scholarly journals Crank--Nicolson Finite Element Approximations for a Linear Stochastic Fourth Order Equation with Additive Space-Time White Noise

2018 ◽  
Vol 56 (2) ◽  
pp. 838-858
Author(s):  
Georgios E. Zouraris
Keyword(s):  
Finite Element ◽  
White Noise ◽  
Fourth Order ◽  
Space Time ◽  
Order Equation ◽  
Fourth Order Equation ◽  
Finite Element Approximations ◽  
Crank Nicolson

scholarly journals THE EASTWOOD–SINGER GAUGE IN EINSTEIN SPACES

2009 ◽  
Vol 06 (04) ◽  
pp. 583-593 ◽  
Author(s):  
GIAMPIERO ESPOSITO ◽  
RAJU ROYCHOWDHURY
Keyword(s):  
Wave Equation ◽  
Fourth Order ◽  
Space Time ◽  
Order Equation ◽  
Scalar Wave ◽  
Scalar Wave Equation ◽  
Fourth Order Equation ◽  
Vector Wave ◽  
Vector Wave Equation ◽  
Einstein Spaces

Electrodynamics in curved space-time can be studied in the Eastwood–Singer gauge, which has the advantage of respecting the invariance under conformal rescalings of the Maxwell equations. Such a construction is here studied in Einstein spaces, for which the Ricci tensor is proportional to the metric. The classical field equations for the potential are then equivalent to first solving a scalar wave equation with cosmological constant, and then solving a vector wave equation where the inhomogeneous term is obtained from the gradient of the solution of the scalar wave equation. The Eastwood–Singer condition leads to a field equation on the potential which is preserved under gauge transformations provided that the scalar function therein obeys a fourth-order equation where the highest-order term is the wave operator composed with itself. The second-order scalar equation is here solved in de Sitter space-time, and also the fourth-order equation in a particular case, and these solutions are found to admit an exponential decay at large time provided that square-integrability for positive time is required. Last, the vector wave equation in the Eastwood–Singer gauge is solved explicitly when the potential is taken to depend only on the time variable.

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.

A Geometrical Solution for the Inverse Kinematics of Three Revolute Manipulators

1993 ◽  
Author(s):  
Eugene F. Fichter
Keyword(s):  
Fourth Order ◽  
Inverse Kinematics ◽  
Solution Procedure ◽  
Order Equation ◽  
Algebraic Solution ◽  
Computer Implementation ◽  
Third Order ◽  
Fourth Order Equation ◽  
Inverse Kinematics Problem

Abstract Points of intersection of a circle and a torus are used to find a solution to the inverse kinematics problem for a three revolute manipulator. Both geometrical and algebraic solution procedures are discussed. The algebraic procedure begins with a third order equation instead of the usual fourth order equation. Since the procedure is basically geometrical it lends itself to a computer implementation which graphically displays each steps in the solution procedure. The potential of this approach for both design and pedagogy is discussed.

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