Estimation of Cauchy data for a first-order nonlinear hyperbolic equation
The authors study the Cauchy problem: \[ u_ t+uu_ x+au=\phi(x,t)\text{ in } \Omega \times (0,t),\;u(x,0)=u_ 0(x), \] where \(\Omega\) is an open interval in \({\mathbb{R}}\). This system is associated with an initial value problem for the system: \[ dx/dt=z,\quad dz/dt=-az+\phi (x,t). \] Suppose that M observations \(\{z_ k\}^ M_{k=1}\) at positions \(\{x_ k\}^ M_{k=1}\) are given at time \(t_ 0\). The goal is to estimate the initial condition \(u_ 0\) from among an admissible set \(Q_{ad}\) of initial conditions. The approach followed by the authors consists firstly in solving that system with the initial conditions \(x(t_ 0)=x_ k\), \(z(t_ 0)=z_ k\), thus obtaining the solutions \(x(t;x_ k,z_ k)\), \(z(t;x_ k,z_ k)\); then in finding \(x(0;x_ k,z_ k)=\xi_ k\) and \(z(0;x_ k,z_ k)=\zeta_ k\) for \(k=1,...,M\); finally in introducing a fit-to-data functional given by \(J(u_ 0)=\sum^{M}_{k=1}(u_ 0(\xi_ k)- \zeta_ k)^ 2\) and considering the problem: \[ \text{Find } \hat u_ 0\in Q_{ad}\text{ such that } J(\hat u_ 0)=\min \{J(u_ 0):\;u_ 0\in Q_{ad}\}. \] A detailed study of the above program is made, and results of numerical experiments are presented.
scientific article; zbMATH DE number 4148724