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Direct numerical method for an inverse problem of hyperbolic equations
Author(s) -
Lin Tao,
Ewing Richard E.
Publication year - 1992
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.1690080605
Subject(s) - mathematics , hyperbolic partial differential equation , inverse problem , computation , mathematical analysis , boundary (topology) , boundary value problem , convergence (economics) , inverse , stability (learning theory) , nonlinear system , partial differential equation , geometry , algorithm , physics , quantum mechanics , machine learning , computer science , economics , economic growth
A coefficient inverse problem of the one‐dimensional hyperbolic equation with overspecified boundary conditions is solved by the finite difference method. The computation is carried out in the x direction instead of the usual t direction. The original boundary condition and the overspecified boundary data are used as the new initial conditions, and the original data at t = 0 are used to compute the coefficient directly. The computation time used by this scheme is almost equal to that for solving the hyperbolic equation in the same region once, even though the inverse problem is essentially nonlinear and hence more difficult to solve. An error estimate is obtained that guarantees the stability of the scheme marching in the x direction. Several numerical experiments are carried out to show the convergence and other properties of the scheme. © 1992 John Wiley & Sons, Inc.