Solving shallow water equations with crisp and uncertain initial conditions

Abstract

Purpose This paper aims to deal with the application of variational iteration method and homotopy perturbation method (HPM) for solving one dimensional shallow water equations with crisp and fuzzy uncertain initial conditions. Design/methodology/approach Firstly, the study solved shallow water equations using variational iteration method and HPM with constant basin depth and crisp initial conditions. Further, the study considered uncertain initial conditions in terms of fuzzy numbers, which leads the governing equations to fuzzy shallow water equations. Then using cut and parametric concepts the study converts fuzzy shallow water equations to crisp form. Then, HPM has been used to solve the fuzzy shallow water equations. Findings Results obtained by both methods HPM and variational iteration method are compared graphically in crisp case. Solution of fuzzy shallow water equations by HPM are presented in the form triangular fuzzy number plots. Originality/value Shallow water equations with crisp and fuzzy initial conditions have been solved.

Homotopy perturbation method for predicting tsunami wave propagation with crisp and uncertain parameters

“…In Section 2, the basic idea of HPM (He, 1999;He, 2000;Karunakar and Chakraverty, 2018c) has been illustrated. Let us consider the following nonlinear differential equation:…”

“…Tapaswini and Chakraverty (2014) used midpoint-based approach for solving nth-order interval differential equations. Karunakar and Chakraverty (2018a, 2018b, 2018c) handled linear and nonlinear SWW equations with internal and fuzzy parameters and established interval and fuzzy solutions by considering basin depth and initial conditions as uncertain using HPM and variational iteration method. Recently, few modifications and improvements in HPM have been introduced in Yu et al (2019) by constructing a homotopy equation with one or more auxiliary parameters embedding in the linear term.…”

Sawi Transform Based Homotopy Perturbation Method for Solving Shallow Water Wave Equations in Fuzzy Environment

“…They used HPM to solve the SWWEs by treating the water depth as uncertain in terms of fuzzy. Additionally, Chakraverty and Karunakar [46] suggested a strategy for dealing with crisp and uncertain differential equations that regulate shallow water equations. They did this by using HPM to solve nonlinear SWWE in a crisp and precise context, specifically for tsunami wave propagation.…”

Variational multi-scale finite element method for the two-phase flow of polymer melt filling process

“…The numerical simulation of a complex flow is always strongly needed in practical applications, though there are some famous analytical methods, for example, the variational iteration method (Anjum and He, 2019; He, 2006; He et al , 2014; He and Ji, 2019a, 2019b; Karunakar and Chakraverty, 2018) and the homotopy perturbation method (He, 2012, 2014; Anjum and He, 2019; Ban and Cui, 2018; Liu et al , 2017; Manafian and Sindi, 2018; Wu and He, 2018a).…”

“…The numerical simulation of a complex flow is always strongly needed in practical applications, though there are some famous analytical methods, for example, the variational iteration method (Anjum and He, 2019;He, 2006;He et al, 2014;Ji, 2019a, 2019b;Karunakar and Chakraverty, 2018) and the homotopy perturbation method (He, 2012(He, , 2014Anjum and He, 2019;Ban and Cui, 2018;Liu et al, 2017;Manafian and Sindi, 2018;Wu and He, 2018a). The injection molding, which is a popular polymer processing method, is a complex process (Cao et al, 2008), and it is difficult to have an analytical solution; therefore, an accurate numerical solution plays a critical role in the optimization of the process.…”