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The main contribution of this work is illustrated as follows: 1- The derivation for embedded singly diagonally implicit Runge-Kutta (SDIRK) method of fourth-order six stages in fifth-order seven stages is introduced to solve ordinary and delay differential equations. The stability region is presented and the numerical results are compared with the other existing methods. 2- Singly diagonally implicit Runge-Kutta Nystróm (SDIRKN) of third-order three stages embedded in fourth-order four stages is constructed. The stability region of the new method is presented and numerical results are compared with the same method of lower order. 3- A new singly diagonally implicit Runge-Kutta-Nystróm general (SDIRKNG) method of third-order embedded in fourth-order is derived to solve second order ordinary differential equations. Analysis the stability region of the new method is discussed and numerical results are presented.

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Alkhasawneh, Raed
Raed A. A. Al-Khasawneh was born on the third of December 1971 in Amman, Jordan. He obtained his first degree in Mathematics from Al-Yarmouk University, Jordan in June, 1994. later on, he pursued his Masters and PhD degrees in the field of applied mathematics at the Department of Mathematics, University Putra Malaysia and graduated in January,2007.

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