Accurate solution of the Thomas-Fermi equation using the fractional order of rational Chebyshev functions. (English) Zbl 1362.34039
Summary: In this paper, the nonlinear singular Thomas-Fermi differential equation for neutral atoms is solved using the fractional order of rational Chebyshev orthogonal functions (FRCs) of the first kind, \(F T_n^\alpha(t, L)\), on a semi-infinite domain, where \(L\) is an arbitrary numerical parameter. First, using the quasilinearization method, the equation be converted into a sequence of linear ordinary differential equations (LDEs), and then these LDEs are solved using the FRCs collocation method. Using 300 collocation points, we have obtained a very good approximation solution and the value of the initial slope \(y^\prime(0) = - 1.5880710226113753127186845094239501095\), highly accurate to 37 decimal places.
MSC:
| 34B16 | Singular nonlinear boundary value problems for ordinary differential equations |
| 34B40 | Boundary value problems on infinite intervals for ordinary differential equations |
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
Keywords:
Thomas-Fermi equation; fractional order of rational Chebyshev functions; quasilinearization method; collocation method; nonlinear ODE; semi-infinite domainReferences:
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