Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research the Institute Computer Science, Czestochowa University Technology

In this paper we prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally well-posed in H s , for s > 1 2 for data small in L 2. To understand the strength of this result one should recall that for s < 1 2 the Cauchy problem is ill-posed, in the sense that uniform continuity with respect to the initial data fails. The result follows from the method of almost cons...

Standard perturbation methods are applied to Euler’s equations of motion governing the capillary-gravity shallow water waves to derive a general higher-order Boussinesq equation involving the small-amplitude parameter, α = a/h0, and long-wavelength parameter, β = (h0/l), where a and l are the actual amplitude and wavelength of the surface wave, and h0 is the height of the undisturbed water surf...

We consider optimal control problems associated to generally non-well posed Cauchy in a general framework. Firstly, we approximate the ill-posed problem with family of well-posed one and show that solutions latter converge former one. Secondly, investigate minimization approximated state equation. prove existence uniqueness minimizers characterize optimality systems. Finally, subjected equation...

In this paper we consider thermoacoustic tomography as the inverse problem of determining from lateral Cauchy data the unknown initial conditions in a wave equation with spatially varying coefficients. This problem also occurs in several applications in the area of medical imaging and non-destructive testing. Using the method of quasi-reversibility, the original ill-posed problem is replaced wi...