On a nonlinear stochastic pseudo-differential equation driven by fractional noise

2017 ◽  
Vol 18 (01) ◽  
pp. 1850002 ◽  
Author(s):  
Junfeng Liu ◽  
Litan Yan
Keyword(s):  
Differential Equation ◽  
Differential Operator ◽  
Mild Solution ◽  
Malliavin Calculus ◽  
Measurable Function ◽  
Transition Density ◽  
Variable Order ◽  
Pseudo Differential Operator ◽  
Fractional Noise ◽  
Pseudo Differential Equation

In this paper, we study the existence, uniqueness and Hölder regularity of the solution to a class of nonlinear stochastic pseudo-differential equation of the following form [Formula: see text] where [Formula: see text] is a pseudo-differential operator with negative definite symbol of variable order which generates a stable-like process with transition density, the coefficient [Formula: see text] is a measurable function, and [Formula: see text] is a double-parameter fractional noise. In addition, the existence and Gaussian type estimates for the density of the mild solution are proved via Malliavin calculus.

SOLVING A NONLINEAR PSEUDO-DIFFERENTIAL EQUATION OF BURGERS TYPE

2008 ◽  
Vol 08 (04) ◽  
pp. 613-624 ◽  
Author(s):  
NIELS JACOB ◽  
ALEXANDER POTRYKUS ◽  
JIANG-LUN WU
Keyword(s):  
Differential Equation ◽  
Differential Operator ◽  
Nonlinear Equations ◽  
Initial Value Problem ◽  
Symbolic Calculus ◽  
Pseudo Differential Operator ◽  
Initial Value ◽  
Semigroup Approach ◽  
Pseudo Differential Equation ◽  
Point Argument

In this paper, we study the initial value problem for a class of nonlinear equations of Burgers type in the following form: [Formula: see text] for u:(t,x) ∈ (0,∞) × ℝn ↦ ℝ, where q(x,D) is a pseudo-differential operator with negative definite symbol. We solve the initial value problem for the equation on ℝn by utilising a fix point argument based upon a combination of semigroup approach and Hoh's symbolic calculus.

The Mehler-Fock-Clifford transform and pseudo-differential operator on function spaces

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2457-2469
Author(s):  
Akhilesh Prasad ◽  
S.K. Verma
Keyword(s):  
Differential Operator ◽  
Function Spaces ◽  
Pseudo Differential Operator ◽  
Test Function ◽  
Basic Properties ◽  
Cone Function ◽  
Translation Operators

In this article, weintroduce a new index transform associated with the cone function Pi ??-1/2 (2?x), named as Mehler-Fock-Clifford transform and study its some basic properties. Convolution and translation operators are defined and obtained their estimates under Lp(I, x-1/2 dx) norm. The test function spaces G? and F? are introduced and discussed the continuity of the differential operator and MFC-transform on these spaces. Moreover, the pseudo-differential operator (p.d.o.) involving MFC-transform is defined and studied its continuity between G? and F?.

scholarly journals Approximation and application of the Riesz-Caputo fractional derivative of variable order with fixed memory

Meccanica ◽  
2021 ◽  
Author(s):  
Tomasz Blaszczyk ◽  
Krzysztof Bekus ◽  
Krzysztof Szajek ◽  
Wojciech Sumelka
Keyword(s):  
Differential Operator ◽  
Fractional Derivative ◽  
Rate Of Convergence ◽  
Continuum Mechanics ◽  
Caputo Fractional Derivative ◽  
Polynomial Interpolation ◽  
Piecewise Linear ◽  
Variable Order ◽  
Piecewise Constant ◽  
Quadratic Interpolation

AbstractIn this paper, the Riesz-Caputo fractional derivative of variable order with fixed memory is considered. The studied non-integer differential operator is approximated by means of modified basic rules of numerical integration. The three proposed methods are based on polynomial interpolation: piecewise constant, piecewise linear, and piecewise quadratic interpolation. The errors generated by the described methods and the experimental rate of convergence are reported. Finally, an application of the Riesz-Caputo fractional derivative of space-dependent order in continuum mechanics is depicted.

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