A New Functional Iterative Algorithm for the Regularized Long-Wave Equation Using an Integral Equation Formalism

Abstract

A fundamental question in computational nonlinear partial differential equations is raised to discover if one could construct a functional iterative algorithm for the regularized long-wave (RLW) equation (or the Benjamin-Bona-Mahony equation) based on an integral equation formalism? Here, the RLW equation is a third-order nonlinear partial differential equation, describing physically nonlinear dispersive waves in shallow water. For the question, the concept of pseudo-parameter, suggested by Jang (Commun Nonlinear Sci Numer Simul 43:118-138, 2017), is introduced and incorporated into the RLW equation. Thereby, dual nonlinear integral equations of second kind involving the parameter are formulated. The application of the fixed point theorem to the integral equations results in a new (semi-analytic and derivative-free) functional iteration algorithm (as required). The new algorithm allows the exploration of new regimes of pseudo-parameters, so that it can be valid for a much wider range (in the complex plane) of pseudo-parameter values than that of Jang (2017). Being fairly simple (or straightforward), the iteration algorithm is found to be not only stable but accurate. Specifically, a numerical experiment on a solitary wave is performed on the convergence and accuracy of the iteration for various complex values of the pseudo-parameters, further providing the regions of convergence subject to some constraints in the complex plane. Moreover, the algorithm yields a particularly relevant physical investigation of the nonlinear behavior near the front of a slowly varying wave train, in which, indeed, interesting nonlinear wave features are demonstrated. As a consequence, the preceding question may be answered.Keywords Regularized long-wave (RLW) equation · Pseudo-parameter · Nonlinear integral equations of second kind · Functional iteration algorithm

An Integral Equation Formalism for Integrating a Nonlinear Initial-Boundary Value Problem for a Boussinesq Equation

“…due to the null initial conditions (13). Applying Laplace transform to (17), combined with the properties of (21) and (22), leads to (20), which proves the lemma.…”

“…One of the main purposes of this paper is to construct an integral equation formalism, equivalent to the initial value problem for (1). Here, the derived formalism is required to be regular and different from the usual existing integral equation ones (e.g., see [17]). For the construction, as a counterpart of the original initial-boundary value problem (defined on the positive real axis), we have first built an auxiliary initial-boundary value problem (defined on the negative real axis).…”

“…ereby, we would like to build an equivalent (nonlinear) pseudodissipative integral formalism in Section 3.5. We begin with integral transforms for (17), which are explained in the next section.…”

The filtering‐based maximum likelihood iterative estimation algorithms for a special class of nonlinear systems with autoregressive moving average noise using the hierarchical identification principle

“…17 They have been applied to many areas. 28 For example, Zhang et al established an implicit iterative algorithm for solving a class of Lyapunov matrix equations. For example, Jafari presented a new parameter estimation approach for the micro-electro-mechanical systems based on the maximum likelihood principle.…”

“…25 The iterative methods are often used in designing the controller 26,27 and finding the solutions of matrix equations or the roots of nonlinear equations. 28 For example, Zhang et al established an implicit iterative algorithm for solving a class of Lyapunov matrix equations. 29 In the fields of parameter estimation and filtering identification, the iterative estimation methods can give more accurate parameter estimates than the recursive estimation methods.…”