Runge Kutta collocation method for the solution of first order ordinary differential equations

Abstract

We introduce one step continuous Runge Kutta collocation method with three free parameters for the solution of stiff first order ordinary differential equations. We adopt interpolation and collocation of the approximate solution at some selected grid points to give system of non linear equations. Using Crammer's rule to solve for the unknown parameters and substituting into the approximate solution gives the continuous method. To determine how best to fix the free parameters, we consider

A Clas of A-Stable Runge-Kutta Collocation Methods for the Solution of First Order Ordinary Dierential Equations

“…To implement (43); following [14] as shown in equation (24), the derived scheme and its internal stages can be written compactly in a partitioned Butcher's table of the form;…”

“…To implement the methods; [14] proposed a prediction equation as shown in equation 24which was employed in Predictor-Corrector mode to obtain the same order of accuracy. The following symmetric explicit predictor scheme of the same order with the corrector scheme are obtained using the same procedure for yn+1 of the the three schemes respectively;…”

Collocation Method for the Numerical Solution of Multi-Order Fractional Differential Equations

“…[1]. Some of the numerical solution of fractional differential equations developed in the literature include: Perturbed collocation method [2], Adomian decompositions method by [3][4][5], Collocation method by [6][7][8][9], Chebyshev-Gelerkin method [10], Bernoulli matrix method [11], Differential transform method [12], Pseudospectral method [13], Bernstein Polynomials method [14,15], the Mellin transform approach [16]. [17] utilized a numerical approach based on the boubaker polynomial to generate approximate numerical solutions to the multi-order fractional differential equations.…”