<abstract><p>In this paper, we consider an initial-boundary value problem for a non-linear fractional diffusion equation on a bounded domain. The fractional derivative is defined in Caputo's sense with respect to the time variable and represents the case of sub-diffusion. Also, the equation involves a second order symmetric uniformly elliptic operator with time-independent coefficients. These initial-boundary value problems arise in applied sciences such as mathematical physics, fluid mechanics, mathematical biology and engineering. By using eigenfunction expansions and Banach fixed point theorem, we establish the existence, uniqueness and regularity properties of the solution of the problem.</p></abstract>
<abstract><p>In this work, we consider a Cauchy problem for the generalized Schrö dinger equation which has important applications in quantum kinetic theory, water wave problems and ferromagnetism. Due to its multidimensionality, it is important from the point of view of modern physics theories such as quantum field theory and string theory. We prove the uniqueness of the solution of the problem in an unbounded domain by using semigeodesic coordinates. The main tool is a pointwise Carleman estimate. To the authors' best knowledge, this is the first study which deals with the solvability of this problem.</p></abstract>
This paper is concerned with the inverse problem of the determination of the right-hand side of a generalized transport equation from boundary measurements. The problem is originally related to an integral geometry problem along a family of curves whose curvature is given by the Christoffel symbols. The existence, uniqueness and stability of the solution of the problem are proven in polar coordinates. Some computational experiments for the approximate solution of the problem are presented by using the Galerkin method and the finite difference method.
In this work, we deal with an ill-posed boundary value problem for multidimensional second-order evolution equations with variable coefficients. By using the given data, we reduce the problem to a functional equation and we obtain a new representation for the solution by means of the Hurwitz formula.
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