scholarly journals Spatial Moduli of Non-Differentiability for Time-Fractional SPIDEs and Their Gradient

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 380
Author(s):  
Wensheng Wang
Keyword(s):  
Differential Equations ◽  
Heat Equation ◽  
White Noise ◽  
Harmonic Analysis ◽  
Fourth Order ◽  
Gaussian Processes ◽  
Space Time ◽  
The Other ◽  
Moduli Of Continuity ◽  
Fractional Pdes

High order and fractional PDEs have become prominent in theory and in modeling many phenomena. In this paper, we study spatial moduli of non-differentiability for the fourth order time fractional stochastic partial integro-differential equations (SPIDEs) and their gradient, driven by space-time white noise. We use the underlying explicit kernels and spectral/harmonic analysis, yielding spatial moduli of non-differentiability for time fractional SPIDEs and their gradient. On one hand, this work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. On the other hand, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of time fractional SPIDEs and their gradient.

scholarly journals Spatial Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1251
Author(s):  
Wensheng Wang
Keyword(s):  
Heat Equation ◽  
White Noise ◽  
Fourth Order ◽  
Gaussian Processes ◽  
Three Dimensional ◽  
Space Time ◽  
Symmetry Analysis ◽  
Stochastic Heat Equation ◽  
Moduli Of Continuity

We investigate spatial moduli of non-differentiability for the fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces. We use the underlying explicit kernels and symmetry analysis, yielding spatial moduli of non-differentiability for L-KS SPDEs and their gradient. This work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. Moreover, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of L-KS SPDEs and their gradient.

scholarly journals Asymptotic Distributions for Power Variations of the Solution to the Spatially Colored Stochastic Heat Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Wensheng Wang ◽  
Xiaoying Chang ◽  
Wang Liao
Keyword(s):  
Heat Equation ◽  
White Noise ◽  
Harmonic Analysis ◽  
Central Limit ◽  
Gaussian Processes ◽  
Limit Theorems ◽  
Colored Noise ◽  
Asymptotic Distributions ◽  
Heat Equations ◽  
Stochastic Heat Equations

Let u α , d = u α , d t , x ,   t ∈ 0 , T , x ∈ ℝ d be the solution to the stochastic heat equations (SHEs) with spatially colored noise. We study the realized power variations for the process u α , d , in time, having infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions. We use the underlying explicit kernels and spectral/harmonic analysis, yielding temporal central limit theorems for SHEs with spatially colored noise. This work builds on the recent works on delicate analysis of variations of general Gaussian processes and SHEs driven by space-time white noise.

Critical Force in the Buckling of Drill Bits

1979 ◽  
Vol 46 (2) ◽  
pp. 461-462 ◽  
Author(s):  
L. E. Bobisud ◽  
C. O. Christenson
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
Fourth Order ◽  
Critical Force ◽  
The Other ◽  
Maximum Force ◽  
System Of Differential Equations ◽  
Linearized Equations ◽  
Drill Bits

We consider a fourth-order nonlinear system of differential equations that describe the slope of a steadily rotating flexible rod, one end of which is clamped, and the other end of which is “hinged.” There is a force directed along the length of the rod. We graph against speed of rotation the maximum force that can be applied before buckling occurs, using linearized equations.

The extended auxiliary equation mapping method to determine novel exact solitary wave solutions of the nonlinear fractional PDEs

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jalil Manafian ◽  
Onur Alp Ilhan ◽  
Laleh Avazpour
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Traveling Wave Solutions ◽  
Mapping Method ◽  
Space Time ◽  
Integration Scheme ◽  
Auxiliary Equation ◽  
Wave Solutions ◽  
Fractional Pdes ◽  
Engineering Sciences

AbstractIn this paper, some new nonlinear fractional partial differential equations (PDEs) have been considered.Three models are including the space-time fractional-order Boussinesq equation, space-time (2 + 1)-dimensional breaking soliton equations, and space-time fractional-order SRLW equation describe the behavior of these equations in the diverse applications. Meanwhile, the fractional derivatives in the sense of β-derivative are defined. Some fractional PDEs will convert to the considered ordinary differential equations by the help of transformation of β-derivative. These equations are analyzed utilizing an integration scheme, namely, the extended auxiliary equation mapping method. The different kinds of traveling wave solutions, solitary, topological, dark soliton, periodic, kink, and rational, fall out as a by-product of this scheme. Finally, the existence of the solutions for the constraint conditions is also shown. The outcome indicates that some fractional PDEs are used as a growing finding in the engineering sciences, mathematical physics, and so forth.

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