Abstract: This paper examines the development of one step, five hybrid point method for the solution of first order initial value problems. We adopted the method of collocation and interpolation of power series approximate solution to generate a continuous linear multistep method. The continuous linear multistep method was evaluated at selected grid points to give the discrete linear multistep method. The method was implemented using a constant order predictor of order seven over an overlapping interval. The basic properties of the derived corrector was investigated and found to be zero stable, consistent and convergent. The region of absolute stability was also investigated. The method was tested on some numerical experiments and found to compete favorably with the existing methods.
Abstract: In this paper, self-starting block hybrid method of
order (5,5,5,5)T is proposed for the solution of the special second
order ordinary differential equations with associated initial or
boundary conditions. The continuous hybrid formulations enable us
to differentiate and evaluate at some grids and off – grid points to
obtain four discrete schemes, which were used in block form for
parallel or sequential solutions of the problems. The computational
burden and computer time wastage involved in the usual reduction of
second order problem into system of first order equations are avoided
by this approach. Furthermore, a stability analysis and efficiency of
the block method are tested on stiff ordinary differential equations,
and the results obtained compared favorably with the exact solution.
Abstract: In this paper, linear multistep technique using power
series as the basis function is used to develop the block methods
which are suitable for generating direct solution of the special second
order ordinary differential equations with associated initial or
boundary conditions. The continuous hybrid formulations enable us
to differentiate and evaluate at some grids and off – grid points to
obtain two different four discrete schemes, each of order (5,5,5,5)T,
which were used in block form for parallel or sequential solutions of
the problems. The computational burden and computer time wastage
involved in the usual reduction of second order problem into system
of first order equations are avoided by this approach. Furthermore, a
stability analysis and efficiency of the block methods are tested on
linear and non-linear ordinary differential equations and the results
obtained compared favorably with the exact solution.
Abstract: In this paper, linear multistep technique using power
series as the basis function is used to develop the block methods
which are suitable for generating direct solution of the special second
order ordinary differential equations of the form y′′ = f(x,y), a < = x < = b with associated initial or boundary conditions. The continuaous hybrid formulations enable us to differentiate and evaluate at some
grids and off – grid points to obtain two different three discrete
schemes, each of order (4,4,4)T, which were used in block form for
parallel or sequential solutions of the problems. The computational
burden and computer time wastage involved in the usual reduction of
second order problem into system of first order equations are avoided
by this approach. Furthermore, a stability analysis and efficiency of
the block method are tested on linear and non-linear ordinary
differential equations whose solutions are oscillatory or nearly
periodic in nature, and the results obtained compared favourably with
the exact solution.