convergence. Results demonstrate that the present method is remarkably effective.
Example 1
([8])
which has the exact solution \({u}(x)=\ln[x^{3}+1]\).
Example 2
We use the proposed method, choose initial approximation \(\overline{u}_{0}(x)=0 \), and take \(N=20, 35 \); \(n=5 \); and \(x_{i}=\frac{i}{N} \), where \(i=1:N \).
Example 3
([5])
We use the proposed scheme, choose initial approximation \(\overline{u}_{0}(x)=0 \), and take \(N=35, 45 \); \(n=7 \); and \(x_{i}=\frac{i}{N} \), where \(i=1:N \).
Example 4
Example 5
5 Conclusion
In this study, we have presented a numerical scheme based on reproducing kernel space for solving high-order nonlinear singular initial value Emden–Fowler equations. The properties of the reproducing kernel space require no more integral computation for some functions, instead of computing some values of a function at some nodes. This simplification of integral computation not only improves the computational speed, but also improves the computational accuracy. It was observed that the errors in the form of maximum absolute error are better than the other developed methods [4, 8].
In addition, it is seen from the figures that for N large enough, the errors decrease. The numerical results show that the present method is an accurate and reliable technique for the high-order linear singular differential-difference equations. One of the considerable advantages of the method is that the approximate solutions are found very easily by using the computer code written in Mathematica 8.0 software package.
Declarations
Acknowledgements
The authors would like to express their thanks to unknown referees for their careful reading and helpful comments.
Authors’ contributions
All authors have made equal contributions. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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