Fractional Differential Equations

EDDINGTON'S “Internal Constitution of the Stars” was published in 1926 and gives what now ranks as a classical account of his own researches and of the general state of the theory at that time. Since then, a tremendous amount of work has appeared. Much of it has to do with the construction of stellar models with different equations of state applying in different zones. Other parts deal with the effects of varying chemical composition, with pulsation and tidal and rotational distortion of stars, and with the precise relations between the interior and the atmosphere of a star. The striking feature of all this work is that so much can be done without assuming any particular mechanism of stellar energy-generation. Only such very comprehensive assumptions are made about the distribution and behaviour of the energy sources that we may expect future knowledge of their mechanism to lead mainly to more detailed results within the framework of the existing general theory. An Introduction to the Study of Stellar Structure By S. Chandrasekhar. (Astrophysical Monographs sponsored by The Astrophysical Journal.) Pp. ix+509. (Chicago: University of Chicago Press; London: Cambridge University Press, 1939.) 50s. net.

An Introduction to the Study of Stellar Structure

Zeros of orthogonal polynomials

Orthogonal approximations in Sobolev spaces stability and convergence spectral methods and pseudospectral methods spectral methods for multi-dimensional and high order problems mixed spectral methods combined spectral methods spectral methods on the spherical surface.

Spectral Methods and Their Applications

Neuro-heuristics for nonlinear singular Thomas-Fermi systems

Accurate solution of the ThomasFermi equation using the fractional order of rational Chebyshev functions

Volterra’s model for population growth in a closed system includes an integral term to indicate accumulated toxicity in addition to the usual terms of the logistic equation, that occurs in ecology. In this paper, a new numerical approximation is introduced for solving this model of arbitrary (integer or fractional) order. The proposed numerical approach is based on the generalized fractional order Chebyshev orthogonal functions of the first kind and the collocation method. Accordingly, we employ a collocation approach, by computing through Volterra’s population model in the integro-differential form. This method reduces the solution of a problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of this method, we compare the numerical results of the present method with some well-known results in order to show that the new method is efficient and applicable.

Solving Volterra’s population growth model of arbitrary order using the generalized fractional order of the Chebyshev functions

In this paper, a new method based on Fractional order of Rational Jacobi (FRJ) functions is proposed that utilizes quasilinearization method to solve non-linear singular Thomas-Fermi equation on unbounded interval $[0,\infty)$
 . The equation is solved without domain truncation and variable changing. First, the quasilinearization method is used to convert the equation to the sequence of linear ordinary differential equations. Then, by using the FRJs collocation method the equations are solved. For the evaluation, comparison with some numerical solutions shows that the proposed solution is highly accurate.

Fractional order of rational Jacobi functions for solving the non-linear singular Thomas-Fermi equation

In this paper, a new algorithm based on the fractional order of rational Euler functions (FRE) is introduced to study the Thomas-Fermi (TF) model which is a nonlinear singular ordinary differential equation on a semi-infinite interval. This problem, using the quasilinearization method (QLM), converts to the sequence of linear ordinary differential equations to obtain the solution. For the first time, the rational Euler (RE) and the FRE have been made based on Euler polynomials. In addition, the equation will be solved on a semi-infinite domain without truncating it to a finite domain by taking FRE as basic functions for the collocation method. This method reduces the solution of this problem to the solution of a system of algebraic equations. We demonstrated that the new proposed algorithm is efficient for obtaining the value of $ y'(0)$
 , $ y(x)$
 and $ y'(x)$
 . Comparison with some numerical and analytical solutions shows that the present solution is highly accurate.

A novel numerical technique to obtain an accurate solution to the Thomas-Fermi equation

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.

Mehdi Delkhosh

Papers

Accurate solution of the ThomasFermi equation using the fractional order of rational Chebyshev functions

Solving Volterra’s population growth model of arbitrary order using the generalized fractional order of the Chebyshev functions

Fractional order of rational Jacobi functions for solving the non-linear singular Thomas-Fermi equation

A novel numerical technique to obtain an accurate solution to the Thomas-Fermi equation

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