Existence of positive solutions for a generalized and fractional ordered Thomas-Fermi theory of neutral atoms

AbstractThe singular boundary value problem we discuss is as follows: D0+αCu(t)=λq(t)f(t,u(t)),0<t<1,α1u(0)+α2u′(0)=a,β1u(1)+β2u′(1)=b, $$\begin{aligned}& {}^{\mathrm{C}}D_{0^{+}}^{\alpha}u(t)=\lambda q(t)f \bigl(t,u(t)\bigr),\quad 0< t< 1, \\& \alpha_{1}u(0)+\alpha_{2}u'(0)=a,\qquad \beta_{1}u(1)+\beta_{2}u'(1)=b, \end{aligned}$$ where 1<α≤2$1<\alpha\leq2$, λ>0$\lambda>0$ is a parameter, D0+αC${}^{\mathrm{C}}D_{0^{+}}^{\alpha}$ is the Caputo fractional derivative. We present the existence of positive solutions for a fractional boundary value problem modeled from the Thomas-Fermi equation subjected to Sturm-Liouville boundary conditions.

[1]  Ewa Pawluszewicz,et al.  Controllability of h-difference linear control systems with two fractional orders , 2012, Proceedings of the 13th International Carpathian Control Conference (ICCC).

[2]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[3]  Wenquan Feng,et al.  Existence of solutions for a singular system of nonlinear fractional differential equations , 2011, Comput. Math. Appl..

[4]  Ewa Pawluszewicz,et al.  Local controllability of nonlinear discrete-time fractional order systems , 2013 .

[5]  D. O’Regan,et al.  An upper and lower solution approach for a generalized Thomas-Fermi theory of neutral atoms. , 2002 .

[6]  Shuqin Zhang,et al.  Solutions to boundary-value problems for nonlinear differential equations of fractional order. , 2009 .

[7]  Á. Nagy,et al.  Density functional. Theory and application to atoms and molecules , 1998 .

[8]  You-Hui Su,et al.  Positive Solution to a Singular -Laplacian BVP with Sign-Changing Nonlinearity Involving Derivative on Time Scales , 2009 .

[9]  J. Machado.  Analysis and design of fractional-order digital control systems , 1997 .

[10]  Meng Zhang,et al.  Positive solutions for boundary value problems of nonlinear fractional differential equations , 2011, Appl. Math. Comput..

[11]  Shurong Sun,et al.  Positive solutions to fractional boundary value problems with nonlinear boundary conditions , 2014 .

[12]  Shurong Sun,et al.  The existence of solutions for boundary value problem of fractional hybrid differential equations , 2012 .

[13]  W. Ames,et al.  Nonlinear problems in abstract cones , 1988 .

[14]  The nonextensive Thomas–Fermi theory in an n-dimensional space , 2013 .

[15]  Krishnan Balachandran,et al.  On the controllability of fractional dynamical systems , 2012, Int. J. Appl. Math. Comput. Sci..

[16]  Daqing Jiang,et al.  Upper and lower solutions method and a second order three-point singular boundary value problem , 2008, Comput. Math. Appl..

[17]  I. Podlubny.  Fractional differential equations , 1998 .

[18]  Tadeusz Kaczorek,et al.  Selected Problems of Fractional Systems Theory , 2011 .

[19]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[20]  Hong‐Rui Sun,et al.  On the Number of Positive Solutions for a Nonlinear Third Order Boundary Value Problem , 2006 .

[21]  Yige Zhao,et al.  The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations , 2011 .

[22]  Chengbo Zhai,et al.  Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter , 2014, Commun. Nonlinear Sci. Numer. Simul..

[23]  Ishak Hashim,et al.  On the rational second kind Chebyshev pseudospectral method for the solution of the Thomas-Fermi equation over an infinite interval , 2014, J. Comput. Appl. Math..

[24]  Qiuping Li,et al.  Existence and uniqueness of solutions for a coupled system of multi-term nonlinear fractional differential equations , 2012, Comput. Math. Appl..

[25]  D. Sierociuk,et al.  Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation , 2006 .