New analytical method of solution to a nonlinear singular fractional Lane–Emden type equation

McSylvester Ejighikeme Omaba; McSylvester Ejighikeme Omaba

doi:10.3934/math.20221072

AIMS Mathematics

2022, Volume 7, Issue 10: 19539-19552. doi: 10.3934/math.20221072

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New analytical method of solution to a nonlinear singular fractional Lane–Emden type equation

McSylvester Ejighikeme Omaba ^,

Department of Mathematics, College of Science, University of Hafr Al Batin, P.O. Box 1803, Hafr Al Batin 31991, Saudi Arabia

Received: 11 August 2022 Revised: 29 August 2022 Accepted: 30 August 2022 Published: 05 September 2022
MSC : 26A33, 28A80, 34A08, 34A09

We consider a nonlinear singular fractional Lane–Emden type differential equation

$ {}^{LC}\mathcal{D}^\alpha_{a^+}\varphi(t)+\frac{\lambda}{t^{\alpha-\beta}}\, \, {}^{LC}\mathcal{D}^\beta_{a^+}\varpi(t, \varphi(t)) = 0, \, \, 0<\beta<\alpha<1, \, \, 0< a<t\leq T, $

with an initial condition $ \varphi(a) = \nu $ assumed to be bounded and non-negative, $ \varpi:[a, T]\times\mathbb{R}\rightarrow \mathbb{R} $ a Lipschitz continuous function, and $ {}^{LC}\mathcal{D}^\alpha_{a^+}, {}^{LC}\mathcal{D}^\beta_{a^+} $ are Liouville–Caputo derivatives of orders $ 0 < \alpha, \beta < 1 $. A new analytical method of solution to the nonlinear singular fractional Lane–Emden type equation using fractional product rule and fractional integration by parts formula is proposed. Furthermore, we prove the existence and uniqueness and the growth estimate of the solution. Examples are given to illustrate our results.
- well-posedness,
- growth bound,
- nonlinear fractional Lane–Emden equation,
- regularized incomplete beta function,
- new analytical method
Citation: McSylvester Ejighikeme Omaba. New analytical method of solution to a nonlinear singular fractional Lane–Emden type equation[J]. AIMS Mathematics, 2022, 7(10): 19539-19552. doi: 10.3934/math.20221072

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Abstract

We consider a nonlinear singular fractional Lane–Emden type differential equation

$ {}^{LC}\mathcal{D}^\alpha_{a^+}\varphi(t)+\frac{\lambda}{t^{\alpha-\beta}}\, \, {}^{LC}\mathcal{D}^\beta_{a^+}\varpi(t, \varphi(t)) = 0, \, \, 0<\beta<\alpha<1, \, \, 0< a<t\leq T, $

with an initial condition $ \varphi(a) = \nu $ assumed to be bounded and non-negative, $ \varpi:[a, T]\times\mathbb{R}\rightarrow \mathbb{R} $ a Lipschitz continuous function, and $ {}^{LC}\mathcal{D}^\alpha_{a^+}, {}^{LC}\mathcal{D}^\beta_{a^+} $ are Liouville–Caputo derivatives of orders $ 0 < \alpha, \beta < 1 $. A new analytical method of solution to the nonlinear singular fractional Lane–Emden type equation using fractional product rule and fractional integration by parts formula is proposed. Furthermore, we prove the existence and uniqueness and the growth estimate of the solution. Examples are given to illustrate our results.

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