Bernstein induced one step hybrid scheme for general solution of second order initial value problems

Abstract

In this paper, a Bernstein polynomial with collocation and interpolation techniques were used to develop one step hybrid scheme with one offgrid point for the direct solution of general second order ordinary differential equations. The basic properties of the derived scheme was investigated and found to be of order four(4), zero stable and convergent. The scheme obtained is used to solve some standard initial value problems. From the numerical results obtained, it was revealed that the proposed method performs better than some of the existing methods in the literature.

Numerical Solution of Some COVID-19 Models

“…Order p = 4 with Error Constant . Thus equation ( 7) can be writing in the form Writing (8) in the form Order p = 3 with Error Constant −, thus equation ( 8) can be writing in the form + )…”

“…Bernstein Polynomial of degree m on interval [0, 1] is defined mathematically as (1) Where the binomial coefficient is See Ojo and Okoro [8] for details.…”

“…The numerical results of the scheme show some efficiency over some existing schemes of the same or higher order". Ojo and Okoro [8] used a "Bernstein polynomial to develop a one-step hybrid scheme with one off-grid point via collocation and interpolation techniques for the direct solution of general second-order ordinary differential equations, and the method was found to be reliable". Farouki, & Goodman [9] and Yasser (2023) showed that Bernstein polynomial recurrence has been proven to be conditionally stable and has an exponential rate of convergence.…”

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“…Ojo and Okoro [3], use a Bernstein polynomial to develop one step hybrid scheme with one off-grid point via collocation and interpolation techniques for the direct solution of general second order ordinary differential equations. Adeniran and Olanegan [4], develop a continuous two-step method using trigonometric function as basis function with two discrete methods as end products, the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.…”