A Numerical Strategy for the Approximate Solution of the Nonlinear Time-Fractional Foam Drainage Equation

Abstract

This study develops a numerical strategy for finding the approximate solution of the nonlinear foam drainage (NFD) equation with a time-fractional derivative. In this paper, we formulate the idea of the Laplace homotopy perturbation transform method (LHPTM) using Laplace transform and the homotopy perturbation method. This approach is free from the heavy calculation of integration and the convolution theorem for the recurrence relation and obtains the solution in the form of a series. Two-dimensional and three-dimensional graphical models are described at various fractional orders. This paper puts forward a practical application to indicate the performance of the proposed method and reveals that all the outputs are in excellent agreement with the exact solutions.

Numerical study of arbitrary order Foam-Drainage equation with cubic B-spline hybrid method

“…Similarly, local derivative has widespread application like, in Boussinesq equation which predicts the nonlinear transverse vibration mechanism [24], Fokker-Planck model in sub-diffusion [25], Burgers' equation in audio signals propagation [26] and fractional Kortewegde-de Vries equation in shallow water waves [27]. Generically, the exact solutions of the NPDEs models are uncertain, and when the invoked derivatives are of arbitrary order then they become more intricate [28]. Therefore, alternative semi-analytical and numerical approaches are essential to obtain the solutions .…”

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

“…Generically, the exact solutions of the NPDEs models are uncertain, and when the invoked derivatives are of arbitrary order then they become more intricate [28]. Therefore, alternative semi-analytical and numerical approaches are essential to obtain the solutions.…”

Approximate Solutions of Multidimensional Wave Problems Using an Effective Approach

“…As a result, many researchers are searching for novel methods to get around these restrictions. Various researchers and scientists have studied multiple novel methods for getting the analytical solution that are reasonably close to the precise solutions such as homotopy analysis method [4], modified extended tanh method [5], new Kudryashov's method [6], Chun-Hui He's iteration method [7], subequation method [8], exp-function method [9], modified exponential rational method [10], homotopy asymptotic method [11], modified extended tanh expansion [12], fractal variational iteration transform method [13], Laplace homotopy perturbation transform method [14], residual power series (RPS) method [15], and Adomian decomposition method [16]. In the past, many experts and researchers established the application of the homotopy perturbation method (HPM) [17,18] in various physical problems, because this approach consistently transforms the challenging issue into a straightforward resolution.…”