scholarly journals On nonlinear evolution equation of second order in Banach spaces

2018 ◽  
Vol 16 (1) ◽  
pp. 268-275
Author(s):  
Kamal N. Soltanov
Keyword(s):  
Differential Equation ◽  
Evolution Equation ◽  
Banach Spaces ◽  
Traffic Flow ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Second Order ◽  
Nonlinear Hyperbolic Equation ◽  
Existence Of A Solution ◽  
Special Case

AbstractHere we study the existence of a solution and also the behavior of the existing solution of the abstract nonlinear differential equation of second order that, in particular, is the nonlinear hyperbolic equation with nonlinear main parts, and in the special case, is the equation of the type of equation of traffic flow.

The Homogeneous Flow

2018 ◽  
Author(s):  
Ercüment H. Ortaçgil
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Second Order ◽  
Initial Condition ◽  
Homogeneous Flow ◽  
Short Time

In this chapter, a second-order nonlinear evolution equation is constructed that starts with any parallelism as initial condition and flows in the direction of a parallelism with vanishing curvature. The existence of unique short-time solutions is proved.

scholarly journals Damped second order flow applied to image denoising

2019 ◽  
Vol 84 (6) ◽  
pp. 1082-1111 ◽  
Author(s):  
G Baravdish ◽  
O Svensson ◽  
M Gulliksson ◽  
Y Zhang
Keyword(s):  
Evolution Equation ◽  
Image Denoising ◽  
Nonlinear Evolution Equation ◽  
Numerical Experiments ◽  
Nonlinear Evolution ◽  
Second Order ◽  
Order Flow ◽  
Numerical Implementation ◽  
Energy Functionals

Abstract In this paper, we introduce a new image denoising model: the damped flow (DF), which is a second order nonlinear evolution equation associated with a class of energy functionals of an image. The existence, uniqueness and regularization property of DF are proven. For the numerical implementation, based on the Störmer–Verlet method, a discrete DF, SV-DDF, is developed. The convergence of SV-DDF is studied as well. Several numerical experiments, as well as a comparison with other methods, are provided to demonstrate the efficiency of SV-DDF.

scholarly journals Progression of Aortic Dissection: An Application of Soliton Solutions of Nonlinear Evolution Equation in (3+1)-Dimensions

2021 ◽  
Author(s):  
VISHAKHA JADAUN ◽  
Nitin Singh
Keyword(s):  
Differential Equation ◽  
Aortic Dissection ◽  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Early Stage ◽  
Nonlinear Partial Differential Equation ◽  
Aortic Tissue ◽  
The Impact

Abstract Aortic dissection is a serious pathology involving the vessel wall of the aorta with significant societal impact. To understand aortic dissection we explain the role of the dynamic pathology in the absence or presence of structural and/or functional abnormalities. We frame a differential equation to evaluate the impact of mean blood pressure on the aortic wall and prove the existence and uniqueness of its solution for homeostatic recoil and relaxation for infinitesimal aortic tissue. We model and analyze generalized (3+1)-dimensional nonlinear partial differential equation for aortic wave dynamics. We use the Lie group of transformations on this nonlinear evolution equation to obtain invariant solutions, traveling wave solutions including solitons. We find that abnormalities in the dynamic pathology of aortic dissection act as triggers for the progression of disease in early-stage through the formation of soliton-like pulses and their interaction. We address the role of unstable wavefields in waveform dynamics when waves are unidirectional. Moreover, the notion of dynamic pathology within the domain of vascular geometry may explain the evolution of aneurysms in cerebral arteries and cardiomyopathies even in the absence of anatomical and physiological abnormalities.

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