2010/10/14 by A. S. Fokas, D. Yang, Fokas, A. S. +2
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Elasticity and Wave Propagation #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in engineering #math-ph #math.AP #math.MP
paper · pdf · doi:10.48550/arxiv.1010.2941
arxiv created 2010/10/14 · openalex publication_date 2010/10/14 · arxiv updated 2010/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a new approach to constructing analytic solutions of the linear PDEs describing elastodynamics. This approach is illustrated for the case of a homogeneous isotropic half-plane body satisfying arbitrary initial conditions and Lamb's boundary conditions. A particular case of this problem, namely the case of homogeneous initial conditions, was first solved by Lamb using the Fourier-Laplace transform. The solution of the general problem can also be expressed in terms of the Fourier transform, but this representation involves transforms of unknown boundary values. This necessitates the formulation and solution of a cumbersome auxiliary problem, which expresses the unknown boundary values in terms of the Laplace transform of the given boundary data. The new approach, which is applicable to arbitrary initial and boundary conditions, bypasses the above auxiliary problem and expresses the solutions directly in terms of the given initial and boundary data.