Rational interpolation method for solving initial value problems (IVPs) in ordinary differential equations

In this paper we designed Rational Interpolation Method for solving Ordinary Differential Equations (ODES) and Stiff initial value problems (IVPs).

This was achieved by considering the Rational Interpolation Formula. $y_{\left(x\right)}=U_{\left(x\right)}=\frac{P_0+P_{1}x+P_2{x}^2+P^3{x}^3+p_4{x}^4+P_5{x}^5}{1+q_{1}x+q_2{x}^2+q_3{x}^3+q_4{x}^4+q_5{x}^5+q_6{x}^6}\text{,}$ satisfying $U\left(X_{n+i}\right)=y_{n+i},i=0,1,2,3,4,5,6$.

We also implemented $k=6$ in Aashikpelokhai (1991) class of rational integration formulas given by $y_{n+1}=\frac{\sum _{i=0}^5{p}_i{X}_{n+1}^i}{1+\sum _{i=1}^6{q}_i{X}_{n+1}^i}$ where, $\text{<mglyph></mglyph>}$$P_j=\frac{\sum _{i=1}^j{h}^{(j+1-i)}{y}_n^{(j+1-i)}}{\sum _{i=1}^j(j+1-i)!X_{n+1}^{(j+1-i)}}{q}_{i-1}+y_n{q}_j\text{,}j=1\left(1\right)5\text{.}$ The results as analyzed with the computer show that the rational interpolation method copes favorably well with ordinary differential equations and stiff initial value problems.