Geodesic
Mixed problem for a linear differential equation of parabolic type with nonlinear impulsive conditions
Nanosistemy: fizika, himiâ, matematika, Tome 15 (2024) no. 2, pp. 160-169.

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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. 
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     author = {Tursun K. Yuldashev and Aziz K. Fayziyev},
     title = {Mixed problem for a linear differential equation of parabolic type with nonlinear impulsive conditions},
     journal = {Nanosistemy: fizika, himi\^a, matematika},
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Tursun K. Yuldashev; Aziz K. Fayziyev. Mixed problem for a linear differential equation of parabolic type with nonlinear impulsive conditions. Nanosistemy: fizika, himiâ, matematika, Tome 15 (2024) no. 2, pp. 160-169. http://geodesic.mathdoc.fr/item/NANO_2024_15_2_a1/