Geodesic
A method for finding coefficients of a quasilinear hyperbolic equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 5, pp. 813-833

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The inverse problem of finding the coefficients $q(s)$ and $p(s)$ in the equation $u_{tt}=a^2u_{xx}+q(u)u_t-p(u)u_x$ is investigated. As overdetermination required in the inverse setting, two additional conditions are set: a boundary condition and a condition with a fixed value of the timelike variable. An iteration method for solving the inverse problem is proposed based on an equivalent system of integral equations of the second kind. A uniqueness theorem and an existence theorem in a small domain are proved for the inverse problem to substantiate the convergence of the algorithm. 
@article{ZVMMF_2006_46_5_a4,
     author = {A. Yu. Shcheglov},
     title = {A~method for finding coefficients of a~quasilinear hyperbolic equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {813--833},
     publisher = {mathdoc},
     volume = {46},
     number = {5},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_5_a4/}
}
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A. Yu. Shcheglov. A method for finding coefficients of a quasilinear hyperbolic equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 5, pp. 813-833. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_5_a4/