Hopf bifurcations in a network of FitzHugh–Nagumo biological neurons

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%

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.

“…[1][2][3][4][5]. In particular, such kind of problems appear in biophysics at micro-and nano-scales [6][7][8][9][10]. Such differential equations with "discontinuities" at fixed or non-fixed time moments are called differential equations with impulsive effects.…”

Section: Introductionmentioning

confidence: 99%

Inverse problem for a second order impulsive system of integro-differential equations with two redefinition vectors and mixed maxima

Yuldashev¹,

Fayziyev²

2023

An inverse problem for a second order system of ordinary integro-differential equations with impulsive effects, mixed maxima and two redefinition vectors is investigated. A system of nonlinear functional integral equations is obtained by applying some transformations. The existence and uniqueness of the solution of the nonlinear inverse problem is reduced to the unique solvability of the system of nonlinear functional integral equations in Banach space P C ([0, T ], R n ). The method of successive approximations in combination with the method of compressing mapping is used in the proof of unique solvability of the nonlinear functional integral equations. Then values of redefinition vectors are founded. KEYWORDS inverse problem, second order system, impulsive integro-differential equations, two-point nonlinear boundary value conditions, two redefinition vectors, mixed maxima, existence and uniqueness of solution. FOR CITATION Yuldashev T.K., Fayziyev A.K. Inverse problem for a second order impulsive system of integrodifferential equations with two redefinition vectors and mixed maxima.