arXiv subjects

Mansur I. Ismailov

Publications and source records attributed to Mansur I. Ismailov.

7 recordsLinked to original sources

arXiv2026

Inverse scattering method for nonlinear negative first order coupled Klein-Gordon equation

The exact N-soliton solutions are derived for the coupled negative first order Klein-Gordon (CNKG) equation subject to vanishing boundary conditions by using the inverse scattering method via Gelfand-Levitan-Marchenko equation. Based on the zero-curvature representation, the conservation laws, integrals of motion, and Hamiltonian structure of the aforementioned coupled nonlinear equations are constructed. The Jost functions and their analyticity properties for the Manakov spectral problem are recalled. The integral equations for the eigenfunctions are then used to formulate the Gel'fand-Levitan-Marchenko equations. Solving these equations establishes a direct correspondence between the kernel functions and the potential, yielding the general N-soliton expressions.

nlin.SI↗

arXiv2024

Inverse scattering method via Gel'fand--Levitan--Marchenko equation for some negative order nonlinear wave equations

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 are examined in the case where linear system becomes the classical Zakharov--Shabat system consists of two equations and admits a real symmetric and 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 (CKG) and negative order modified Korteweg-de Vries (nmKdV) equation are obtained by using the inverse scattering method via the Gel'fand--Levitan--Marchenko equation.

nlin.SI↗

arXiv2023

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.

math.AP↗

arXiv2017

Inverse Problem of Finding the Coefficient of the Lowest Term in Two-dimensional Heat Equation with Ionkin-type Boundary Condition

We consider an inverse problem of determining the time-dependent lowest order coefficient of two-dimensional (2D) heat equation with Ionkin boundary and total energy integral overdetermination condition. The well-posedness of the problem is obtained by generalized Fourier method combined by the Banach fixed poind theorem. For obtaining a numerical solution of the inverse problem, we propose the discretization method from a new combination. On the one hand, it is known the traditional method of uniform finite difference combined with numerical integration on a uniform grid (trapezoidal and Simpson's), on the other hand, we give the method of non-uniform finite difference is combined by a numerical integration on a non-uniform grid (with Gauss-Lobatto nodes). Numerical examples illustrate how to implement the method.

math.AP↗

arXiv2013

Inverse scattering problem on the half-axis for a first order system of ordinary differential equations

In this article, the inverse scattering problem (ISP) of recovering the matrix coefficient of a first order system of ordinary differential equations on the half-axis from its scattering matrix is considered. In the case of a triangular structure of the matrix coefficient, this system has a Volterra-type integral transformation operator at infinity. Such type of transformation operator allows to determine the scattering matrix on the half-axis via the matrix Riemann-Hilbert factorization in the case, where contour is real axis, normalization is canonical and all the partial indices are zero. The ISP on the half-axis is solved by reducing it to ISP on the whole axis for the considered system with the coefficients that are extended to the whole axis as zero.

math.SP↗

arXiv2013

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.

math-ph↗

arXiv2010

Inverse Problem of Finding the Time-dependent Coefficient of Heat Equation from Integral Overdetermination Condition Data

In this paper we consider the problem of simultaneously determining the time-dependent thermal diffusivity and the temperature distribution in one-dimensional heat equation in the case of nonlocal boundary and integral overdetermination conditions. We establish conditions for the existence and uniqueness of a classical solution of the problem under considerations. We present some results on the numerical solution with an example.

math.AP↗