A class of hybrid collocation methods for third-order ordinary differential equations

Abstract

A class of numerical methods is proposed for solving general third-order ordinary differential equations directly by collocation at the grid points x = x n+j , i = 0(1)k and at an off grid point x = x n+u , where k is the step number of the method and u is an arbitrary rational number in (x n , x n+k ). A predictor of order 2k − 1 is also proposed to cater for y n+k in the main method. Taylor series expansion is employed for the calculation of y n+1 , y n+2 , y n+u and their higher derivatives. Evaluation of the resulting method at x = x n+k for any value of u in the specified open interval yields a particular discrete scheme as a special case of the method. The efficiency of the method is tested on some general initial value problems of third-order ordinary differential equations.

Direct Block Predictor-Corrector Method for the Solution of General Fourth Order Odes

“…Direct method of solving higher order ordinary differential equations in predictor-corrector mode have been studied by many scholars including: Awoyemi (2003), Adee et al (2005), Awoyemi and Idowu (2005), Fatunla (1994), Kayode (2009), Awoyemi and Kayode (2005) to mention but a few. Awoyemi (2003) investigated a p-stable linear multistep method for general third order initial value problems of ordinary differential equations.…”

“…Awoyemi (2003) investigated a p-stable linear multistep method for general third order initial value problems of ordinary differential equations. Awoyemi and Idowu (2005) developed a class of hybrid collocation method for third order ordinary differential equations. These authors respectively proposed some linear multistep methods with continuous coefficients using Taylor series expansion to supply the starting values.…”

Symmetric 2-Step 4-Point Hybrid Method for the Solution of General Third Order Differential Equations

“…Attempts have also been made by these scholars [12][13][14]6,7,15]. Olabode [13] proposed a 5-step block scheme for the solution of special type of eqn.…”

A New Block Method for Special Third Order Ordinary Differential Equations

“…Some authors have proposed solution to higher order initial value problems of ordinary differential equations using different approaches [1][2][3][4][5] . In particular Awoyemi and Idowu [2] developed a class hybrid collocation method for third order of ordinary differential equations.…”

“…Some authors have proposed solution to higher order initial value problems of ordinary differential equations using different approaches [1][2][3][4][5] . In particular Awoyemi and Idowu [2] developed a class hybrid collocation method for third order of ordinary differential equations. Awoyemi [1] derived a p-stable linear multistep method for general third order initial value problems of ordinary differential equations which is to be used in form of predictor-corrector forms and like most linear multistep methods, they require starting values from Runge-Kutta methods or any other onestep methods.…”