Off Grid Collocation Four Step Initial Value Solver for Second Order Ordinary Differential Equations (Published)
The derivation and application of a four step Block Linear Multistep Method is hereby presented. To achieve this, Chebyshev polynomial was employed as basis function. Chebyshev polynomial was adopted as basis function based on its level of accuracy among other monomials in the interval [-1, 1]. Block method was adopted in this presentation based on its accuracy over the popular Predictor – Corrector method. The method under consideration gives solution at each grid point within the interval of integration. The method was arrived at by interpolating the polynomial equation and collocating the differential equation at some selected points. The order and error constant of the method were investigated likewise the consistency and zero stability which is one of the desirability property of linear multistep method were equally investigated. The method was applied to solve some second order ordinary differential equations and compare its level of accuracy with the analytical solution and equally compare its level of accuracy with some other existing methods.
Keywords: Block Method., Interpolation, chebyshev polynomial., collocation, corrector, predictor
Integrator Block Off – Grid Points Collocation Method For Direct Solution Of Second Order Ordinary Differential Equations Using Chebyshev Polynomials As Basis Function (Published)
The numerical computation of differential equations cannot be overemphasized as it is evident in the literatures. It has been observed that analytical solution of some differential equations are intractable, hence there is need to seek for an alternative solution to such equations. Circumventing this problem resulted into an approximate solution otherwise known as numerical solution. There are so many numerical methods that can be used in solving differential equations which include predictor – corrector method which is linear multistep in nature and not self-starting method. In this presentation the focus is on presenting a self-starting multistep method for direct solution of Second Order Ordinary Differential Equations as against the popular predictor – corrector method which needs additional value for starting point which may alter the accuracy of the method. The method is a mixture of grid and off grid collocation point and often refer to as Block linear multistep method.
Keywords: Block Method., Chebyshev Polynomials, Integrator Off –Grid, Interpolation, Predictor – Corrector, collocation
Integrator Block Off – Grid Points Collocation Method For Direct Solution Of Second Order Ordinary Differential Equations Using Chebyshev Polynomials As Basis Function. (Published)
The numerical computation of differential equations cannot be overemphasized as it is evident in the literatures. It has been observed that analytical solution of some differential equations are intractable, hence there is need to seek for an alternative solution to such equations. Circumventing this problem resulted into an approximate solution otherwise known as numerical solution. There are so many numerical methods that can be used in solving differential equations which include predictor – corrector method which is linear multistep in nature and not self-starting method. In this presentation the focus is on presenting a self-starting multistep method for direct solution of Second Order Ordinary Differential Equations as against the popular predictor – corrector method which needs additional value for starting point which may alter the accuracy of the method. The method is a mixture of grid and off grid collocation point and often refer to as Block linear multistep method.
Keywords: Block Method., Chebyshev Polynomials, Integrator Off –Grid, Interpolation, Predictor – Corrector, collocation
DEGREE OF BEST ONE-SIDED INTERPOLATION BY HERMITE-FEJER POLYNOMIALS IN WEIGHTED SPACES (Published)
In this search we find a degree of best one-sided approximation of the function f that lies in weighted space (L_(p,α)-space) by interpolation-operators which based on Hermite-Fejer interpolation polynomial constructed on the zeroes of Stieltjes polynomial E_(n+1)^((λ)) and the product E_(n+1)^((λ)) P_n^((λ)) for 0≤λ≤1 and 0≤λ≤1/2 (resp.). Here we denoted of these interpolation-operators by H_(n+1)^(*,±) (f,x_n) and H_(2n+1)^(**,±) (f,x_n) where x_n is the set of zeroes of E_(n+1)^((λ)) and E_(n+1)^((λ)) P_n^((λ)) (resp.) such that P_n^((λ)) is ultra-spherical polynomial with respect to ω_λ (x)=(1-x^2 )^(λ-1⁄2).The result which we end in it that the limit of differences between H_(n+1)^(*,+) and H_(n+1)^(*,_) is zero and H_(2n+1)^(**,+),H_(2n+1)^(**,_)(resp.)i.e. (⏟((lim)()┬(n→∞) H_(n+1)^(*,+)-H_(n+1)^(*,-))=0)(resp. for H_(2n+1)^(**,±) (f,x_n)).Also in this search we shall prove inverse theorem by using equivalent result between E_n (f) and E ̃_n (f) and inverse theorem in a best approximation case such that both pervious theorems are in weighted space where the weight function is a generalized Jacobi (GL) weighted u(x). Finally we try to estimate degree of best one-sided approximation of the derivative of the function f in weighted space
Keywords: Hermite-Fejer, Interpolation, Polynomials, Weighted Spaces