Numerical solutions of stiff ordinary differential equations and differential algebraic equations using one-step implicit hybrid methods

The numerical solutions of stiff ordinary differential equations and differential algebraic equations have been studied in this thesis. New one-step implicit hybrid methods are developed to solve stiff ordinary differential equations (ODEs) and semiexplicit index-1 differential algebraic equations (DAEs). These methods are formulated by using Lagrange interpolating polynomial. The developed one-step methods will solve ODEs and DAEs with the introduction of off-step points by constant step size. The source codes were written in C language. Stiff equations in Mathematics indicate that for a certain numerical method to solve differential equations that may give unstable results unless the step size taken is extremely small. Newton’s iteration is implemented together with the developed method to solve stiff equations. The numerical results showed that the performance of the methods outperformed compared to existing method in terms of maximum error and average error. Further, this study is extended by using the developed method to solve DAEs. Semiexplicit index-1 DAEs is the system of ordinary differential equations with algebraic constrains. Newton’s iteration is implemented with the developed methods to solve DAEs. The numerical results showed the performance of the developed methods is more efficient then existing methods in terms of maximum error and average error. In conclusion, the proposed one-step implicit hybrid methods are suitable for solving stiff ordinary differential equations and semi-explicit index-1 differential algebraic equations.

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