Properties of an oriented ring of neurons with the FitzHugh-Nagumo model

In this paper, we consider a linear parabolic type partial differential equation in the space of generalized functions as the equation of neutron diffusion in the presence of neutron absorption by the atomic nucleus with nonlinear impulsive effects. Spectral equation is obtained from the Dirichlet boundary value conditions and this spectral problem is studied. The Fourier method of variables separation is used. Countable system of nonlinear functional integral equations is obtained with respect to the Fourier coefficients of unknown function. Theorem on a unique solvability of the countable system of functional integral equations is proved. The method of successive approximations is used in combination with the method of contracting mapping. Criteria of uniqueness and existence of generalized solution of the impulsive mixed problem is obtained. Solution of the mixed problem is derived in the form of the Fourier series. It is shown that the Fourier series converges uniformly. KEYWORDS Mixed problem, impulsive parabolic equation, nonlinear impulsive conditions, involution, unique solvability FOR CITATION Yuldashev T.K., Fayziyev A.K. Mixed problem for a linear differential equation of parabolic type with nonlinear impulsive conditions.

Section: Formulation Of the Problem Statementmentioning

confidence: 99%

“…Problem ( 5), ( 6) will be solved by the method of separation of variables: U (t, x) = u(t) ϑ(x). After separation of variables, from (5), (6), we arrive at the following spectral problem for an ordinary differential equation…”

Section: Formal Solution Of the Mixed Problemmentioning

confidence: 99%

Mixed problem for a linear differential equation of parabolic type with nonlinear impulsive conditions

Yuldashev,

Fayziyev

2024

“…[1][2][3][4][5]. In particular, such kind of problems appears in biophysics at micro-and nanoscales [6][7][8][9][10]. A lot of publications of studying the differential equations with impulse effects related to various natural and technical processes are appearing [11][12][13][14][15][16][17][18][19][20].…”

Section: Problem Statementmentioning

confidence: 99%

Determination of the coefficient function in a Whitham type nonlinear differential equation with impulse effects

Yuldashev¹,

Fayziyev²

2023

In the article, the problems of unique solvability and determination of the redefinition coefficient function in the initial inverse problem for a nonlinear Whitham type partial differential equation with impulse effects are studied. The modified method of characteristics allows partial differential equations of the first order to be represented as ordinary differential equations that describe the change of unknown function along the line of characteristics. The unique solvability of the initial inverse problem is proved by the method of successive approximations and contraction mappings. The determination of the unknown coefficient is reduced to solving the nonlinear integral equation. KEYWORDS inverse problem, Whitham type equations, determination of the coefficient function, method of successive approximations, unique solvability.

“…The differential and integro-differential equations have applications in biological, chemical and physical sciences, ecology, biotechnology, industrial robotics, pharmacokinetics, biophysics at micro-and nano-scales [1][2][3][4][5][6][7][8][9][10][11][12][13]. Today, for ordinary integro-differential equations, new problems are posed and a large number of papers, devoting to study of integrodifferential equations, are published.…”

Section: Problem Statementmentioning

confidence: 99%

Inverse problem for Fredholm integro-differential equation with final redefinition conditions at the end of the interval

Yuldashev¹,

Zarifzoda²

2022

The questions of solvability and construction of solutions of an inverse problem for second-order Fredholm integro-differential equation with degenerate kernel, final conditions at the end of the interval, two parameters, and two redefinition data are considered. The sets of regular parameter values are determined and the corresponding solutions are constructed. The specific features of the inverse problem are studied. Criteria for the unique solvability of the posed inverse problem are established. KEYWORDS integro-differential equation, degenerate kernel, final conditions at the end of a segment, parameters, redefinition data, solvability.