A collocation approach to solve the Riccati-type differential equation systems

Abstract

In this paper, a collocation method is presented for the solutions of the system of the Riccati-type differential equations with variable coefficients. The proposed approach consists of reducing the problem to a nonlinear algebraic equation system by expanding the approximate solutions in terms of the Bessel polynomials with unknown coefficients. The unknown coefficients of the Bessel polynomials are found by using the matrix operations of derivatives together with the collocation method. The proposed method gives the analytic solutions when the exact solutions are polynomials. Also, an error analysis technique based on the residual function is introduced for the suggested method. The reliability and efficiency of the proposed scheme are demonstrated by some numerical examples. Comparing the methodology with some known techniques shows that the presented approach is relatively easy and highly accurate. All of the numerical calculations have been done by using a program written in Maple.

Keywords:

• system of the Riccati-type differential equations
• approximate solutions
• collocation method
• collocation points
• Bessel functions of first kind
• Bessel collocation method

2010 AMS Subject Classifications:

• 65L60
• 65L10
• 65L80
• 34B15
• 34A34
• 34K28

Acknowledgements

The author thanks the editor and the referees for carefully reading the paper and for their many constructive comments and suggestions to improve the paper.