A new solution procedure for a nonlinear infinite beam equation of motion

• Published in: Communications in nonlinear science & numerical simulation Vol. 39; pp. 321 - 331
• Main Author: Jang, T.S.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.2016
• Subjects: A fourth-order nonlinear partial differential equation; A nonlinear integral equation of second kind; A pseudo-parameter; Construction; Equations of motion; Integral equations; Iterative methods; Mathematical analysis; Mathematical models; Nonlinearity; Partial differential equations; The fixed point; A pseudo-parameter; A fourth-order nonlinear partial differential equation; The fixed point; A nonlinear integral equation of second kind
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2016.02.039

•We derive a new nonlinear integral equation, equivalent to a nonlinear beam equation.•The fixed point approach, applied to the integral equation, gives a solution procedure.•Using the procedure, we present a new method for integrating the nonlinear equation.•We prove the convergence and uniqueness of the presented method. Our goal of this paper is of a purely theoretical question, however which would be fundamental in computational partial differential equations: Can a linear solution-structure for the equation of motion for an infinite nonlinear beam be directly manipulated for constructing its nonlinear solution? Here, the equation of motion is modeled as mathematically a fourth-order nonlinear partial differential equation. To answer the question, a pseudo-parameter is firstly introduced to modify the equation of motion. And then, an integral formalism for the modified equation is found here, being taken as a linear solution-structure. It enables us to formulate a nonlinear integral equation of second kind, equivalent to the original equation of motion. The fixed point approach, applied to the integral equation, results in proposing a new iterative solution procedure for constructing the nonlinear solution of the original beam equation of motion, which consists luckily of just the simple regular numerical integration for its iterative process; i.e., it appears to be fairly simple as well as straightforward to apply. A mathematical analysis is carried out on both natures of convergence and uniqueness of the iterative procedure by proving a contractive character of a nonlinear operator. It follows conclusively,therefore, that it would be one of the useful nonlinear strategies for integrating the equation of motion for a nonlinear infinite beam, whereby the preceding question may be answered. In addition, it may be worth noticing that the pseudo-parameter introduced here has double roles; firstly, it connects the original beam equation of motion with the integral equation, second, it is related with the convergence of the iterative method proposed here.