Lie algebra classification, conservation laws and invariant solutions for modification of the generalization of the Emden--Fowler equation.

We obtain the optimal system’s generating operators associated to a modification of the generalization of the Emden–Fowler Equation. equation. Using those operators we characterize all invariant solutions associated to a generalized. Moreover, we present the variational symmetries and the corresponding conservation laws, using Noether’s theorem and Ibragimov’s method. Finally, we classify the Lie algebra associated to the given equation.

Shifted Jacobi polynomials for nonlinear singular variable-order time fractional Emden–Fowler equation generated by derivative with non-singular kernel

In this work, a nonlinear singular variable-order fractional Emden–Fowler equation involved with derivative with non-singular kernel (in the Atangana–Baleanu–Caputo type) is introduced and a computational method is proposed for its numerical solution. The desired method is established upon the shifted Jacobi polynomials and their operational matrix of variable-order fractional differentiation (which is extracted in the present study) together with the spectral collocation method. The presented method transforms obtaining the solution of the main problem into obtaining the solution of an algebraic system of equations. Several numerical examples are examined to show the validity and the high accuracy of the established method.

Existence of a solution of discrete Emden-Fowler equation caused by continuous equation

<p style='text-indent:20px;'>The paper studies the asymptotic behaviour of solutions to a second-order non-linear discrete equation of Emden–Fowler type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0 $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ u\colon \{k_0, k_0+1, \dots\}\to \mathbb{R} $\end{document}</tex-math></inline-formula> is an unknown solution, <inline-formula><tex-math id="M2">\begin{document}$ \Delta^2 u(k) $\end{document}</tex-math></inline-formula> is its second-order forward difference, <inline-formula><tex-math id="M3">\begin{document}$ k_0 $\end{document}</tex-math></inline-formula> is a fixed integer and <inline-formula><tex-math id="M4">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ m $\end{document}</tex-math></inline-formula> are real numbers, <inline-formula><tex-math id="M6">\begin{document}$ m\not = 0, 1 $\end{document}</tex-math></inline-formula>.</p>