Farhod D. Rakhmonov’s research while affiliated with National University of Uzbekistan and other places

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Publications (4)


MIXED PROBLEM FOR A NONLINEAR IMPULSIVE DIFFERENTIAL EQUATION OF PARABOLIC TYPE
  • Article

March 2024

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3 Reads

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2 Citations

Челябинский физико-математический журнал

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A. K. Fayziyev

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F. D. Rakhmonov

In this paper, we consider a nonlinear impulsive parabolic type partial differential equation with nonlinear impulsive conditions. Dirichlet type boundary value conditions with respect to spatial variable is used, and eigenvalues and eigenfunctions of the spectral problem are founded. The Fourier method of the separation of variables is applied. A countable system of nonlinear functional equations is obtained with respect to the Fourier coefficients of the unknown function. A theorem on a unique solvability of the countable system of nonlinear functional equations is proved by the method of successive approximations. A criteria of uniqueness and existence of a solution for the nonlinear impulsive mixed problem is obtained. A solution of the mixed problem is derived in the form of the Fourier series. The absolute and uniform convergence of the Fourier series is proved.



Nonlocal inverse problem for a pseudohyperbolic-pseudoelliptic type differential equation

July 2021

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25 Reads

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1 Citation

AIP Conference Proceedings

The problems of solvability of a nonlocal inverse boundary value problem for a mixed type pseudohyperbolic-pseudoelliptic differential equation with a spectral parameter and integral conditions are considered. The equation is characterized by the fact that it has three unknown functions: with respect to the main unknown function the equation is a mixed type differential equation; and with respect to redefinition functions it is Fredholm integral equations of the second kind. The cases of regular and irregular values of the spectral parameter are studied. By the aid of the method of the Fourier series, a nonlinear system of two countable systems of Fredholm ordinary integral equations of the second kind is obtained. From this system we prove the existence and uniqueness of the Fourier coefficients of redefinition functions. In the proof of the unique solvability of this system of two countable systems of Fredholm ordinary integral equations of the second kind the method of compressive mappings is applied. In this proof we use also the Cauchy-Schwartz inequality and the Bessel inequality. For regular values of the spectral parameter, a criterion of unique solvability of the inverse boundary value problem is established. Solutions of the inverse boundary value problem are obtained in the form of Fourier series. The convergence of the obtained Fourier series and the possibility of term-by-term differentiation of the main Fourier series are shown. In the case of irregular values of spectral parameter, it is constructed an infinite number of solutions of the inverse problem in the view of Fourier series. The results are formulated as a theorem.


Nonlocal problem for a nonlinear fractional mixed type integro-differential equation with spectral parameters

July 2021

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11 Reads

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10 Citations

AIP Conference Proceedings

In this article, we consider a boundary value problem for a nonlinear partial integro-differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with positive spectral parameter in a negative rectangular domain. The partial integro-differential equation of mixed type depends on another real spectral parameter in integral part of the mixed equation. With respect to first variable this equation is a fractional integro-differential equation in the positive part of the considering segment, and is a second-order integro-differential equation with spectral parameter in the negative part of this segment. Using the Fourier series method of separation variables and Fredholm method of degenerate kernels, the solutions of nonlinear boundary value problems are constructed in the form of a Fourier series. Theorems on the existence and uniqueness of solution of the problem are proved for regular values of the spectral parameters.

Citations (3)


... where B n,m is an arbitrary constant. To find this constant, by virtue of the condition (33), from (38) and (39) we have where ...

Reference:

Optimal Control Problem for a Linear Pseudoparabolic Equation with Final Condition, Degeneration and Gerasimov-Caputo Operator
MIXED PROBLEM FOR A NONLINEAR IMPULSIVE DIFFERENTIAL EQUATION OF PARABOLIC TYPE
  • Citing Article
  • March 2024

Челябинский физико-математический журнал

... From this follows the solvability of algebraic system (21) [26]. Estimate (23) allows (by virtue of Theorem 1 on weak compactness [26,28]) to pass to the limit as N → ∞ and conclude that some subsequence u N k ε (x, t) converges weakly, by virtue of the uniqueness of the solution to the problem (Theorem 1), to the sought-for solution u ε (x, t) of problem (8)- (11) in space V (Q), possessing the properties specified in Theorem 2. For u ε (x, t), by virtue of (23), the following inequality is valid: ...

On a Benney–Luke Type Differential Equation with Nonlinear Boundary Value Conditions
  • Citing Article
  • December 2021

Lobachevskii Journal of Mathematics

... Поэтому по данному разделу математики до сих пор появляются большое количество научных публикаций (см. например [1][2][3][4][5][6][7][8][9][10][11][12][13][14]). ...

Nonlocal problem for a nonlinear fractional mixed type integro-differential equation with spectral parameters
  • Citing Conference Paper
  • July 2021

AIP Conference Proceedings