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Solving nonlinear differential equations in astrophysics and fluid mechanics using the generalized pseudospectral method

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Abstract

In this study, a combined numerical method is introduced and used to solve nonlinear differential equations. Because of the use of this method of generalized Lagrange functions and their derivative operational matrices, they were first introduced, and then using these new functions, the generalized pseudospectral method as a new numerical method was combined with the quasilinearization method, and an efficient method is produced. Due to the use of derivative operational matrices and conversion of a nonlinear differential equation to a sequence of linear differential equations, in performing this method, it is not necessary to calculate the analytical derivative and solve the system of nonlinear algebraic equations, which reduces computational costs. The efficiency and convergence of the method have been demonstrated by implementing it on two important equations in astrophysics and fluid mechanics, and comparing the results with other methods and graphical graphs.

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Abbreviations

\(\varepsilon \) and \(\delta \) :

The material fluid parameters

M :

The polytropic index

\(L^{\phi }_j(x)\) :

The generalized Lagrange functions (basis functions)

\({\varvec{\mathrm {D}}}^{\mathrm {(}m\mathrm {)}}\) :

The m-order derivative operational matrix

\(\phi (x)\) :

The generalizer function

\({\delta }_{ij}\) :

The Kronecker delta

N :

The number of basis functions

[a, b]:

The interval of basis functions

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Islamic Azad University, Bardaskan Branch, Bardaskan, Iran

    Mehdi Delkhosh

  2. Department of Mathematics and Computer Science, Kashmar Higher Education Institute, Kashmar, Iran

    Amanj Rahmanzadeh & Seyyedeh-Fatemeh Shafiei

Authors
  1. Mehdi Delkhosh
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  2. Amanj Rahmanzadeh
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  3. Seyyedeh-Fatemeh Shafiei
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Correspondence to Mehdi Delkhosh.

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Delkhosh, M., Rahmanzadeh, A. & Shafiei, SF. Solving nonlinear differential equations in astrophysics and fluid mechanics using the generalized pseudospectral method. SeMA 78, 457–474 (2021). https://doi.org/10.1007/s40324-021-00246-1

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