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Numerical solutions of fractional differential equations of Lane-Emden type by an accurate technique
Advances in Difference Equations volume 2015, Article number: 220 (2015)
Abstract
In this work, we study the fractional order Lane-Emden differential equations by using the reproducing kernel method. The exact solution is shown in the form of a series in the reproducing kernel Hilbert space. Some numerical examples are given in order to demonstrate the accuracy of the present method. The results obtained from the method are compared with the exact solutions and another method. The obtained numerical results are better than the ones provided by the collocation method. Results of numerical examples show that the presented method is simple, effective, and easy to use.
1 Introduction
The fractional differential equations of various types plays important roles and tools not only in mathematics but also in physics, control systems, dynamical systems and engineering to create the mathematical modeling of many physical phenomena. As is well known, the Lane-Emden differential equations are important for mathematical modeling [1]. Therefore, the goal of our manuscript is to research the effectiveness of reproducing kernel method (RKM) to solve fractional differential equations of Lane-Emden type. To demonstrate this, we solve several examples in the succeeding sections. We consider the following equation:
with the initial conditions
where \(0< t \leq1\), \(k \geq0\), \(1 < \alpha\leq2\), \(0 < \beta\leq 1\), A, B are constants, \(f(t,y)\) is a continuous real valued function and \(g(t) \in C[0,1]\) [2].
Lane-Emden differential equations are singular initial value problems relating to second order differential equations (ODEs) utilized to model successfully several real world phenomena in mathematical physics and astrophysics. The Lane-Emden equation describes plenty of phenomena including aspects of stellar structure, the thermal history of a spherical cloud of gas, isothermal gas spheres, and thermionic currents. We recall that the ordinary Lane-Emden equation does not always give a correct description of the dynamics of systems in complex media. Thus, in order to bypass this obstacle and to better describe the dynamical processes in a fractal medium, numerous generalizations of Lane-Emden equation were suggested. Thus, taking into account the memory effects are better described within the fractional derivatives, the fractional Lane-Emden equations are extracting hidden aspects for the complex phenomena they described in various field of the applied mathematics, mathematical physics, and astrophysics [2, 3].
Fractional order Lane-Emden differential equations involve multi-term fractional ODEs. The multi-term fractional differential equations have been considered by many authors and some numerical methods have been proposed [4–7].
Fractional calculus has a large variety of implementations in lots of several scientific and engineering disciplines. The main notions of fractional calculus and implementations are given in [8, 9].
We recall that a general solution technique for fractional differential equations has not yet been constituted. Some methods have been enhanced for particular sorts of problems. Consequently, a single standard method for problems regarding fractional calculus has not appeared. Thus, finding credible and affirmative solution methods along with fast application techniques is beneficial and enables examination of the field. The power series method [10], the differential transform [11] and [12], the homotopy analysis method [13], the variational iteration method [14], the homotopy perturbation method [15] and the sinc-Galerkin method [16] are some well known methods for solving fractional differential equations. For more details see [17–19].
The theory of reproducing kernels [20], was utilized for the first time at the beginning of the 20th century by Zaremba in his work on boundary value problems. Recently, much attention was devoted to the further investigations of RKM in order to be applied to various scientific models. Since RKM accurately computes the series solution it is of great interest for applied sciences. The method provides the solution in a rapidly convergent series with components that can easily be calculated. In [21] an overview of RKM is shown. For more details of this method the reader can see [22–29].
The organization of the manuscript is as follows.
Section 2 gives the basic theorems of fractional calculus. Section 3 introduces several reproducing kernel spaces. The representation in \({}^{o} \varpi_{2}^{3}[0,1]\) and a related linear operator are presented in Section 4. Section 5 exhibits the main results. The exact and approximate solutions of (1)-(2) are given in this section. We verify that the approximate solution converges uniformly to the exact solution. Two examples are shown in Section 6. Some conclusions are given in the final section.
2 Preliminaries
Definition 1
[16]
The left and right Riemann-Liouville fractional derivatives of order α of \(h(t)\) are given as
and
The left and right Caputo fractional derivatives of order α of \(h(t)\) are
and
such that \(h:[a,b]\rightarrow\mathbb{R}\) is a function, α is a positive real number, n is the integer satisfying \(n-1 \leq\alpha \leq n\), and Γ is the Euler gamma function.
Definition 2
[16]
If \(0 < \alpha< 1\) and h is a function such that \(h(a)=h(b)=0\), we can write
and
3 Reproducing kernel functions
Definition 3
[21]
Let \(F\neq\emptyset\). A function \(R:F\times F\to\mathbb{C}\) is called a reproducing kernel function of the Hilbert space H if and only if
-
(a)
\(R(\cdot,v)\in H\) for all \(v\in F\),
-
(b)
\(\langle\varrho,R(\cdot,v) \rangle=\varrho(v)\) for all \(v\in F\) and all \(\varrho\in H\).
Definition 4
[21]
A Hilbert space H which is defined on a non-empty set F is called a reproducing kernel Hilbert space if there exists a reproducing kernel function \(R:F\times F\to\mathbb{C}\).
Definition 5
[21]
We describe the space \(\varpi_{2}^{1}[0,1]\) by
The inner product and the norm in \(\varpi_{2}^{1}[0,1]\) are defined by
and
The space \(\varpi_{2}^{1}[0,1]\) is a reproducing kernel Hilbert space. The reproducing kernel function \(T_{t}\) of this space is given as
Definition 6
[21]
We denote the space \({}^{o} \varpi_{2}^{3}[0,1]\) by
The inner product and the norm in \({}^{o} \varpi_{2}^{3}[0,1]\) are defined as
and
Theorem 3.1
The reproducing kernel function \(V_{y}\) of the reproducing kernel Hilbert space \({}^{o} \varpi_{2}^{3}[0,1]\) is obtained as
Proof
Let \(\zeta \in {}^{o} \varpi_{2}^{3}[0,1]\) and \(0\leq y \leq1\). By using Definition 6 and integration by parts, we get
After substituting the values of \(V_{y}(0)\), \(V'_{y}(0)\), \(V''_{y}(0)\), \(V_{y}^{(3)}(0)\), \(V_{y}^{(4)}(0)\), \(V_{y}^{(3)}(1)\), \(V_{y}^{(4)}(1)\) into the above equation we conclude that
This completes the proof. □
4 Bounded linear operator in \({}^{o} \varpi_{2}^{3}[0,1]\)
The solution of (1)-(2) is presented in the reproducing kernel Hilbert space \({}^{o} \varpi_{2}^{3}[0,1]\). Let us describe the linear operator \(A: {}^{o} \varpi_{2}^{3}[0,1] \rightarrow\varpi_{2}^{1}[0,1]\) by
Model problem (1)-(2) alters to the problem
after homogenizing the initial conditions.
Theorem 4.1
The linear operator A is a bounded linear operator.
Proof
We should prove \(\Vert A\zeta \Vert _{\varpi_{2}^{1}}^{2}\leq N \Vert \zeta \Vert _{{}^{o} \varpi_{2}^{3}}^{2}\), where \(N>0\) is a positive constant. By making use of (9) and (10), we get
By the reproducing property, we conclude that
and
Therefore, we get
where \(N_{1}>0\) is a positive constant. Thus, we obtain
Taking into account that \((A\zeta)'(t) = \langle\zeta(\cdot),(AV_{t})'(\cdot) \rangle _{{}^{o} \varpi_{2}^{3}}\), then we get
where \(N_{2}>0\) is a positive constant. Thus, we obtain
and
Therefore, we get
where \(N=N_{1}^{2}+N_{2}^{2}>0\). □
5 Exact and approximate solutions
Let us put \(\varrho_{i}(t)=T_{t_{i}}(t)\) and \(\eta_{i}(t)=A^{\ast}\varrho_{i}(t)\), where \(A^{\ast}\) is conjugate operator of A. The orthonormal system \(\{\hat{\eta}_{i}(t) \}_{i=1}^{\infty}\) of \({}^{o} \varpi _{2}^{3}[0,1]\) can be obtained from Gram-Schmidt orthogonalization process of \(\{\eta_{i}(t)\}_{i=1}^{\infty }\) and
Theorem 5.1
Let \(\{ t_{i} \}_{i=1}^{\infty}\) be dense in \([0,1]\) and \(\eta_{i}(t)= A_{y}V_{t}(y)\vert_{y=t_{i}}\). Then the sequence \(\{ \eta_{i}(t) \} _{i=1}^{\infty}\) is a complete system in \({}^{o} \varpi_{2}^{3}[0,1]\).
Proof
We recall that
Thus, \(\eta_{i}(t)\in {}^{o} \varpi_{2}^{3}[0,1]\). For each fixed \(\zeta(t)\in {}^{o} \varpi_{2}^{3}[0,1]\), let \(\langle \zeta(t),\eta_{i}(t) \rangle=0\) (\(i=1,2,\ldots\)), i.e.,
\(\{t_{i} \}_{i=1}^{\infty}\) is dense in \([0,1]\). Therefore, \((A\zeta)(t)=0\) and \(\zeta\equiv0\). This completes the proof. □
Theorem 5.2
If \(\zeta(t)\) is the exact solution of (14), then we have
where \(\{(t_{i})\}_{i=1}^{\infty}\) is dense in \([0,1]\).
Proof
We know
from (15). By uniqueness of the solution of (14), we obtain
This completes the proof. □
The approximate solution \(\zeta_{n}(t)\) is obtained as
Lemma 5.3
[30]
If \(\Vert \zeta_{n}-\zeta \Vert _{{}^{o} \varpi_{2}^{3}}\rightarrow 0\), \(t_{n} \rightarrow t\) (\(n\rightarrow\infty\)) and \(z(t,\zeta)\) is continuous for \(t\in[0,1]\), then
Theorem 5.4
For any fixed \(\zeta_{0}(t)\in {}^{o} \varpi_{2}^{3}[0,1]\) assume \(\zeta_{n}(t)=\sum_{i=1}^{n}A_{i}\hat{\eta}_{i}(t)\), \(A_{i}=\sum_{k=1}^{i}\sigma_{ik}z(t_{k}, \zeta_{k-1}(t_{k}))\), \(\Vert \zeta_{n} \Vert _{{}^{o} \varpi_{2}^{3}}\) is bounded, \(\{t_{i} \} _{i=1}^{\infty}\) is dense in \([0,1]\), \(z(t,\zeta)\in\varpi _{2}^{1}[0,1]\) for any \(\zeta(t)\in {}^{o} \varpi_{2}^{3}[0,1]\). Then \(\zeta _{n}(t)\) converges to the exact solution of (16) in \({}^{o} \varpi _{2}^{3}[0,1]\) and
Proof
Let us prove the convergence of \(\zeta_{n}(t)\). We have
from the orthonormality of \(\{\hat{\eta}_{i}\}_{i=1}^{\infty}\). Thus, we get
from the boundedness of \(\Vert \zeta_{n}\Vert _{{}^{o} \varpi _{2}^{3}}\). Then we obtain
i.e.,
Let \(m>n\), in view of \((\zeta_{m}-\zeta_{m-1} ) \perp (\zeta_{m-1}-\zeta _{m-2} ) \perp\cdots\perp ( \zeta_{n+1}-\zeta_{n} )\), we get
There exists \(\zeta(t)\in {}^{o} \varpi_{2}^{3}[0,1]\), such that
by completeness of \({}^{o} \varpi_{2}^{3}[0,1]\). We have
In virtue of
we get
If \(n=1\), then we get
If \(n=2\), then we have
It is obvious from (20) and (21) that
By induction, we conclude that
Using the convergence of \(\zeta_{n}(t)\) and Lemma 5.3 gives
i.e., \(\zeta ( t ) \) is the solution of (14) and
This completes the proof. □
Theorem 5.5
If \(\zeta\in {}^{o} \varpi_{2}^{3}[0,1]\) then \(\Vert \zeta_{n}-\zeta \Vert _{{}^{o} \varpi_{2}^{3}} \rightarrow0\), \(n\rightarrow\infty\), and the sequence \(\Vert \zeta_{n}-\zeta \Vert _{{}^{o} \varpi_{2}^{3}}\) is monotonically decreasing in n.
Proof
We obtain
by (16) and (17). Therefore, we have
and
Consequently, \(\Vert \zeta_{n}-\zeta \Vert _{{}^{o} \varpi_{2}^{3}} \) is monotonically decreasing in n. □
6 Numerical experiments
Two examples are given in this section. A comparison of the absolute errors is shown in Tables 1 and 2.
Example 6.1
Let us consider
with the initial conditions
where
and \(\alpha=\frac{3}{2}\), \(\beta=\frac{1}{2}\). The exact solution of (23)-(24) is given as [2]
Using the above method, we obtain Tables 1 and 3.
Example 6.2
We regard
with the initial conditions
where
and \(\alpha=\frac{3}{2}\), \(\beta=1\). The exact solution of (25)-(26) is given as [2]
Using the above method, we draw Table 2.
Remark
We found numerical results of Examples 6.1 and 6.2 by RKM and we show our results in Tables 1-3 and Figure 1. We used Maple 16 to obtain these results. We mention the time in Table 3. This proves that we can find good results in very short times.
Absolute error of \(\pmb{y(t)}\) for Example 6.1 .
7 Conclusion
Fractional differential equations of Lane-Emden type were investigated by RKM. We explained the technique and managed it in some illustrative examples. The results obtained indicated that RKM can solve the problem with few computations. Numerical examples demonstrate that our method supports the theoretical results.
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Akgül, A., Inc, M., Karatas, E. et al. Numerical solutions of fractional differential equations of Lane-Emden type by an accurate technique. Adv Differ Equ 2015, 220 (2015). https://doi.org/10.1186/s13662-015-0558-8
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DOI: https://doi.org/10.1186/s13662-015-0558-8
Keywords
- reproducing kernel method
- fractional differential equations
- Lane-Emden differential equations