A Regular Integral Equation Formalism for Solving the Standard Boussinesq’s Equations for Variable Water Depth

• Published in: Journal of scientific computing Vol. 75; no. 3; pp. 1721 - 1756
• Main Author: Jang, T. S.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2018; Springer Nature B.V
• Subjects: Algebra; Algorithms; Beaches; Boussinesq equations; Computational Mathematics and Numerical Analysis; Formalism; Formulas (mathematics); Fourier transforms; Integral equations; Iterative methods; Laplace transforms; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Ordinary differential equations; Partial differential equations; Questions; Shallow water; Solitary waves; Theoretical; Topography; Velocity; Water depth; Water waves; A regular integral equation formalism; The standard Boussinesq’s equations for variable water depth; A pseudo-water depth parameter; The successive approximation
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-017-0605-6

This paper begins with a question of existence of a regular integral equation formalism, but different from the existing usual ones, for solving the standard Boussinesq’s equations for variable water depth (or Peregrine’s model). For the question, a pseudo -water depth parameter, suggested by Jang (Commun Nonlinear Sci Numer Simul 43:118–138, 2017 ), is introduced to alter the standard Boussinesq’s equations into an integral formalism. This enables us to construct a regular (nonlinear) integral equations of second kind (as required), being equivalent to the standard Boussinesq’s equations (of Peregrine’s model). The (constructed) integral equations are, of course, inherently different from the usual integral equation formalisms. For solving them, the successive approximation (or the fixed point iteration) is applied (Jang 2017 ), whereby a new iterative formula is immediately derived, in this paper, for numerical solutions of the standard Boussinesq’s equations for variable water depth. The formula, semi-analytic and derivative-free, is shown to be useful to observe especially the nonlinear wave phenomena of shallow water waves on a beach. In fact, a numerical experiment is performed on a solitary wave approaching a sloping beach. It shows clearly the main feature of nonlinear wave characteristics, which has reached good agreement with the known (numerical) solutions. Hence, while being theoretical but fundamental in nonlinear computational partial differential equations, the question raised in the study may be solved.