Direct and Inverse Problems for the Heat Equation with a Dynamic type Boundary Condition

This paper considers the initial-boundary value problem for the heat equation with a dynamic type boundary condition. Under some regularity, consistency and orthogonality conditions, the existence, uniqueness and continuous dependence upon the data of the classical solution are shown by using the generalized Fourier method. This paper also investigates the inverse problem of finding a time-dependent coefficient of the heat equation from the data of integral overdetermination condition

Inverse Scattering Problem for LinearSystem of Four-Wave Interaction Problemon the Half-Line with a General BoundaryCondition

The first order hyperbolic system of four equations on the semi-axis in thecase of equal numbers of incident and scattered waves are considered whenthe velocities of the scattering waves are coincident. It is determined thecriteria for inverse scattering problem (the problem of finding the potentialwith respect to scattering operator) in terms of transmission matrices intwo different boundary conditions. The uniqueness of the inverse scatteringproblem is studied by utilizing it to Gelfand–Levitan–Marchenko type linearintegral equation

An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions

This paper investigates the inverse problem of finding a time-dependent coefficient in a heat equation with nonlocal boundary and integral overdetermination conditions. Under some regularity and consistency conditions on the input data, the existence, uniqueness and continuous dependence upon the data of the solution are shown by using the generalized Fourier method

Inverse problems of identifying the time-dependent source coefficient for subelliptic heat equations

We discuss inverse problems of determining the time-dependent source coefficient for a general class of subelliptic heat equations. We show that a single data at an observation point guarantees the existence of a (smooth) solution pair for the inverse problem. Moreover, additional data at the observation point implies an explicit formula for the time-dependent source coefficient. We also explore an inverse problem with nonlocal additional data, which seems a new approach even in the Laplacian case

Inverse Scattering Method via Riemann–Hilbert Problem for Nonlinear Klein–Gordon Equation Coupled with a Scalar Field

A class of negative order Ablowitz–Kaup–Newell–Segur nonlinear evolution equations are obtained by applying the Lax hierarchy of the first order linear system of three equations. The inverse scattering problem on the whole axis is examined in the case where linear system becomes the classical Zakharov–Shabat system consists of two equations and admits a real anti-symmetric potential. Referring to these results, the N-soliton solutions for the integro-differential version of the nonlinear Klein–Gordon equation coupled with a scalar field are obtained by using the inverse scattering method via the Riemann–Hilbert problem

Direct and Inverse Problems for Thermal Groovingbu Surface Diffusion with time Dependent Mullins Coefficient

We consider the Mullins’ equation of a single surface grooving when the surface diffusion is not considered as very slow. This problem can be formed by a surface grooving of profiles in a finite space region. The finiteness of the space region allows to apply the Fourier series analysis for one groove and also to consider the Mullins coefficient as well as slope of the groove root to be time-dependent. We also solve the inverse problem of finding time-dependent Mullins coefficient from total mass measurement. For both of these problems, the grooving side boundary conditions are identical to those of Mullins, and the opposite boundary is accompanied by a zero position and zero curvature which both together arrive at self adjoint boundary conditions