Auto-B�cklund transformations for the nonlinear Schr�dinger equation with variable coefficients

In the paper a theoretical study the both the quantized energies of excitonic states and their wave functions in grapheneand in materials with "Mexican hat" band structure dispersion as well as in zinc-blende GaN is presented. An integral twodimensionalSchrÃ¶dinger equation of the electron-hole pairing for a particles with electron-hole symmetry of reflection isexactly solved. The solutions of SchrÃ¶dinger equation in momentum space in studied materials by projection the twodimensionalspace of momentum on the three-dimensional sphere are found exactly. We analytically solve an integral twodimensionalSchrÃ¶dinger equation of the electron-hole pairing for particles with electron-hole symmetry of reflection. Instudied materials the electron-hole pairing leads to the exciton insulator states. Quantized spectral series and lightabsorption rates of the excitonic states which distribute in valence cone are found exactly. If the electron and hole areseparated, their energy is higher than if they are paired. The particle-hole symmetry of Dirac equation of layered materialsallows perfect pairing between electron Fermi sphere and hole Fermi sphere in the valence cone and conduction cone andhence driving the Cooper instability. The solutions of Coulomb problem of electron-hole pair does not depend from a widthof band gap of graphene. It means the absolute compliance with the cyclic geometry of diagrams at justification of theequation of motion for a microscopic dipole of graphene where >1 s r . The absorption spectrums for the zinc-blendeGaN/(Al,Ga)N quantum well as well as for the zinc-blende bulk GaN are presented. Comparison with availableexperimental data shows good agreement.

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Convergence of Finite Fifference Method for Parabolic Problem

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2003-0005 ◽

2003 ◽

Vol 3 (1) ◽

pp. 45-58 ◽

Cited By ~ 2

Author(s):

Dejan Bojović

Keyword(s):

Convergence Rate ◽

Boundary Value Problem ◽

Heat Equation ◽

Parabolic Problem ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Variable Coefficients ◽

Rate Estimate ◽

Convergence Rate Estimate ◽

Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

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MODELING OF NONLINEAR VIBRATION PROBLEMS WITH FLUID FLOWS THROUGH PIPELINES

Irrigatsiya va Melioratsiya ◽

10.35938/2018.1.12.11 ◽

2018 ◽

pp. 44-47

Author(s):

F.J. Тurayev

Keyword(s):

Differential Equations ◽

Nonlinear Vibration ◽

Viscoelastic Properties ◽

Construction Material ◽

Fluid Flows ◽

Variable Coefficients ◽

Algebraic Equations ◽

Flowing Liquid ◽

Vibration Problems

In this paper, mathematical model of nonlinear vibration problems with fluid flows through pipelines have been developed. Using the Bubnov–Galerkin method for the boundary conditions, the resulting nonlinear integro-differential equations with partial derivatives are reduced to solving systems of nonlinear ordinary integro-differential equations with both constant and variable coefficients as functions of time.A system of algebraic equations is obtained according to numerical method for the unknowns. The influence of the singularity of heredity kernels on the vibrations of structures possessing viscoelastic properties is numerically investigated.It was found that the determination of the effect of viscoelastic properties of the construction material on vibrations of the pipeline with a flowing liquid requires applying weakly singular hereditary kernels with an Abel type singularity.

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