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On a fractional problem of Lane–Emden type: Ulam type stabilities and numerical behaviors

Abstract

In this work, we study some types of Ulam stability for a nonlinear fractional differential equation of Lane–Emden type with anti periodic conditions. Then, by using a numerical approach for the Caputo derivative, we investigate behaviors of the considered problem.

Introduction

The theory of singular fractional boundary value problems has become an area of research investigation in the last three decades (see [1, 3, 6, 7, 16, 21]). One of the equations describing this type of problems is the very important Lane–Emden equation, which was published by Lane in 1870 [18] and detailed by Emden [8]. Lane–Emden differential equations are singular initial value problems of the second order, they describe a variety of phenomena in mathematical physics and astrophysics such as aspects of the stellar structure. For more information and some applications, one can consult Refs. [2, 13, 23].

The classical Lane–Emden equation has the following form [5, 8]:

$$ x^{{\prime \prime }} ( t ) +\frac{a}{t}x^{{\prime }} ( t ) +f \bigl( t,x ( t ) \bigr) =g ( t ) , \quad t\in [ 0,1 ] , $$

under the conditions

$$ x ( 0 ) =A, \qquad x^{{\prime }} ( 0 ) =B, $$

where A and B are constants and f and g are continuous real functions.

The above problem has attracted many researchers attention. In fact, in [20], the authors have used the method of collocation to study the following Lane–Emden problem:

$$ \textstyle\begin{cases} D^{\alpha }y ( t ) + \frac{k}{t^{\alpha -\beta }}D^{\beta }y ( t ) +f ( t,y ( t ) ) =g ( t ) , \quad t\in [ 0,1 ] , \\ k\geq 0,\qquad 1< \alpha \leq 2,\qquad 0< \beta \leq 1, \end{cases}$$

Ibrahim [15] has been concerned with the stability of Ulam Hyers for the following fractional Lane–Emden problem:

$$ \textstyle\begin{cases} D^{\beta } ( D^{\alpha }+\frac{a}{t} ) u ( t ) +f ( t,u ( t ) ) =g ( t ) , \\ u ( 0 ) =\mu , \qquad u ( 1 ) =\nu , \\ 0< \alpha ,\beta \leq 1,\qquad 0\leq t\leq 1,\qquad a\geq 0,\end{cases}$$

under the conditions: \(D^{\gamma }\) is the Caputo derivative, f is a continuous function and \(g\in C ( [ 0,1 ] ) \).

Very recently, Y. Gouari et al. [10] have investigated the following nonlocal fractional problem of Lane–Emden type:

$$ \textstyle\begin{cases} D^{\beta }(D^{\alpha }+\frac{k}{t^{\lambda }})y(t)+\Delta _{1}f(t,y(t),D^{ \delta }y(t))+\Delta _{2}g(t,y(t),I^{\rho }y(t))+h(t,y(t))=l(t), \\ y(0)=0, \qquad y(1)=b\int _{0}^{\eta }y(s)\,ds,\quad 0< \eta < 1, \qquad I^{q}y(u)=y(1),\quad 0< u< 1, \\ k>0,\qquad 0< \lambda \leq 1,\qquad 1\leq \beta \leq 2,\qquad 0\leq \alpha , \delta \leq 1, \qquad t\in\, ]0,1[, \end{cases} $$

Motivated by the above cited papers, in [25] we have proved the existence and uniqueness of solutions by application of the Banach contraction principle for the following anti periodic fractional differential problem:

$$ \textstyle\begin{cases} D^{\alpha }D^{\beta }y(t)+\frac{k}{t^{\lambda }}D^{\alpha }y(t)+a_{1}F (t,y(t),D^{\gamma }y(t),J^{p}y(t) ) \\ \quad {}+a_{2}G (t,y(t),D^{\gamma }y(t) )+a_{3}H (t,y(t) )=L(t). \\ y(0)+y(1)=0, \qquad y^{\prime }(0)+y^{\prime }(1)=0,\qquad D^{\gamma }(0)+D^{\gamma }(1)=0, \\ k>0,\qquad 1\leq \beta \leq 2, \qquad 0\leq \gamma \leq \alpha \leq 1, \qquad 0< \lambda < 1,\qquad p>0,\qquad t\in {}[ 0,1], \end{cases} $$
(1)

where \(I:=[0,1]\), the derivatives of the problem are in the sense of Caputo, \(J^{p}\) denotes the Riemann–Liouville integral of order p and \(F:I\times \mathbb{R}^{3}\rightarrow \mathbb{R}\), \(G:I\times \mathbb{R}^{2}\rightarrow \mathbb{R}\), \(H:I\times \mathbb{R}^{2}\rightarrow \mathbb{R}\) and \(L:I\rightarrow \mathbb{R} \) are four given functions, and \(\mathbb{R} \) being the set of real numbers.

In this work, we continue studying the above problem by investigating certain types of Ulam stability for the problem (1). Then, using a numerical approach of the derivative Caputo, we analyze certain behavior of the problem by means of the fourth-order Runge–Kutta integrator method.

Preliminaries

We present some necessary lemmas and theorems which will be used in this paper.

As it is proved in our last work [25], the integral solution of (1) is given by the following auxiliary result.

Lemma 1

Let \(L_{1}\in C([0,1])\), \(t\in I\), \(0\leq \gamma \leq \alpha \leq 1\), \(1<\beta <2\). Then the integral solution of the problem

$$ \textstyle\begin{cases} D^{\alpha }(D^{\beta })y(t)+(\frac{k}{t^{\lambda }})D^{\alpha }y(t)=L_{1}(t), \\ y(0)+y(1)=0, \qquad y^{\prime }(0)+y^{\prime }(1)=0,\qquad D^{\gamma }(0)+D^{\gamma }(1)=0,\end{cases} $$
(2)

is given by the following expression:

$$\begin{aligned} y(t) =& \int _{0}^{t} \frac{(t-s)^{\beta -1}}{\Gamma (\beta )}\int _{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \biggl[ L_{1}(u)-\frac{k}{u^{\lambda }}D^{\alpha }y(u) \biggr] \,du\,ds \\ &{}+ \bigl[ K_{1}t^{\beta }+K_{2}t+K_{3} \bigr] \biggl[ \int _{0}^{1} \frac{(1-s)^{\beta -2}}{\Gamma (\beta -1)}\int _{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \biggl[ L_{1}(u)-\frac{k}{u^{\lambda }}D^{\alpha }y(u) \biggr] \,du\,ds \biggr] \\ &{}+ \bigl[ K_{4}t^{\beta }+K_{5}t-K_{6} \bigr] \biggl[ \int _{0}^{1} \frac{(1-s)^{\beta -\gamma -1}}{\Gamma (\beta -\gamma )}\int _{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \biggl[ L_{1}(u)-\frac{k}{u^{\lambda }}D^{\alpha }y(u) \biggr] \,du\,ds \biggr] \\ &{}- [ K_{7} ] \biggl[ \int _{0}^{1} \frac{(1-s)^{\beta -1}}{\Gamma (\beta )} \int _{0}^{s} \frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \biggl[ L_{1}(u)-\frac{k}{u^{\lambda }}D^{\alpha }y(u) \biggr] \,du\,ds \biggr] , \end{aligned}$$
(3)

where

$$\begin{aligned} L_{1}(u) =&L(u)-a_{1}F \bigl(u,y(u),D^{\gamma }y(u),J^{p}y(u) \bigr)-a_{2}G \bigl(u,y(u),D^{ \gamma }y(u) \bigr) \\ &{}-a_{3}H \bigl(u,y(u) \bigr)-\frac{k}{u^{\lambda }}D^{\alpha }y(u) \end{aligned}$$

and

$$\begin{aligned}& K_{1}= \frac{\Gamma ( \beta -\gamma +1 ) }{ \beta [ \Gamma ( \beta -\gamma +1 ) -2\Gamma ( \beta ) \Gamma ( 2-\gamma ) ] }, \\& K_{2}= \frac{\Gamma ( \beta ) \Gamma ( 2-\gamma ) }{2\Gamma ( \beta ) \Gamma ( 2-\gamma ) -\Gamma ( \beta -\gamma +1 ) }, \\& K_{3}= \frac{\Gamma ( \beta +1 ) \Gamma ( 2-\gamma ) -\Gamma ( \beta -\gamma +1 ) }{2\beta \Gamma ( \beta -\gamma +1 ) -4\Gamma ( \beta +1 ) \Gamma ( 2-\gamma ) }, \\& K_{4}= \frac{2\Gamma ( 2-\gamma ) \Gamma ( \beta -\gamma +1 ) }{ \beta [ \Gamma ( \beta -\gamma +1 ) -2\Gamma ( \beta ) \Gamma ( 2-\gamma ) ] }, \\& K_{5}=\frac{\Gamma ( 2-\gamma ) \Gamma ( \beta -\gamma +1 ) }{2\Gamma ( \beta ) \Gamma ( 2-\gamma ) -\Gamma ( \beta -\gamma +1 ) }, \\& K_{6}= \frac{2\Gamma ( 2-\gamma ) \Gamma ( \beta -\gamma +1 ) -\beta \Gamma ( 2-\gamma ) \Gamma ( \beta -\gamma +1 ) }{2\beta \Gamma ( \beta -\gamma +1 ) -4\Gamma ( \beta +1 ) \Gamma ( 2-\gamma ) }, \\& K_{7}= \frac{\Gamma ( \beta -\gamma +1 ) -2\Gamma ( \beta ) \Gamma ( 2-\gamma ) }{2\Gamma ( \beta -\gamma +1 ) -4\Gamma ( \beta ) \Gamma ( 2-\gamma ) }. \end{aligned}$$

Before presenting our main results, we shall introduce also the Banach space

$$ X:= \bigl\{ y\in C(I,\mathbb{R}),D^{\alpha }y\in C(I,\mathbb{R}),D^{ \gamma }y \in C(I,\mathbb{R}) \bigr\} , $$

and the norm

$$ \Vert y \Vert _{X}=\max \bigl\{ \Vert y \Vert _{\infty }, \bigl\Vert D^{\alpha }y \bigr\Vert _{ \infty }, \bigl\Vert D^{\gamma }y \bigr\Vert _{\infty } \bigr\} , $$

where

$$ \Vert x \Vert _{\infty }=\sup_{t\in I} \bigl\vert x(t) \bigr\vert , \qquad \bigl\Vert D^{\alpha }x \bigr\Vert _{\infty }= \sup_{t\in I} \bigl\vert D^{\alpha }x(t) \bigr\vert ,\qquad \bigl\Vert D^{ \gamma }x \bigr\Vert _{\infty }=\sup_{t\in I} \bigl\vert D^{\gamma }x(t) \bigr\vert . $$

Also, we consider the following hypotheses:

  1. (H1):

    There exist nonnegative constants \(W_{i}\), \(i=1,\ldots,6\), such that for each \(t\in I\) and for all \(x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}\in \mathbb{R}\) we have

    $$\begin{aligned}& \bigl\vert F(t,x_{1},x_{2},x_{3})-F(t,y_{1},y_{2},y_{3}) \bigr\vert \leq W_{1} \vert x_{1}-y_{1} \vert +W_{2} \vert x_{2}-y_{2} \vert +W_{3} \vert x_{3}-y_{3} \vert , \\& \bigl\vert G(t,x_{1},x_{2})-G(t,y_{1},y_{2}) \bigr\vert \leq W_{4} \vert x_{1}-y_{1} \vert +W_{5} \vert x_{2}-y_{2} \vert , \\& \bigl\vert H(t,x_{1})-H(t,y_{1}) \bigr\vert \leq W_{6} \vert x_{1}-y_{1} \vert . \end{aligned}$$

The following quantities are also needed in this paper:

$$\begin{aligned}& \begin{aligned} N_{1} ={}&(a_{1}W_{1,2}+a_{2}W_{4,5}+a_{3}W_{6}) \biggl[ \frac{1+ \vert K_{7} \vert }{\Gamma (\alpha +\beta +1)} \\ &{}+ \frac{ \vert K_{1} \vert + \vert K_{2} \vert + \vert K_{3} \vert }{ \Gamma (\alpha +\beta )}+ \frac{ \vert K_{4} \vert + \vert K_{5} \vert + \vert K_{6} \vert }{\Gamma (\alpha +\beta -\gamma +1)} \biggr] \\ &{}+a_{1}W_{3} \biggl[ \frac{1+ \vert K_{7} \vert }{\Gamma (\alpha +\beta +p+1)}+ \frac{ \vert K_{1} \vert + \vert K_{2} \vert + \vert K_{3} \vert }{\Gamma (\alpha +\beta +p)}+ \frac{ \vert K_{4} \vert + \vert K_{5} \vert + \vert K_{6} \vert }{\Gamma (\alpha +\beta -\gamma +p+1)} \biggr] \\ &{}+ \vert k \vert \Gamma (1-\lambda ) \biggl[ \frac{1+ \vert K_{7} \vert }{\Gamma (\alpha +\beta -\lambda +1)}+ \frac{ \vert K_{1} \vert + \vert K_{2} \vert + \vert K_{3} \vert }{\Gamma (\alpha +\beta -\lambda )}+ \frac{ \vert K_{4} \vert + \vert K_{5} \vert + \vert K_{6} \vert }{\Gamma (\alpha +\beta -\gamma -\lambda +1)} \biggr], \end{aligned} \\& \begin{aligned} N_{2} ={}&(a_{1}W_{1,2}+a_{2}W_{4,5}+a_{3}W_{6}) \biggl[\frac{1+ \vert K_{7} \vert }{\Gamma (\beta +1)} +\frac{ \vert K_{1} \vert \Gamma (\beta +1)\Gamma (2-\alpha )+ \vert K_{2} \vert \Gamma (\beta -\alpha +1)}{\Gamma (\beta )\Gamma (\beta -\alpha +1)\Gamma (2-\alpha )} \\ &{}+\frac{ \vert K_{4} \vert \Gamma (\beta +1)\Gamma (2-\alpha )+ \vert K_{5} \vert \Gamma (\beta -\alpha +1)}{ \Gamma (\beta -\gamma +1)\Gamma (\beta -\alpha +1)\Gamma (2-\alpha )} \biggr] \\ &{}+a_{1}W_{3} \biggl[ \frac{1+ \vert K_{7} \vert }{\Gamma (\beta +p+1)} + \frac{ \vert K_{1} \vert \Gamma (\beta +1)\Gamma (2-\alpha ) + \vert K_{2} \vert \Gamma (\beta -\alpha +1)}{\Gamma (\beta +p)\Gamma (\beta -\alpha +1)\Gamma (2-\alpha )} \\ &{}+\frac{ \vert K_{4} \vert \Gamma (\beta +1)\Gamma (2-\alpha ) + \vert K_{5} \vert \Gamma (\beta -\alpha +1)}{\Gamma (\beta -\gamma +p+1)\Gamma (\beta -\alpha +1) \Gamma (2-\alpha )} \biggr] \\ &{}+ \vert k \vert \Gamma (1-\lambda ) \biggl[\frac{1+ \vert K_{7} \vert }{\Gamma (\beta -\lambda +1)} + \frac{ \vert K_{1} \vert \Gamma (\beta +1)\Gamma (2-\alpha ) + \vert K_{2} \vert \Gamma (\beta -\alpha +1)}{\Gamma (\beta -\lambda )\Gamma (\beta -\alpha +1)\Gamma (2-\alpha )} \\ &{}+\frac{ \vert K_{4} \vert \Gamma (\beta +1)\Gamma (2-\alpha ) + \vert K_{5} \vert \Gamma (\beta -\alpha +1)}{\Gamma (\beta -\gamma -\lambda +1)\Gamma (\beta -\alpha +1)\Gamma (2-\alpha )} \biggr], \end{aligned} \\& N_{3} =(a_{1}W_{1,2}+a_{2}W_{4,5}+a_{3}W_{6}) \biggl[\frac{1+ \vert K_{7} \vert }{\Gamma (\alpha +\beta -\gamma +1)} \\& \hphantom{N_{3} ={}}{}+\frac{ \vert K_{1} \vert \Gamma (\beta +1)\Gamma (2-\gamma ) + \vert K_{2} \vert \Gamma (\beta -\gamma +1)}{\Gamma (\alpha +\beta -\gamma )\Gamma (\beta -\gamma +1) \Gamma (2-\gamma )} \\& \hphantom{N_{3} ={}}{}+\frac{ \vert K_{4} \vert \Gamma (\beta +1)\Gamma (2-\gamma ) + \vert K_{5} \vert \Gamma (\beta -\gamma +1)}{\Gamma (\alpha +\beta -2\gamma +1)\Gamma (\beta -\gamma +1) \Gamma (2-\gamma )} \biggr] \\& \hphantom{N_{3} ={}}{}+a_{1}W_{3} \biggl[ \frac{1+ \vert K_{7} \vert }{\Gamma (\alpha +\beta -\gamma +p+1)} + \frac{ \vert K_{1} \vert \Gamma (\beta +1)\Gamma (2-\gamma )+ \vert K_{2} \vert \Gamma (\beta -\gamma +1)}{\Gamma (\alpha +\beta -\gamma +p)\Gamma (\beta -\gamma +1) \Gamma (2-\gamma )} \\& \hphantom{N_{3} ={}}{}+\frac{ \vert K_{4} \vert \Gamma (\beta +1)\Gamma (2-\gamma )+ \vert K_{5} \vert \Gamma (\beta -\gamma +1)}{\Gamma (\alpha +\beta -2\gamma +p+1)\Gamma (\beta -\gamma +1) \Gamma (2-\gamma )} \biggr] \\& \hphantom{N_{3} ={}}{}+ \vert k \vert \Gamma (1-\lambda ) \biggl[\frac{1+ \vert K_{7} \vert }{\Gamma (\alpha +\beta -\gamma -\lambda +1)} + \frac{ \vert K_{1} \vert \Gamma (\beta +1)\Gamma (2-\gamma ) + \vert K_{2} \vert \Gamma (\beta -\gamma +1)}{\Gamma (\alpha +\beta -\gamma -\lambda ) \Gamma (\beta -\gamma +1)\Gamma (2-\gamma )} \\& \hphantom{N_{3} ={}}{}+\frac{ \vert K_{4} \vert \Gamma (\beta +1)\Gamma (2-\gamma ) + \vert K_{5} \vert \Gamma (\beta -\gamma +1)}{\Gamma (\alpha +\beta -2\gamma -\lambda +1) \Gamma (\beta -\gamma +1)\Gamma (2-\gamma )} \biggr] , \\& \quad \text{where }W_{1,2}:=\max (W_{1},W_{2})\text{ and } W_{4,5}:=\max (W_{4},W_{5}). \end{aligned}$$

We recall the following result [25], which allows us to study the stability phenomena of the considered problem.

Theorem 2

([25])

Assume that (H1) holds and suppose that \(0< N<1\), where \(N=\max (N_{1},N_{2},N_{3})\). Then the problem (1) has a unique solution on I.

Ulam type stabilities

The notion of the stability problem of functional equations originated from a problem of Stanislaw Ulam [26], posed in 1940: When can we assert that approximate solution of a functional equation can be approximated by a solution of the corresponding equation. In 1941, Hyers [14] solved it. This approach can guarantee that there exists a close exact solution useful in many applications. For more details on the recent advances on the Hyers–Ulam stability (see for example [9, 11, 24, 27]).

In order to study some types of Ulam stability for the problem (1), we consider the following fractional differential equation:

Let \(1\leq \beta \leq 2\), \(0\leq \gamma \leq \alpha \leq 1\) and ϵ a positive real numbers and the function \(T\in C(I,\mathbb{R}^{+})\). We consider the following fractional differential equation:

$$\begin{aligned}& D^{\alpha }D^{\beta }y(t)+\frac{k}{t^{\lambda }}D^{\alpha }y(t)+a_{1}F \bigl(t,y(t),D^{\gamma }y(t),J^{p}y(t) \bigr) \\& \quad {}+a_{2}G \bigl(t,y(t),D^{\gamma }y(t) \bigr)+a_{3}H \bigl(t,y(t) \bigr)=L(t),\quad t\in I, \end{aligned}$$
(4)

and the following fractional differential inequality:

$$\begin{aligned}& \biggl\vert D^{\alpha }D^{\beta }x(t)+ \frac{k}{t^{\lambda }}D^{\alpha }x(t)+a_{1}F \bigl(t,x(t),D^{\gamma }x(t),J^{p}x(t) \bigr)+a_{2}G \bigl(t,x(t),D^{\gamma }x(t) \bigr) \\& \quad {}+a_{3}H \bigl(t,x(t) \bigr)-L(t) \biggr\vert \leq \epsilon ,\quad t\in I, \end{aligned}$$
(5)
$$\begin{aligned}& \biggl\vert D^{\alpha }D^{\beta }x(t)+ \frac{k}{t^{\lambda }}D^{\alpha }x(t)+a_{1}F \bigl(t,x(t),D^{\gamma }x(t),J^{p}x(t) \bigr)+a_{2}G \bigl(t,x(t),D^{\gamma }x(t) \bigr) \\& \quad {}+a_{3}H \bigl(t,x(t) \bigr)-L(t) \biggr\vert \leq \epsilon T ( t ) ,\quad t\in I, \end{aligned}$$
(6)
$$\begin{aligned}& \biggl\vert D^{\alpha }D^{\beta }x(t)+ \frac{k}{t^{\lambda }}D^{\alpha }x(t)+a_{1}F \bigl(t,x(t),D^{\gamma }x(t),J^{p}x(t) \bigr)+a_{2}G \bigl(t,x(t),D^{\gamma }x(t) \bigr) \\& \quad {}+a_{3}H \bigl(t,x(t) \bigr)-L(t) \biggr\vert \leq T ( t ) ,\quad t\in I. \end{aligned}$$
(7)

Definition 3

The problem (1) is Ulam–Hyers stable, if there exists a real number \(S>0\), such that, for each \(\epsilon >0\), \(t\in I\), and for each \(x\in X\) solution of (5), there exists a solution \(y\in X\) of (4) (with the same conditions as in (1)), such that

$$ \Vert x-y \Vert _{X}\leq S\epsilon ,\quad t\in I. $$

Definition 4

The problem (1) is generalized Ulam–Hyers stable, if there exists an increasing function \(Z\in C(\mathbb{R}^{+},\mathbb{R}^{+})\), \(Z(0)=0\), such that, for all \(\epsilon >0\), and for each solution \(x\in X\) of (5), there exists a solution \(y\in X\) of (4) (with the same conditions as in (1)), such that

$$ \Vert x-y \Vert _{X}\leq Z(\epsilon ),\quad t\in I. $$

Definition 5

The problem (1) is Ulam–Hyers–Rassias stable, if there exists a function \(T\in C(\mathbb{I},\mathbb{R}^{+})\) and \(\sigma >0\), such that for each \(\epsilon >0\) and for all solutions \(x\in X\) of (6) there exists a solution \(y\in X\) of (4) (with the same conditions as in (1)), such that

$$ \bigl\vert x ( t ) -y ( t ) \bigr\vert \leq \sigma \epsilon T ( t ) ,\quad t \in I. $$

Definition 6

The problem (1) is generalized Ulam–Hyers–Rassias stable, if there exists a function \(T\in C(\mathbb{I},\mathbb{R}^{+})\) and \(\sigma >0\), such that for all solutions \(x\in X\) of (7) there exists a solution \(y\in X\) of (4) (with the same conditions as in (1)), such that

$$ \bigl\vert x ( t ) -y ( t ) \bigr\vert \leq \sigma T ( t ) ,\quad t\in I. $$

Now, we are ready to prove the following result.

Theorem 7

Assume that (H1) is fulfilled and \(N=\max (N_{1},N_{2},N_{3})<1\). Then the problem (1) is Ulam–Hyers stable in X.

Proof

Let us note

$$\begin{aligned}& O = \biggl\vert \int _{0}^{t}\frac{(t-s)^{\beta -1}}{\Gamma (\beta )}\int _{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \biggl[ L(s)-a_{1}F \bigl(s,x(s),D^{\gamma }x(s),J^{p}x(s) \bigr) \\& \hphantom{O ={}}{}-a_{2}G \bigl(s,x(s),D^{\gamma }x(s) \bigr)-a_{3}H \bigl(s,x(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }x(s) \biggr] \,du\,ds \\& \hphantom{O ={}}{}+ \bigl[ K_{1}t^{\beta }+K_{2}t+K_{3} \bigr] \int _{0}^{1}\frac{(1-s)^{\beta -2}}{\Gamma (\beta -1)}\int _{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \\& \hphantom{O ={}}{}\times\biggl[ L(s)-a_{1}F \bigl(s,x(s),D^{\gamma }x(s),J^{p}x(s) \bigr) \\& \hphantom{O ={}}{}-a_{2}G \bigl(s,x(s),D^{\gamma }x(s) \bigr)-a_{3}H \bigl(s,x(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }x(s) \biggr] \,du\,ds \\& \hphantom{O ={}}{}+ \bigl[ K_{4}t^{\beta }+K_{5}t-K_{6} \bigr] \int _{0}^{1} \frac{(1-s)^{\beta -\gamma -1}}{\Gamma (\beta -\gamma )} \int _{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \\& \hphantom{O ={}}{}\times\biggl[ L(s)-a_{1}F \bigl(s,x(s),D^{\gamma }x(s),J^{p}x(s) \bigr) \\& \hphantom{O ={}}{}-a_{2}G \bigl(s,x(s),D^{\gamma }x(s) \bigr)-a_{3}H \bigl(s,x(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }x(s) \biggr] \,du\,ds \\& \hphantom{O ={}}{}- [ K_{7} ] \int _{0}^{t} \frac{(t-s)^{\beta -1}}{\Gamma (\beta )} \int _{0}^{s} \frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \biggl[ L(s)-a_{1}F \bigl(s,x(s),D^{\gamma }x(s),J^{p}x(s) \bigr) \\& \hphantom{O ={}}{}-a_{2}G \bigl(s,x(s),D^{\gamma }x(s) \bigr)-a_{3}H \bigl(s,x(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }x(s) \biggr] \,du\,ds \\& \hphantom{O ={}}{}- \int _{0}^{t}\frac{(t-s)^{\beta -1}}{\Gamma (\beta )}\int _{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \biggl[ L(s)-a_{1}F \bigl(s,y(s),D^{\gamma }y(s),J^{p}y(s) \bigr) \\& \hphantom{O ={}}{}-a_{2}G \bigl(s,y(s),D^{\gamma }y(s) \bigr)-a_{3}H \bigl(s,y(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }y(s) \biggr] \,du\,ds \\& \hphantom{O ={}}{}- \bigl[ K_{1}t^{\beta }+K_{2}t+K_{3} \bigr] \int _{0}^{1}\frac{(1-s)^{\beta -2}}{\Gamma (\beta -1)}\int _{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \biggl[ L(s)-a_{1}F \bigl(s,y(s),D^{\gamma }y(s),J^{p}y(s) \bigr) \\& \hphantom{O ={}}{}-a_{2}G \bigl(s,y(s),D^{\gamma }y(s) \bigr)-a_{3}H \bigl(s,y(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }y(s) \biggr] \,du\,ds \\& \hphantom{O ={}}{}- \bigl[ K_{4}t^{\beta }+K_{5}t-K_{6} \bigr] \int _{0}^{1} \frac{(1-s)^{\beta -\gamma -1}}{\Gamma (\beta -\gamma )} \int _{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \\& \hphantom{O ={}}{}\times\biggl[ L(s)-a_{1}F \bigl(s,y(s),D^{\gamma }y(s),J^{p}y(s) \bigr) \\& \hphantom{O ={}}{}-a_{2}G \bigl(s,y(s),D^{\gamma }y(s) \bigr)-a_{3}H \bigl(s,y(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }y(s) \biggr] \,du\,ds \\& \hphantom{O ={}}{}+ [ K_{7} ] \times \int _{0}^{t} \frac{(t-s)^{\beta -1}}{\Gamma (\beta )} \int _{0}^{s} \frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \biggl[ L(s)-a_{1}F \bigl(s,y(s),D^{\gamma }y(s),J^{p}y(s) \bigr) \\& \hphantom{O ={}}{}-a_{2}G \bigl(s,y(s),D^{\gamma }y(s) \bigr)-a_{3}H \bigl(s,y(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }y(s) \biggr] \,du\,ds \biggr\vert , \\& \begin{aligned} M_{1} ={} & \biggl\vert x(t)- \int _{0}^{t}\frac{(t-s)^{\beta -1}}{\Gamma (\beta )}\int _{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \biggl[ L(s)-a_{1}F \bigl(s,x(s),D^{\gamma }x(s),J^{p}x(s) \bigr) \\ &{}-a_{2}G \bigl(u,x(s),D^{\gamma }x(s) \bigr)-a_{3}H \bigl(s,x(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }x(s) \biggr] \,du\,ds \\ &{}- \bigl[ K_{1}t^{\beta }+K_{2}t+K_{3} \bigr] \\ &{}\times \int _{0}^{1} \frac{(1-s)^{\beta -2}}{\Gamma (\beta -1)}\int _{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \biggl[ L(s)-a_{1}F \bigl(s,x(s),D^{\gamma }x(s),J^{p}x(s) \bigr) \\ &{}-a_{2}G \bigl(u,x(s),D^{\gamma }x(s) \bigr)-a_{3}H \bigl(s,x(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }x(s) \biggr] \,du\,ds \\ &{}- \bigl[ K_{4}t^{\beta }+K_{5}t-K_{6} \bigr] \\ &{}\times \int _{0}^{1} \frac{(1-s)^{\beta -\gamma -1}}{\Gamma (\beta -\gamma )} \int _{0}^{s} \frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \biggl[ L(s)-a_{1}F \bigl(s,x(s),D^{\gamma }x(s),J^{p}x(s) \bigr) \\ &{}-a_{2}G \bigl(u,x(s),D^{\gamma }x(s) \bigr)-a_{3}H \bigl(s,x(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }x(s) \biggr] \,du\,ds \\ &{}+ [ K_{7} ] \\ &{}\times \int _{0}^{t} \frac{(t-s)^{\beta -1}}{\Gamma (\beta )}\int _{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \biggl[ L(s)-a_{1}F \bigl(s,x(s),D^{\gamma }x(s),J^{p}x(s) \bigr) \\ &{}-a_{2}G \bigl(u,x(s),D^{\gamma }x(s) \bigr)-a_{3}H \bigl(s,x(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }x(s) \biggr] \,du\,ds \biggr\vert , \end{aligned}\\& \begin{aligned} M_{2} ={} & \biggl\vert D^{\alpha }x(t)- \int _{0}^{t} \frac{(t-s)^{\beta -1}}{\Gamma (\beta )} \biggl[L(s)-a_{1}F \bigl(s,x(s),D^{\gamma }x(s),J^{p}x(s) \bigr) \\ &{}-a_{2}G \bigl(u,x(s),D^{\gamma }x(s) \bigr)-a_{3}H \bigl(s,x(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }x(s) \biggr] \,ds \\ &{}- \biggl[ \frac{K_{1}\Gamma (\beta +1)t^{\beta -\alpha }}{\Gamma (\beta -\alpha +1)}+ \frac{K_{2}t^{1-\alpha }}{\Gamma (2-\alpha )} \biggr] \\ &{}\times \int _{0}^{1} \frac{(1-s)^{\beta -2}}{\Gamma (\beta -1)} \biggl[ L(s)-a_{1}F \bigl(s,x(s),D^{\gamma }x(s),J^{p}x(s) \bigr) \\ &{}-a_{2}G \bigl(s,x(s),D^{\gamma }x(s) \bigr)-a_{3}H \bigl(s,x(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }x(s) \biggr] \,ds \\ &{}- \biggl[ \frac{K_{4}\Gamma (\beta +1)t^{\beta -\alpha }}{\Gamma (\beta -\alpha +1)}+ \frac{K_{5}t^{1-\alpha }}{\Gamma (2-\alpha )} \biggr] \\ &{}\times \int _{0}^{1} \frac{(1-s)^{\beta -\gamma -1}}{\Gamma (\beta -\gamma )} \biggl[ L(s)-a_{1}F \bigl(s,x(s),D^{\gamma }x(s),J^{p}x(s) \bigr) \\ &{}-a_{2}G \bigl(s,x(s),D^{\gamma }x(s) \bigr)-a_{3}H \bigl(s,x(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }x(s) \biggr] \,ds \\ &{}+ [ K_{7} ] \int _{0}^{t} \frac{(t-s)^{\beta -1}}{\Gamma (\beta )} \biggl[ L(s)-a_{1}F \bigl(s,x(s),D^{\gamma }x(s),J^{p}x(s) \bigr) \\ &{}-a_{2}G \bigl(s,x(s),D^{\gamma }x(s) \bigr)-a_{3}H \bigl(s,x(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }x(s) \biggr] \,ds \biggr\vert , \end{aligned}\\ & \begin{aligned} M_{3} ={} & \biggl\vert D^{\gamma }x(t)- \int _{0}^{t} \frac{(t-s)^{\alpha +\beta -\gamma -1}}{\Gamma (\alpha +\beta -\gamma )} \biggl[ L(s)-a_{1}F \bigl(s,x(s),D^{\gamma }x(s),J^{p}x(s) \bigr)-a_{2} \\ &{}\times G \bigl(u,x(s),D^{\gamma }x(s) \bigr)-a_{3}H \bigl(s,x(s) \bigr)- \frac{k}{s^{\lambda }}D^{ \alpha }x(s) \biggr] \,ds \\ &{}- \biggl[ \frac{K_{1}\Gamma (\beta +1)t^{\beta -\gamma }}{\Gamma (\beta -\gamma +1)}+ \frac{K_{2}t^{1-\gamma }}{\Gamma (2-\gamma )} \biggr] \\ &{}\times \int _{0}^{1} \frac{(1-s)^{\alpha +\beta -\gamma -2}}{\Gamma (\alpha +\beta -\gamma -1)} \biggl[ L(s)-a_{1}F \bigl(s,x(s),D^{\gamma }x(s),J^{p}x(s) \bigr) \\ &{}-a_{2}G \bigl(s,x(s),D^{\gamma }x(s) \bigr)-a_{3}H \bigl(s,x(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }x(s) \biggr] \,ds \\ &{}- \biggl[ \frac{K_{4}\Gamma (\beta +1)t^{\beta -\gamma }}{\Gamma (\beta -\gamma +1)}+ \frac{K_{5}t^{1-\gamma }}{\Gamma (2-\gamma )} \biggr] \\ &{}\times \int _{0}^{1} \frac{(1-s)^{\alpha +\beta -2\gamma -1}}{\Gamma (\alpha +\beta -2\gamma )} \biggl[ L(s)-a_{1}F \bigl(s,x(s),D^{\gamma }x(s),J^{p}x(s) \bigr) \\ &{}-a_{2}G \bigl(s,x(s),D^{\gamma }x(s) \bigr)-a_{3}H \bigl(s,x(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }x(s) \biggr] \,ds \\ &{}+ [ K_{7} ] \int _{0}^{t} \frac{(t-s)^{\alpha +\beta -\gamma -1}}{\Gamma (\alpha +\beta -\gamma )} \biggl[ L(s)-a_{1}F \bigl(s,x(s),D^{\gamma }x(s),J^{p}x(s) \bigr) \\ &{}-a_{2}G \bigl(s,x(s),D^{\gamma }x(s) \bigr)-a_{3}H \bigl(s,x(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }x(s) \biggr] \,ds \biggr\vert . \end{aligned} \end{aligned}$$

Let now \(x\in X\) be a solution of (5). Then, by integrating (5), we obtain

$$ M_{1} \leq \frac{\epsilon t^{\alpha +\beta }}{\Gamma (\alpha +\beta +1)}. $$

Thanks to Theorem 2, the unique solution of (1) is given by

$$\begin{aligned} y(t) =& \int _{0}^{t}\frac{(t-s)^{\beta -1}}{\Gamma (\beta )}\int _{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \biggl[ L(s)-a_{1}F \bigl(s,y(s),D^{\gamma }y(s),J^{p}y(s) \bigr) \\ &{}-a_{2}G \bigl(s,y(s),D^{\gamma }y(s) \bigr)-a_{3}H \bigl(s,y(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }y(s) \biggr] \,du\,ds \\ &{}+ \bigl[ K_{1}t^{\beta }+K_{2}t+K_{3} \bigr] \\ &{}\times \int _{0}^{1}\frac{(1-s)^{\beta -2}}{\Gamma (\beta -1)}\int _{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \biggl[ L(s)-a_{1}F \bigl(s,y(s),D^{\gamma }y(s),J^{p}y(s) \bigr) \\ &{}-a_{2}G \bigl(s,y(s),D^{\gamma }y(s) \bigr)-a_{3}H \bigl(s,y(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }y(s) \biggr] \,du\,ds \\ &{}+ \bigl[ K_{4}t^{\beta }+K_{5}t-K_{6} \bigr] \\ &{}\times \int _{0}^{1} \frac{(1-s)^{\beta -\gamma -1}}{\Gamma (\beta -\gamma )} \int _{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \biggl[ L(s)-a_{1}F \bigl(s,y(s),D^{\gamma }y(s),J^{p}y(s) \bigr) \\ &{}-a_{2}G \bigl(s,y(s),D^{\gamma }y(s) \bigr)-a_{3}H \bigl(s,y(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }y(s) \biggr] \,du\,ds \\ &{}- [ K_{7} ] \times \int _{0}^{t} \frac{(t-s)^{\beta -1}}{\Gamma (\beta )} \int _{0}^{s} \frac{(s-u)^{\alpha -1}}{\Gamma (\alpha )} \biggl[ L(s)-a_{1}F \bigl(s,y(s),D^{\gamma }y(s),J^{p}y(s) \bigr) \\ &{}-a_{2}G \bigl(s,y(s),D^{\gamma }y(s) \bigr)-a_{3}H \bigl(s,y(s) \bigr)- \frac{k}{s^{\lambda }}D^{\alpha }y(s) \biggr] \,du\,ds. \end{aligned}$$

Then, from all \(t\in I\), we get

$$\begin{aligned} \bigl\vert x(t)-y(t) \bigr\vert \leq &\frac{\epsilon t^{\alpha +\beta }}{\Gamma (\alpha +\beta +1)}+ O. \end{aligned}$$

This implies that

$$ \Vert x-y \Vert _{\infty }\leq \frac{\epsilon }{\Gamma (\alpha +\beta +1)}+N_{1} \Vert x-y \Vert _{X}. $$
(8)

By integrating and differentiating (5), we get

$$ M_{2} \leq \frac{\epsilon t^{\beta }}{\Gamma (\beta +1)}. $$

Similarly, we show that

$$ \bigl\Vert D^{\alpha }x-D^{\alpha }y \bigr\Vert _{\infty }\leq \frac{\epsilon }{\Gamma (\beta +1)}+N_{2} \Vert x-y \Vert _{X}. $$
(9)

On the other hand, we have

$$ M_{3} \leq \frac{\epsilon t^{\alpha +\beta -\gamma }}{\Gamma (\alpha +\beta -\gamma +1)}. $$

Also, we have

$$ \bigl\Vert D^{\gamma }x-D^{\gamma }y \bigr\Vert _{\infty }\leq \frac{\epsilon }{\Gamma (\alpha +\beta -\gamma +1)}+N_{3} \Vert x-y \Vert _{X}. $$
(10)

Using the inequalities (8), (9) and (10), we get

$$ \Vert x-y \Vert _{X}\leq \max \biggl( \frac{\epsilon }{\Gamma (\alpha +\beta +1)}, \frac{\epsilon }{\Gamma (\beta +1)}, \frac{\epsilon }{\Gamma (\alpha +\beta -\gamma +1)} \biggr) +N \Vert x-y \Vert _{X}. $$

Thus,

$$ \Vert x-y \Vert _{X}\leq S \epsilon , $$

such that

$$ S= \frac{\max ( \frac{1}{\Gamma (\alpha +\beta +1)},\frac{1}{\Gamma (\beta +1)},\frac{1}{\Gamma (\alpha +\beta -\gamma +1)} ) }{1-N} > 0. $$

Consequently, the problem (1) shows the Ulam–Hyers stability. □

Taking \(Z(\epsilon )=S\epsilon \), we can state that the problem (1) is generalized Ulam–Hyers stable.

In the following, we introduce the following hypothesis to study Rassias stability.

  1. (H2):

    \(T\in C(\mathbb{I},\mathbb{R}^{+})\) is continuous, nondecreasing function, and there exists \(\lambda _{T,\alpha }>0\) such that \(J^{\alpha }T ( t ) \leq \lambda _{T,\alpha }T ( t ) \) for each \(t\in I\).

We present the following result.

Theorem 8

Assume that (H1)(H2) are satisfied and \(N:=\max (N_{1},N_{2},N_{3})<1\).

Then the problem (1) is Ulam–Hyers–Rassias stable in X.

Proof

Let \(x\in X\) be a solution of (6). Then, by integrating (6), we obtain

$$ M_{1} \leq \epsilon J^{\beta }J^{\alpha }T ( t ). $$

Let y be the unique solution of the problem (1). Then, for each \(t\in I\), we have

$$ \bigl\vert x(t)-y(t) \bigr\vert \leq \epsilon J^{\beta }J^{\alpha }T ( t ) + O. $$

In view of (H2), we have

$$\begin{aligned}& \bigl\vert x(t)-y(t) \bigr\vert \leq \epsilon J^{\beta }J^{\alpha }T ( t ) +N_{1} \Vert x-y \Vert _{X}\leq \epsilon \lambda _{T,\beta +\alpha }T ( t ) +N_{1} \Vert x-y \Vert _{X}, \\& \quad \text{which implies that } \bigl\vert x(t)-y(t) \bigr\vert \leq \epsilon \lambda _{T,\beta +\alpha }T ( t ) +N_{1} \Vert x-y \Vert _{X}. \end{aligned}$$
(11)

On the other hand, by integrating and differentiating (6), we get

$$ M_{2} \leq \epsilon J^{\beta }T ( t ) . $$

Also, we can show that

$$ \bigl\vert D^{\alpha }x-D^{\alpha }y \bigr\vert \leq \epsilon \lambda _{T,\beta }T ( t ) +N_{2} \Vert x-y \Vert _{X} . $$
(12)

We have also

$$ M_{3} \leq \epsilon J^{\alpha +\beta -\gamma }T ( t ). $$

By the same arguments as before, we observe that

$$ \bigl\vert D^{\gamma }x ( t ) -D^{\gamma }y ( t ) \bigr\vert \leq \epsilon \lambda _{T,\alpha +\beta - \gamma }T ( t ) +N_{3} \Vert x-y \Vert _{X} . $$
(13)

Using the inequalities (11), (12) and (13) yields

$$ \textstyle\begin{cases} \vert x(t)-y(t) \vert \leq \epsilon \max ( \lambda _{T,\alpha +\beta }, \lambda _{T,\beta },\lambda _{T,\alpha +\beta -\gamma } ) T ( t ) +N_{1} \Vert x-y \Vert _{X},& t\in I, \\ \vert D^{\alpha }x ( t ) -D^{\alpha }y ( t ) \vert \leq \epsilon \max ( \lambda _{T,\alpha + \beta },\lambda _{T,\beta },\lambda _{T,\alpha +\beta -\gamma } ) T ( t ) +N_{2} \Vert x-y \Vert _{X},& t\in I, \\ \vert D^{\gamma }x ( t ) -D^{\gamma }y ( t ) \vert \leq \epsilon \max ( \lambda _{T,\alpha + \beta },\lambda _{T,\beta },\lambda _{T,\alpha +\beta -\gamma } ) T ( t ) +N_{3} \Vert x-y \Vert _{X},& t\in I. \end{cases}$$

Hence, it follows that there exists a real number

$$ \sigma = \frac{\max ( \lambda _{T,\alpha +\beta },\lambda _{T,\beta },\lambda _{T,\alpha +\beta -\gamma } )}{1-N}, $$

such that

$$ \Vert x-y \Vert _{X}\leq \sigma \epsilon T ( t ),\quad t\in I. $$

Consequently, the problem (1) shows the Ulam–Hyers–Rassias stability. □

Numerical simulations

In this section, we recall a numerical approach for the Caputo derivative. Then, for some fixed parameters, we investigate behavior of the above fractional Lane–Emden problem. To do this, we shall first obtain a reduced fractional differential system that is equivalent to our studied problem. Using a fourth-order Runge–Kutta integrator, the numerical simulations recover the convective behavior of the integer model in astrophysics [4]. In order to ensure the effect of the fractional order in Lane–Emden dynamics, we consider judicious values for α and β.

  • Hydrodynamic simulations of giant stars, where the stellar profiles can be modeled in [12, 22, 28] as

    $$ \frac{1}{t}\frac{d}{d t} \biggl(t^{2} \frac{dy}{d t} + t^{2} \frac{g_{c}(a t)}{4 a \pi G p_{0} } \biggr)+ y^{n}=0 , $$

    where y is the polytropic temperature with index n, \(t\equiv \frac{r}{a} \), and \(p_{0}\) the central gas density. For \(r \leq \frac{h}{2}\) and \(x \equiv \frac{r}{h}\), the smoothed gravitational force of the core is defined by

    $$ g_{c}(r):= G m_{c} \frac{x (\frac{32}{3}+x^{2} (\frac{-192}{5}+32x ) )}{h^{2}} . $$
  • Self-similar profiles of nonlinear wave equations in flat space-time were modeled in [4, 17] as

    $$ \bigl(1-t^{2} \bigr)\frac{d^{2}y}{d t^{2}}+ \biggl( \frac{A}{t}+ B t \biggr) \frac{dy}{d t} - C y + D y^{E} =0 . $$

Numerical approach for Caputo derivative

In this subsection, we presented an important numerical approach for the Riemann–Liouville fractional integral and the Caputo derivative; we recall the theorems of [6, 19].

Theorem 9

Assume that \(y\in \mathcal{C}^{1}([0,1],\mathbb{R})\). The fractional integration approach is given by

$$ J^{\alpha }y(t_{i})\simeq \frac{h^{\alpha }}{\Gamma (\alpha +2)}\sum _{j=0}^{i}y(t_{j}) \sigma _{j}(\alpha ), \quad i=0, \dots ,n+1, $$

where

$$ \sigma _{j}(\alpha )= \textstyle\begin{cases} (n+2-j)^{(\alpha +1)}+(n-j)^{(\alpha +1)}-2(n-j+1)^{(\alpha +1)},& j=1,\dots, i-1, \\ (n)^{(\alpha +1)}-(n-\alpha )(n+1)^{\alpha },& j=0, \textit{and } 1, j=i . \end{cases}$$

Theorem 10

Assume that \(y\in \mathcal{C}^{1}([0,1],\mathbb{R})\) and \(0<\alpha \leq 1\). Then we have

$$ D^{\alpha }y(t_{i})\simeq \frac{h^{1-\alpha }}{\Gamma (1-\alpha +2)}\sum _{j=0}^{i}y^{(j)}(t_{j}) \sigma _{j}(1-\alpha ) ,\quad i=0,\dots ,n, $$

where

$$ y^{(j)}= \textstyle\begin{cases} \frac{y_{1}-y_{0}}{h}, \quad j=0,\qquad \frac{y_{j+1}-y_{j-1}}{2h},\quad j=1,\dots, i-1,\qquad \frac{y_{i}-y_{i-1}}{h},\quad j=i. \end{cases} $$

Simulation for Lane–Emden behaviors

We note that the problem (1) can be reduced to the following system:

$$\begin{aligned}& D^{\beta }y(t) = z(t), \\& \begin{aligned} D^{\alpha }z(t) ={}& {-}\frac{k}{t^{\lambda }}D^{\alpha }y(t)-a_{1}F \bigl(t,y(t),D^{\gamma }y(t),J^{p}y(t) \bigr) \\ &{} - a_{2}G \bigl(t,y(t),D^{\gamma }y(t) \bigr)-a_{3}H \bigl(t,y(t) \bigr)+L(t). \end{aligned} \end{aligned}$$

In order to achieve the mentioned phenomena, we take \(1<\alpha +\beta \leq 2\), and \(\lambda =1\). Taking into account our problem parameters, three cases can be observed:

  1. Case 1:

    \(\alpha =\beta =1\), we get

    $$\begin{aligned}& Dy(t) = z(t), \\& \begin{aligned} Dz(t) ={}& {-}\frac{k}{t^{\lambda }}Dy(t)-a_{1}F \bigl(t,y(t),D^{\gamma }y(t),J^{p}y(t) \bigr) \\ &{} - a_{2}G \bigl(t,y(t),D^{\gamma }y(t) \bigr) -a_{3}H \bigl(t,y(t) \bigr)+L(t). \end{aligned} \end{aligned}$$
  2. Case 2:

    \(0<\alpha \leq 1\), \(\beta =1\), we obtain

    $$\begin{aligned}& Dy(t) = z(t), \\& \begin{aligned} D^{\alpha }z(t) ={}& {-}\frac{k}{t^{\lambda }}D^{\alpha }y(t)-a_{1}F \bigl(t,y(t),D^{\gamma }y(t),J^{p}y(t) \bigr) \\ &{} - a_{2}G \bigl(t,y(t),D^{\gamma }y(t) \bigr) -a_{3}H \bigl(t,y(t) \bigr)+L(t). \end{aligned} \end{aligned}$$

    As a consequence,

    $$\begin{aligned}& Dy(t) = z(t), \\& \begin{aligned} Dz(t) = {}& D^{1-\alpha } \biggl( -\frac{k}{t^{\lambda }}D^{\alpha }y(t)-a_{1}F \bigl( t,y(t),D^{\gamma }y(t),J^{p}y(t) \bigr) \\ &{} - a_{2}G \bigl(t,y(t),D^{\gamma }y(t) \bigr) -a_{3}H \bigl(t,y(t) \bigr)+L(t) \biggr). \end{aligned} \end{aligned}$$
  3. Case 3:

    \(0<\alpha \leq 1\), \(1\leq \beta \leq 2\), we have

    $$\begin{aligned}& D^{\beta }y(t) = z(t), \\& \begin{aligned} D^{\alpha }z(t) ={}& {-}\frac{k}{t^{\lambda }}D^{\alpha }y(t)-a_{1}F \bigl(t,y(t),D^{\gamma }y(t),J^{p}y(t) \bigr) \\ &{} - a_{2}G \bigl(t,y(t),D^{\gamma }y(t) \bigr) -a_{3}H \bigl(t,y(t) \bigr)+L(t), \end{aligned} \end{aligned}$$

    that is,

    $$\begin{aligned}& J^{2-\beta }D \bigl[Dy(t) \bigr] = z(t), \\& \begin{aligned} D^{\alpha }z(t) = {}& {-}\frac{k}{t^{\lambda }}D^{\alpha }y(t)-a_{1}F \bigl(t,y(t),D^{\gamma }y(t),J^{p}y(t) \bigr) \\ &{} - a_{2}G \bigl(t,y(t),D^{\gamma }y(t) \bigr) -a_{3}H \bigl(t,y(t) \bigr)+L(t). \end{aligned} \end{aligned}$$

    Therefore,

    $$\begin{aligned}& Dy(t) = z(t), \\& J^{2-\beta }D z = w(t), \\& \begin{aligned} D^{\alpha }w(t) ={}& {-}\frac{k}{t^{\lambda }}D^{\alpha }y(t)-a_{1}F \bigl(t,y(t),D^{\gamma }y(t),J^{p}y(t) \bigr) \\ &{} - a_{2}G \bigl(t,y(t),D^{\gamma }y(t) \bigr) -a_{3}H \bigl(t,y(t) \bigr)+L(t). \end{aligned} \end{aligned}$$

    Consequently,

    $$\begin{aligned}& Dy(t) = z(t), \\& Dz(t) = D^{2-\beta } w(t), \\& \begin{aligned} Dw(t) = {}& D^{1-\alpha } \biggl( -\frac{k}{t^{\lambda }}D^{\alpha }y(t)-a_{1}F \bigl( t,y(t),D^{\gamma }y(t),J^{p}y(t) \bigr) \\ &{} - a_{2}G \bigl(t,y(t),D^{\gamma }y(t) \bigr) -a_{3}H \bigl(t,y(t) \bigr)+L(t) \biggr). \end{aligned} \end{aligned}$$

I: As a first simulation, we consider the hydrodynamic simulations of giant stars, where \(k=2\), \(p =\gamma =0.01\), and f, H, G, H, L are given by

$$\begin{aligned}& a_{1}F \bigl(t,y(t),D^{\gamma }y(t),J^{p}y(t) \bigr) = \frac{16a^{4}m_{c}}{\pi p_{0}h^{6}} t^{4}+\frac{289}{51t} \bigl(J^{p}y(t) \bigr)^{n} , \\& a_{2}G \bigl(t,y(t),D^{\gamma }y(t) \bigr) = - \frac{48a^{3}m_{c}}{\pi p_{0}h^{5}} t^{3}- \frac{663}{255t} \bigl(D^{\gamma }y(t) \bigr)^{n}, \\& a_{3}H \bigl(t,y(t) \bigr) = \frac{32a^{4}m_{c}}{\pi p_{0}h^{6}} t^{4}-\frac{527}{255t} \bigl(y(t) \bigr)^{n}, \\& L(t) = \frac{8am_{c}}{\pi p_{0}h^{3}} t. \end{aligned}$$

For the first case, with initial conditions \((0,0)\), \(h =0.001\), and \(n =\{ 1, 1.5, 2, 2.5 \}\), the numerical simulations are carried out only by the fourth-order Runge–Kutta method, for specific parameters, we have Fig. 1.

Figure 1
figure 1

Numerical simulation of Case 1 for different values of the polytropic index n and \(\alpha = \beta =1\)

Remark 11

Through ongoing evaluation, we observe that the change in value of n has no impact on the attitude of the remaining cases.

For the following simulation we take \(n=1.5\) as it is more adequate. Now, to ensure that all three cases are convenient, we should be looking for a suitable fractional order.

For the second case, with initial conditions \((0,0)\), \(h=0.001\), and \(\alpha =\{0.55,0.35,0.2\}\), numerical simulations are realized by a combination of the Caputo approach and the fourth-order Runge–Kutta method, we acquire Fig. 2. By comparing the above result with the one of the first case, we conclude that both cases are adequate for \(\alpha =0.35\) (see Fig. 3).

Figure 2
figure 2

Numerical simulations of Case 2 for \(\alpha =\{0.55,0.35,0.2\}\) and \(\beta =1\)

Figure 3
figure 3

Comparative simulation (Case 1–Case 2)

For the third case, with initial conditions \((0,0,0.5)\), \(h=0.001\), \(\beta =\{1.45,1.3,1.05\}\), for any β value, we take \(\alpha =\{0.55,0.35,0.2\}\). Numerical simulations are carried out by a combination of the Caputo approach and the fourth-order Runge–Kutta method, we see, according to Figs. 46, that \(\beta =1.3\) is the valid value. It is obvious from Fig. 7 that \(\alpha =0.35\) is the appropriate value.

Figure 4
figure 4

Numerical simulations of Case 3 for \(\beta =1.45\) and \(\alpha =\{0.55,0.35,0.2\}\)

Figure 5
figure 5

Numerical simulations of Case 3 for \(\beta =1.3\) and \(\alpha =\{0.55,0.35,0.2\}\)

Figure 6
figure 6

Numerical simulations of Case 3 for \(\beta =1.05\) and \(\alpha =\{0.55,0.35,0.2\}\)

Figure 7
figure 7

Comparative simulation (Case 1–Case 3)

II: As a second simulation, we consider self-similar profiles of nonlinear wave equation in flat space-time, where \(k=A\), \(p =\gamma =0.01\), and f, H, G, H, L are given by

$$\begin{aligned}& a_{1}F \bigl(t,y(t),D^{\gamma }y(t),J^{p}y(t) \bigr) = \frac{C}{1-t^{2}}J^{p}y(t), \\& a_{2}G \bigl(t,y(t),D^{\gamma }y(t) \bigr) = - \frac{B t}{1-t^{2}}D^{\gamma }y(t), \\& a_{3}H \bigl(t,y(t) \bigr) = -\frac{D}{1-t^{2}} \bigl(y(t) \bigr)^{E}, \\& L(t) = 0, \end{aligned}$$

with initial conditions \((0.576037116,0.24090)\), and \(A=2\), \(B=\frac{-25}{12}\), \(C=\frac{1}{4}\), \(D=1\), \(E=2\), \(h=0.01\). The integration for the first case is carried out by the fourth-order Runge–Kutta method, now, we are trying to determine an appropriate fractional order (see Fig. 8).

Figure 8
figure 8

Numerical simulation of Case 1

For the second case, we take the same data as above, and \(\alpha =\{0.95,0.9,0.8\}\). Numerical simulations are realized by a combination of the Caputo approach and the fourth-order Runge–Kutta method (see Fig. 9).

Figure 9
figure 9

Numerical simulation of Case 2

Comparing our outcome to that in the first case, we summarize that the two cases are consistent in terms of \(\alpha =0.95\) (see Fig. 10).

Figure 10
figure 10

Comparative simulation (Case 1–Case 2)

For the third case, with initial conditions \((0.576037116,0.24090,0)\), and \(h=0.01\), \(\beta =\{1.2,1.15,1.05\}\), for each β, we take \(\alpha =\{0.95,0.9,0.8\}\). Numerical simulations are carried out by a combination of the Caputo approach and the fourth-order Runge–Kutta method.

It appeared from Figs. 1113 that \(\beta =1.2\) and \(\alpha =0.8\) are the most acceptable values too.

Figure 11
figure 11

Numerical simulations of Case 3 for \(\beta =1.2\) and different values of α, on the left side. Comparative simulation (Case 1–Case 3), on the right side

Figure 12
figure 12

Numerical simulations of Case 3 for \(\beta =1.15\) and different values of α, on the left side. Comparative simulation (Case 1–Case 3), on the right side

Figure 13
figure 13

Numerical simulations of Case 3 for \(\beta =1.05\) and different values of α, on the left side. Comparative simulation (Case 1–Case 3), on the right side

Conclusions

In this manuscript, we study some types of Ulam stability for a nonlinear fractional differential equation of Lane–Emden type with antiperiodic conditions. Then, by using a numerical approach for the Caputo derivative, we investigate the behaviors of the considered system.

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Acknowledgements

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

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Correspondence to Mehmet Zeki Sarikaya.

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Tablennehas, K., Dahmani, Z., Belhamiti, M.M. et al. On a fractional problem of Lane–Emden type: Ulam type stabilities and numerical behaviors. Adv Differ Equ 2021, 324 (2021). https://doi.org/10.1186/s13662-021-03483-w

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MSC

  • 30C45
  • 39B72
  • 39B82

Keywords

  • Lane–Emden equation
  • Caputo derivative
  • Random differential equation
  • Ulam stability
  • Fourth Runge–Kutta method