A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions

In this paper, the fractional order of rational Bessel functions collocation method (FRBC) to solve Thomas-Fermi equation which is defined in the semi-infinite domain and has singularity at $x = 0$ and its boundary condition occurs at infinity, have been introduced. We solve the problem on semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration solves it by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results in other to show that the new method is accurate, efficient and applicable

Existence and approximation of solutions to three-point boundary value problems for fractional differential equations

In this paper, we study existence and approximation of solutions to some three-point boundary value problems for fractional differential equations of the type \begin{equation*}\begin{split} {}^{c}\mathcal{D}_{0+}^{q}u(t)+f(t,u(t))&=0, t\in(0,1), 1<q<2\\ u^{'}(0)=0, \xi u(\eta)&=u(1), \end{split}\end{equation*} where $\xi, \eta\in(0,1)$ and ${}^{c}\mathcal{D}_{0+}^{q}$ is the fractional derivative in the sense of Caputo. For the existence of solution, we develop the method of upper and lower solutions and for the approximation of solutions, we develop the generalized quasilinearization technique (GQT). The GQT generates a monotone sequence of solutions of linear problems that converges monotonically and quadratically to solution of the original nonlinear problem

Quasilinearization Applied to Boundary Value Problems at Resonance for Riemann-Liouville Fractional Differential Equations

The quasilinearization method is applied to a boundary value problem at resonance for a Riemann-Liouville fractional differential equation. Under suitable hypotheses, the method of upper and lower solutions is employed to establish uniqueness of solutions. A shift method, coupled with the method of upper and lower solutions, is applied to establish existence of solutions. The quasilinearization algorithm is then applied to obtain sequences of lower and upper solutions that converge monotonically and quadratically to the unique solution of the boundary value problem at resonance

Quasilinearization and Boundary Value Problems at Resonance for Caputo Fractional Differential Equations

The quasilinearization method is applied to a boundary value problem at resonance for a Caputo fractional differential equation. The method of upper and lower solutions is first employed to obtain the uniqueness of solutions of the boundary value problem at resonance. The shift argument is applied to show the existence of solutions. The quasilinearization algorithm is then developed and sequences of approximate solutions are constructed that converge monotonically and quadratically to the unique solution of the boundary value problem at resonance. Two applications are provided to illustrate the main results

A numerical solution for nonlinear heat transfer of fin problems using the Haar wavelet quasilinearization method

The aim of this paper is to study the new application of Haar wavelet quasilinearization method (HWQM) to solve one-dimensional nonlinear heat transfer of fin problems. Three different types of nonlinear problems are numerically treated and the HWQM solutions are compared with those of the other method. The effects of temperature distribution of a straight fin with temperature-dependent thermal conductivity in the presence of various parameters related to nonlinear boundary value problems are analyzed and discussed. Numerical results of HWQM gives excellent numerical results in terms of competitiveness and accuracy compared to other numerical methods. This method was proven to be stable, convergent and, easily coded

(R2030) Generalized Quasilinearization Method for a Initial Value Problem on Time Scales

We have investigated that the generalized quasilinearization method under some convenient conditions for nonlinear initial value problem (IVP) of dynamic equation on time scale constructed by monotone sequences of function by using comparison theorem that is the solution of linear IVP of dynamic equation on time scale which converge uniformly and monotonically to the unique solution of the original problem, and the convergence is quadratic

Reproducing Kernel Method for Fractional Riccati Differential Equations

This paper is devoted to a new numerical method for fractional Riccati differential equations. The method combines the reproducing kernel method and the quasilinearization technique. Its main advantage is that it can produce good approximations in a larger interval, rather than a local vicinity of the initial position. Numerical results are compared with some existing methods to show the accuracy and effectiveness of the present method