Journal of Applied Mathematics
Volume 2 (2002), Issue 8, Pages 407-435

An asymptotic approach to inverse scattering problems on weakly nonlinear elastic rods

Shinuk Kim and Kevin L. Kreider

Department of Theoretical and AppliedMathematics, The University of Akron, Akron 44325-4002, OH, USA

Received 27 September 2001; Revised 28 May 2002

Copyright © 2002 Shinuk Kim and Kevin L. Kreider. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Elastic wave propagation in weakly nonlinear elastic rods is considered in the time domain. The method of wave splitting is employed to formulate a standard scattering problem, forming the mathematical basis for both direct and inverse problems. A quasi-linear version of the Wendroff scheme (FDTD) is used to solve the direct problem. To solve the inverse problem, an asymptotic expansion is used for the wave field; this linearizes the order equations, allowing the use of standard numerical techniques. Analysis and numerical results are presented for three model inverse problems: (i) recovery of the nonlinear parameter in the stress-strain relation for a homogeneous elastic rod, (ii) recovery of the cross-sectional area for a homogeneous elastic rod, (iii) recovery of the elastic modulus for an inhomogeneous elastic rod.