Abstract
This paper presents the derivation techniques of block method for solving higher order initial value problems of ordinary differential equations directly. The method was developed via interpolation and collocation of the shifted Legendre polynomials as basis function. The method is capable of providing the numerical solution at several points simultaneously.
Highlights
This work considers solving an ordinary differential equation (ODE) of the th order ( ≥ 2)
The first is to reduce it to a system of first order ordinary differential equations and solve using predictor corrector or Runge-Kutta method
The method generates simultaneous solutions at all grid points as suggested by many researchers such as Anake (2011), Onumanyi and Okunuga (1985), Onumanyi and Yusuph (2002), Onumanyi et al, (1993, 1994), Awoyemi (2001, 2005 and 1991), Areo et al (2008), Fatunla (1991, 1995), Lambert (1991), Awoyemi and Kayode (2005), Awari et al (2014), Okunuga and Ehigie (2009), Adesanya et al, (2009, 2008), Serisina et al, (2004), Owolabi (2015), Yahaya and Badmus (2009) and much recently by Warren and Zill (2013) and Omar and Kuboye (2015). These methods solve higher order initial value problems of ordinary differential equations without going through the process of reduction. This present method is aimed at developing a general block method for the direct solution of higher order ( ≥ 2) initial value problems of ordinary differential equations, with the block approach, the non-self-starting nature associated with the predictor corrector method has been eliminated
Summary
This work considers solving an ordinary differential equation (ODE) of the th order ( ≥ 2). The method generates simultaneous solutions at all grid points as suggested by many researchers such as Anake (2011), Onumanyi and Okunuga (1985), Onumanyi and Yusuph (2002), Onumanyi et al, (1993, 1994), Awoyemi (2001, 2005 and 1991), Areo et al (2008), Fatunla (1991, 1995), Lambert (1991), Awoyemi and Kayode (2005), Awari et al (2014), Okunuga and Ehigie (2009), Adesanya et al, (2009, 2008), Serisina et al, (2004), Owolabi (2015), Yahaya and Badmus (2009) and much recently by Warren and Zill (2013) and Omar and Kuboye (2015) These methods solve higher order initial value problems of ordinary differential equations without going through the process of reduction. Unlike the approach in predictor corrector method where additional equations were supplied from a different formulation, all our additional equations are obtained from the same continuous formulation
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