Scientific Journals of Osh State University
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Inverse Problem for a Nonlinear Pseudiparabolic Differential Equation with Final Condition and Gerasimov–Caputo Operator
No full textIn this paper in rectangle domain an inverse problem for a fractional analogue ofthe pseudoparabolic differential operator with mixed conditions, degeneration and identificationsource is considered. Fractional operator is the Gerasimov–Caputo type and the solution of thenonlinear differential equation with two spatial variables is studied in the class of generalizedfunctions. The nonlinear Fourier series method is used and by the aid of Kilbas–Saigo functiona nonlinear countable system of functional integral equation is obtained. In the proof of uniquesolvability of the countable system is applied the method of successive approximations in combination with the method of compressing mapping. We use the Cauchy–Schwarz inequality andthe Bessel inequality in proving the absolute and uniform convergence of the obtained Fourierseries. Then we derive the desire redefinition function also in the form of Fourier series
On G. E. Hutchinson Population Model for Fractional Differential Equations with Maxima
No full textThe article is devoted to the G. E. Hutchinson population model for fractionalfunctional-differential equations with maxima. The functional-differential equation with maximaand Gerasimov–Caputo fractional operator is considered under the initial-final conditions. Theproposed functional-differential equation we consider as a mathematical model of populationdynamics of a species. The generalized spectral Jakobi–Galerkin method is used. The uniquesolvability theorem is proved
Boundary Value Problem for a Seventh Order Nonhomogeneous Partial Differential Equation
No full textIn this paper, we consider a boundary value problem for a seventh order partial differential equation. It is used Samarskii–Ionkin type boundary value conditions on spatial variablex. The non-self-adjoint spectral problem and adjoint spectral problem are studied. The systemsof eigenvalues and eigenfunctions are determined. By the biorthogonal systems of eigenvalues,the Fourier series method of separation of variables is applied. Consequently, the unique solution of the boundary value problem is obtained in the form of Fourier series. Absolutely anduniformly convergence of Fourier series is proved
Optimal Control Problem for an Impulsive Systems of FunctionalDifferential Equations with a Nonlinear Function under the Sign of a First-Order Differential
No full textIn this paper the optimal control problems for an impulsive systems of functionaldifferential equations with a nonlinear function under the sign of the first-order differential andwith maxima are investigated. The Initial value problem is reduced to a system of nonlinearfunctional-integral equations in a Banach space BD[0, T], Rn. In the fixed values of controlfunction, by the method of contracting mappings proves the existence and uniqueness of statefunction for a nonlinear systems of functional-integral equations with maxima. Then usingfunctional of quality, we are built Pontryagin’s function and obtain criteria of optimality. Thenwe are proved existence and uniqueness of control function
Mathematical Modeling of a Reaction-Diffusion System with a Free Boundary
No full textThis article presents the application of a Stefan-type two-phase free boundary problem to model dynamics of the prosthesis-tissue interface in dentistry and prosthetics. Addressingissues such as stress concentrations and tissue damage caused by biomechanical incompatibility,a mathematical model based on reaction-diffusion equations is proposed to describe the temporalevolution of the free boundary. The existence and uniqueness of global classical solution of themodel are rigorously proven. The regularity of the free boundary is examined, and a computational scheme is introduced to visualize the interface dynamics. The findings are directed towardsoptimizing the long-term stability and osseointegration of dental prostheses
ИНТУИЦИЯ В ПРОЦЕССЕ ПОЗНАНИЯ
No full textThis article analyzes the manifestation of intuition in the world of science, the role of feeling and logical observation in decision-making. The role of the flow state in intuitive cognition in the process of memory, attention, imagination, fantasy, dreams, and creativity isalso described.Бул макалада илим дүйнөсүндөгү интуициянын көрүнүшү жана чечим кабыл алууда сезим менен логикалык байкоо жүргүзүүнүн ролу талданат. Ошондой эле, эс тутум, көңүл буруу, элестетүү, кыялдануу жана чыгармачылык процесстериндеги интуитивдик таанып билүүдөгү агымдын ролу сүрөттөлөт.В данной статье анализируется проявление интуиции в мире науки, роль чувства и логического наблюдения в принятии решений. Также описывается роль состояния потока в интуитивном познании в процессе памяти, внимания, воображения, сновидений, творчества
Boundary Value Problem for a System of Fredholm Integro-Differential Equations with Maxima
No full textA Dzhumabayev’s parameterization method is proposed to solve a ”linear” two-pointboundary value problem for a Fredholm integro-differential equation with maxima. The systemof differential equations is changed with the system of integral equations. Then by the method ofcontracting mapping is studied the integral equation in the space BD [0, T], Rn. As a practical way of solving the original problem, it is transformed into a multipoint boundary value problemwith parameters. Introduction of additional parameters yields a special Cauchy problem for asystem of integro-differential equations with parameters on the subintervals. Using the solutionto this problem, the boundary condition and continuity conditions of solutions at the interiorpoints of the partition, we construct a system of linear algebraic equations in parameters. Wegive the algorithms of how to calculate the solutions of multipoint boundary value problem
Boundary Value Problems with Mixed Dirichlet and Neumann Conditions for Three-Dimensional Degenerate Elliptic Equation
No full textThis article investigates two problems with mixed Dirichlet and Neumann conditionsfor a three-dimensional degenerate elliptic equation. Fundamental solutions of the named equation are expressed through a triple Lauricella hypergeometric function and explicit solutions ofthe mixed problems in the first octant are written out through a double Appell hypergeometricfunction. The energy integral method is used to prove the uniqueness of the solutions to theproblems under consideration. In the course of proving the existence of the problem solution,differentiation formulas, decomposition formulas, some adjacent relations formulas and the autotransformation formula of hypergeometric functions are used. The Gauss–Ostrogradsky formulais used to express problem’s solutions in an explicit form
Nonlinear Fractional-Differential-Integral Equation with Product of Two Nonlinear Functions, Degeneration and Maxima
No full textIn this article a nonlinear initial and final values problem for a Gerasimov–Caputotype fractional differential equation with degeneration is considered in the case of differentiationorder is 0 < α ≤ 1. The right-hand side of the equation consists product of two nonlinearfunctions, Fredholm integral term and construction of maxima from unknown function. Thesolution of this fractional differential-integral equation is studied in the Banach space. A nonlinear integral equation is obtained by the aid of Mittag–Leffler function. The method of successiveapproximations in combination with the method of contracting mapping is applied in proof ofone valued solvability of the problem. The continuous dependence of solution of the problem oninitial data also is studied
Density-Type Properties of the Space of the Solutions of an Ordinary Differential Equation
No full textIn this paper we provide a detailed analysis of the density-type properties of thespace of solutions of an ordinary differential equation. We show that the density, local density,weak density, and local weak density of the space of solutions of an ordinary differential equationare countable. Furthermore, we prove that these properties are preserved under functors ofhyperspace and superextensions of the space of solutions of an ordinary differential equation