## Numerical solution of two-point nonlinear boundary value problems via Legendre-Picard iteration method.(English)Zbl 07538453

Summary: The aim of the present work is to introduce an effective numerical method for solving two-point nonlinear boundary value problems. The proposed iterative scheme, called the Legendre-Picard iteration method is based on the Picard iteration technique, shifted Legendre polynomials and Legendre-Gauss quadrature formula. In the Legendre-Picard iteration method, the boundary value problem is reduced to an iterative formula for updating the coefficients of the approximate solution in each step and with a straightforward manner, the integrals of the shifted Legendre polynomials are calculated. In addition, to reduce the CPU time, a vector-matrix scheme of the Legendre-Picard iteration method is constructed. The convergence analysis of the method is studied. Five nonlinear boundary value problems are given to illustrate the validity of the Legendre-Picard iteration method. Numerical results indicate the good performance and the precision of the proposed procedure.

### MSC:

 65-XX Numerical analysis 76-XX Fluid mechanics
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### References:

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