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

We are concerned with linear and nonlinear multi-term fractional differential equations (FDEs). The shifted Chebyshev operational matrix (COM) of fractional derivatives is derived and used together with spectral methods for solving FDEs. Our approach was based on the shifted Chebyshev tau and collocation methods. The proposed algorithms are applied to solve two types of FDEs, linear and nonlinear, subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Numerical results with comparisons are given to confirm the reliability of the proposed method for some FDEs.

http://www.kau.edu.sa/Files/130/Researches/62005_33049.pdf

A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order

A new jacobi operational matrix: an application for solving fractional differential equations

In this paper, two finite difference/element approaches for the time-fractional subdiffusion equation with Dirichlet boundary conditions are developed, in which the time direction is approximated by the fractional linear multistep method and the space direction is approximated by the finite element method. The two methods are unconditionally stable and convergent of order $O(\tau^q+h^{r+1})$ in the $L^2$ norm, where $q=2-\beta$ or 2 when the analytical solution to the subdiffusion equation is sufficiently smooth, $\beta\,(0<\beta<1)$ is the order of the fractional derivative, $\tau$ and $h$ are the step sizes in time and space, respectively, and $r$ is the degree of the polynomial space. The corresponding schemes for the subdiffusion equation with Neumann boundary conditions are presented as well, where the stability and convergence are shown. Numerical examples are provided to verify the theoretical analysis. Comparisons between the algorithms derived in this paper and the existing algorithms are given, ...

The Use of Finite Difference/Element Approaches for Solving the Time-Fractional Subdiffusion Equation

https://www.kau.edu.sa/Files/130/Researches/61664_32628.pdf

Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations

A new comparative study between homotopy analysis transform method and homotopy perturbation transform method on a semi infinite domain

Two new classes of optimal Jarratt-type fourth-order methods

In this study, we improve the algebraic formulation of the fractional partial differential equations (FPDEs) by using the matrix-vector multiplication representation of the problem. This representation allows us to investigate an operational approach of the Tau method for the numerical solution of FPDEs. We introduce a converter matrix for the construction of converted Chebyshev and Legendre polynomials which is applied in the operational approach of the Tau method. We present the advantages of using the method and compare it with several other methods. Some experiments are applied to solve FPDEs including linear and nonlinear terms. By comparing the numerical results obtained from the other methods, we demonstrate the high accuracy and efficiency of the proposed method.

Tau approximate solution of fractional partial differential equations

In this paper, a numerical method which produces an approximate polynomial solution is presented for solving Lane-Emden equations as singular initial value problems. Firstly, we use an integral operator (Yousefi (2006) [4]) and convert Lane-Emden equations into integral equations. Then, we convert the acquired integral equation into a power series. Finally, transforming the power series into Pade series form, we obtain an approximate polynomial of arbitrary order for solving Lane-Emden equations. The advantages of using the proposed method are presented. Then, an efficient error estimation for the proposed method is also introduced and finally some experiments and their numerical solutions are given; and comparing between the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method.

On the numerical solution of differential equations of Lane-Emden type

This paper proposes two classes of three-step without memory iterations based on the well known second-order method of Steffensen. Per computing step, the methods from the developed classes reach the order of convergence eight using only four evaluations, while they are totally free from derivative evaluation. Hence, they agree with the optimality conjecture of Kung-Traub for providing multi-point iterations without memory. As things develop, numerical examples are employed to support the underlying theory developed for the contributed classes of optimal Steffensen-type eighth-order methods.

S. Karimi Vanani

Papers

A new comparative study between homotopy analysis transform method and homotopy perturbation transform method on a semi infinite domain

Two new classes of optimal Jarratt-type fourth-order methods

Tau approximate solution of fractional partial differential equations

On the numerical solution of differential equations of Lane-Emden type

Optimal Steffensen-type methods with eighth order of convergence