Abstract
This paper is aimed at discussing and comparing the performance of standard method with its hybrid method of the same step number for the solution of first order initial value problems of ordinary differential equations. The continuous formulation for both methods was obtained via interpolation and collocation with the application of the shifted Legendre polynomials as approximate solution which was evaluated at some selected grid points to generate the discrete block methods. The order, consistency, zero stability, convergent and stability regions for both methods were investigated. The methods were then applied in block form as simultaneous numerical integrators over non-overlapping intervals. The results revealed that the hybrid method converges faster than the standard method and has minimum absolute error values.
Highlights
Most physical phenomena in science and engineering used mathematical models to help in the understanding of the physical world problems
Two different methods for solving first order initial value problems of ordinary differential equations have been proposed in this work, the conventional or standard method and the hybrid method with the hybrid having more advantages over the conventional method the hybrid higher order and accuracy
The results revealed that the hybrid method converges faster than the standard method, since it has minimum absolute error values
Summary
Most physical phenomena in science and engineering used mathematical models to help in the understanding of the physical world problems These models often yield equations that contain some derivatives of an unknown function of one or several variables. Consider a numerical method for solving general first order initial value problems of ordinary differential equations of the form: y′ = f (x, y), y(0) = y0. [5], introduced the application of two step continuous hybrid Butcher’s method in block form for the solution of first order initial value problems; this approach eliminates requirements for a starting value. [10], introduced a hybrid linear collocation multistep scheme for solving first order initial value problems of ordinary differential equations. [9], developed a three step implicit hybrid linear multistep method for the solution of third order ordinary differential equations Order Initial Value Problems of Ordinary Differential Equations the Continuous implicit hybrid one step methods for the solution of initial value problems of general second order ordinary differential equations. [5], introduced the application of two step continuous hybrid Butcher’s method in block form for the solution of first order initial value problems; this approach eliminates requirements for a starting value. [2], introduced a new hybrid method for systems of stiff equations. [11], developed a new Butcher type two-step block hybrid multistep method for accurate and efficient parallel solution of ordinary differential equations. [1], used hybrid formula of order four to generate starting values for Numerov method. [3], developed linear multistep hybrid methods with continuous coefficients for solving stiff ordinary differential equations. [10], introduced a hybrid linear collocation multistep scheme for solving first order initial value problems of ordinary differential equations. [9], developed a three step implicit hybrid linear multistep method for the solution of third order ordinary differential equations
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