# New Technique for Solving Lane-Emden Equation with Vieta-Fibonacci Polynomials

### Abstract

In the present work, new efficient iterative algorithm is presented for solving Lane-Emden equation with initial conditions. The solution is considered as a linear combination of Vieta-Fibonacci polynomials with unknown coefficients. First, the Vieta-Fibonacci polynomials are presented with some new important properties. A new exact formula for finding operation matrix of derivative for the Vieta-Fibonacci polynomials is constructed. Using such new properties the original Lane-Emden equation is transformed to the solution of algebraic equation with small-unknown coefficients. The obtained solution is in the form of a power series with easily computable coefficients. Comparison with some known exact and approximate solutions shows that the proposed solution is highly accurate.

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### References

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*Journal of Al-Qadisiyah for Computer Science and Mathematics*,

*13*(2), Math Page 22-. https://doi.org/10.29304/jqcm.2021.13.2.791