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

The operational matrix of fractional integration for shifted Chebyshev polynomials

Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations

Numerical solution for the variable order linear cable equation with Bernstein polynomials

In this paper, the problem of the magneto-hemodynamic laminar viscous flow of a conducting physiological fluid in a semi-porous channel under a transverse magnetic field is investigated numerically. Using a Berman’s similarity transformation, the two-dimensional momentum conservation partial differential equations can be written as a system of nonlinear ordinary differential equations incorporating Lorentizian magneto-hydrodynamic body force terms. A new computational method based on the operational matrix of derivative of orthonormal Bernstein polynomials for solving the resulting differential systems is introduced. Moreover, by using the residual correction process, two types of error estimates are provided and reported to show the strength of the proposed method. Graphical and tabular results are presented to investigate the influence of the Hartmann number (Ha) and the transpiration Reynolds number (Re on velocity profiles in the channel. The results are compared with those obtained by previous works to confirm the accuracy and efficiency of the proposed scheme.

Investigation of magneto-hemodynamic flow in a semi-porous channel using orthonormal Bernstein polynomials

Computational method based on Bernstein operational matrices for nonlinear Volterra–Fredholm–Hammerstein integral equations

This paper introduces an approach for obtaining the numerical solution of the nonlinear Volterra-Fredholm integro-differential (NVFID) equations using hybrid Legendre polynomials and Block-Pulse functions. These hybrid functions and their operational matrices are used for representing matrix form of these equations. The main characteristic of this approach is that it reduces NVFID equations to a system of algebraic equations, which greatly simplifying the problem. Numerical examples illustrate the validity and applicability of the proposed method.

Hybrid Legendre polynomials and Block-Pulse functions approach for nonlinear Volterra-Fredholm integro-differential equations

A Bernstein operational matrix approach for solving a system of high order linear Volterra–Fredholm integro-differential equations

Hybrid functions approach for the nonlinear Volterra–Fredholm integral equations

In this paper, we present a practical matrix method for solving nonlinear Volterra-Fredholm integro-differential equations under initial conditions in terms of Bernstein polynomials on the interval [0,1]. The nonlinear part is approximated in the form of matrices’ equations by operational matrices of Bernstein polynomials, and the differential part is approximated in the form of matrices’ equations by derivative operational matrix of Bernstein polynomials. Finally, the main equation is transformed into a nonlinear equations system, and the unknown of the main equation is then approximated. We also give some numerical examples to show the applicability of the operational matrices for solving nonlinear Volterra-Fredholm integro-differential equations (NVFIDEs).

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Behrooz Basirat

Papers

Computational method based on Bernstein operational matrices for nonlinear Volterra–Fredholm–Hammerstein integral equations

Hybrid Legendre polynomials and Block-Pulse functions approach for nonlinear Volterra-Fredholm integro-differential equations

A Bernstein operational matrix approach for solving a system of high order linear Volterra–Fredholm integro-differential equations

Hybrid functions approach for the nonlinear Volterra–Fredholm integral equations

Numerical Solution of Nonlinear Integro-Differential Equations with Initial Conditions by Bernstein Operational Matrix of Derivative