Approximate Optimal Controller for Weakly Nonlinear Evolutionary Equation of Parabolic Type

2021 ◽  
Author(s):  
N. V. Gorban ◽  
O. A. Kapustian ◽  
O. V. Kapustyan
Keyword(s):  
Parabolic Type ◽  
Optimal Controller ◽  
Evolutionary Equation ◽  
Weakly Nonlinear ◽  
Nonlinear Evolutionary Equation

scholarly journals The Global Existence of Nonlinear Evolutionary Equation with Small Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Xunwu Yin
Keyword(s):  
Global Existence ◽  
Local Existence ◽  
Evolutionary Equation ◽  
Fractional Powers ◽  
Gronwall’S Inequality ◽  
Sectorial Operators ◽  
Local Existence And Uniqueness ◽  
Small Delay ◽  
Fixed Point Principle ◽  
Nonlinear Evolutionary Equation

We investigate the global existence of the delayed nonlinear evolutionary equation∂tu+Au=f(u(t),u(t−τ)). Our work space is the fractional powers spaceXα. Under the fundamental theorem on sectorial operators, we make use of the fixed-point principle to prove the local existence and uniqueness theorem. Then, the global existence is obtained by Gronwall’s inequality.

scholarly journals Novel Soliton Molecules and Wave Interactions for A (3+1)-Dimensional Nonlinear Evolutionary Equation

2021 ◽  
Author(s):  
Xiaoyan Tang ◽  
Chao Jie Cui ◽  
Zu feng Liang ◽  
Wei Ding
Keyword(s):  
Nonlinear Evolution ◽  
Evolutionary Equation ◽  
New Wave ◽  
Variable Separation ◽  
Wave Interactions ◽  
Elastic Interactions ◽  
Localized Excitations ◽  
Soliton Lattice ◽  
Wave Patterns ◽  
Nonlinear Evolutionary Equation

Abstract New wave excitations are revealed for a (3+1)-dimensional nonlinear evolution equation to enrich nonlinear wave patterns in nonlinear systems. Based on a new variable separation solution with two arbitrary variable separated functions obtained by means of the multilinear variable separation approach, localized excitations of N dromions, N x M lump lattice and N x M ring soliton lattice are explored. Interestingly, it is observed that soliton molecules can be composed of diverse "atoms" such as the dromions, lumps and ring solitons, respectively. Elastic interactions between solitons and soliton molecules are graphically demonstrated.

scholarly journals Integration of the Harry Dym equation with an integral type source

2021 ◽  
Vol 31 (2) ◽  
pp. 285-295
Author(s):  
G.U. Urazboev ◽  
A.K. Babadjanova ◽  
D.R. Saparbaeva
Keyword(s):  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattering Data ◽  
Evolutionary Equation ◽  
Integral Type ◽  
Self Consistent ◽  
The Cauchy Problem ◽  
Nonlinear Evolutionary Equation ◽  
Dym Equation ◽  
Harry Dym Equation

In the work, we deduce the evolution of scattering data for a spectral problem associated with the nonlinear evolutionary equation of Harry Dym with a self-consistent source of integral type. The obtained equalities completely determine the scattering data for any $t$, which makes it possible to apply the method of the inverse scattering problem to solve the Cauchy problem for the Harry Dym equation with an integral type source.

scholarly journals A Nonlocal in Time Problem for Evolutionary Singular Equations in Generalized Spaces of Type S∘

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Hanna Verezhak ◽  
Vasyl Gorodetskyi
Keyword(s):  
Initial Function ◽  
Infinite Order ◽  
Parabolic Type ◽  
Generalized Functions ◽  
Bessel Operator ◽  
Evolutionary Equation ◽  
Correct Solvability ◽  
Singular Equations ◽  
Time Problem ◽  
Space Of Generalized Functions

In this paper, we establish the correct solvability of a nonlocal multipoint in time problem for the evolutionary equation of a parabolic type with the Bessel operator of infinite order in the case where the initial function is an element of the space of generalized functions of type S∘′.

The Evolution Equation for Wave Processes of the Shift Deformation

2012 ◽  
Vol 249-250 ◽  
pp. 132-136 ◽  
Author(s):  
Yulia Ivanova ◽  
Victoria Ragozina
Keyword(s):  
Shock Wave ◽  
Wave Process ◽  
Evolutionary Equation ◽  
Dynamic Shear ◽  
Wave Processes ◽  
Volume Deformation ◽  
One Dimensional ◽  
Plane Transverse ◽  
Nonlinear Evolutionary Equation ◽  
Subsequent Motion

One-dimensional process of formation and the subsequent motion of a plane transverse shock wave is studied on the basis of solutions of the corresponding nonlinear evolutionary equation. This equation defines behaviour of the solution in front area of wave process and follows from inner series of a matched asymptotic expansions method. A comparative analysis of the volume deformation and forming processes will be carried out and their basic differences are specified. In the capacity of model examples solutions of some concrete boundary value problems of a dynamic shear deformation are observed.

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