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An efficient hybrid method to solve nonlinear differential equations in applied sciences

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Abstract

In this study, by combining the generalized pseudospectral method, which is a new numerical method, with the quasi-linearization method (QLM), an efficient hybrid method to solve nonlinear differential equations in applied sciences and engineering is presented. Given that this method requires generalized Lagrange functions and their derivative operational matrices, we first introduce them and then implement the generalized pseudospectral method at each iteration of the quasi-linearization method and obtain a system of linear algebraic equations. In the presented method, derivative operational matrices are used, and also, the nonlinear differential equation is converted into a sequence of linear differential equations using QLM, so there is no need to calculate the analytical derivative and solve nonlinear systems during the implementation of the method, which reduces computational costs and increasing the efficiency of the method. The efficiency and accuracy of the method have been demonstrated by applying it to several important applied equations; the Blasius equation, the Falkner–Skan problem over an isothermal moving wedge, and the third-grade fluid in a porous half-space. Then the obtained results are compared with other researchers.

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Abbreviations

\(N\) :

The number of points

\(\left[a, b\right]\) :

The interval

\({u}_{N}\left(x\right)\) :

The approximated solution

\(M\) :

The Falkner–Skan power-law parameter

\(L\) :

The shape parameter

\(\phi (x)\) :

An arbitrary function on \([a, b]\)

\({L}_{j}^{\phi }\left(x\right)\) :

Generalized Lagrange functions on \([a, b]\)

\({\mathbf{D}}^{(m)}\) :

The derivative operational matrix

\(\lambda \) :

The moving wedge parameter

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

The Kronecker delta

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Authors and Affiliations

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

    Mehdi Delkhosh & Hossein Cheraghian

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  1. Mehdi Delkhosh
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  2. Hossein Cheraghian
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Correspondence to Mehdi Delkhosh.

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Communicated by Justin Wan.

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Delkhosh, M., Cheraghian, H. An efficient hybrid method to solve nonlinear differential equations in applied sciences. Comp. Appl. Math. 41, 322 (2022). https://doi.org/10.1007/s40314-022-02024-9

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