Solution of the foam-drainage equation with cubic B-spline hybrid approach

This work presents a robust and efficient numerical stratagem for the study of integer and fractional order non-linear Foam-Drainage (FD) model. The scheme first uses, usual forward difference and the L ₁ formula, in integer and fractional cases, respectively. Then, the collocation approach together with cubic B-splines (CBS) basis are employed to estimate the unknown solution and its derivatives. With the help of these discretizations and Quasi-linearization, solving non-linear FD model transforms to the system of linear algebraic equations. The solution of the linear system approximates the CBS coefficients which further leads to the numerical solutions. Moreover, by Von Neumann stability it is proved that the proposed scheme is unconditionally stable. To evaluate the performance and accuracy of the technique, absolute error (AE), L ₂, and L _∞ norms are presented. The obtained outcomes are also matched with some existing results in literature. It is noted from simulations that the proposed method gives quite accurate solutions.