Connection of the solution of a stochastic differential inclusion with the solution of an ordinary differential equation

1993 ◽  
Vol 67 (4) ◽  
pp. 3197-3203
Author(s):  
T. N. Kravets
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Inclusion ◽  
Stochastic Differential Inclusion

scholarly journals Accurate approximated solution to the differential inclusion based on the ordinary differential equation

2021 ◽  
Vol 73 (1) ◽  
pp. 117-127
Author(s):  
T. H. Nguyen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Inclusion ◽  
Control Parameter ◽  
Normal Cone ◽  
Convex Set ◽  
Applied Mathematics ◽  
Small Error ◽  
The Cauchy Problem ◽  
Closed Convex Set

UDC 517.9 Many problems in applied mathematics can be transformed and described by the differential inclusion involving which is a normal cone to a closed convex set at The Cauchy problem of this inclusion is studied in the paper. Since the change of leads to the change of solving the inclusion becomes extremely complicated. In this paper, we consider an ordinary differential equation containing a control parameter When is large enough, the studied equation gives a solution approximating to a solution of the inclusion above. The theorem about the approximation of these solutions with arbitrary small error (this error can be controlled by increasing ) is proved in this paper.  

scholarly journals Classification of extinction profiles for a one-dimensional diffusive Hamilton–Jacobi equation with critical absorption

2018 ◽  
Vol 148 (3) ◽  
pp. 559-574
Author(s):  
Razvan Gabriel Iagar ◽  
Philippe Laurençot
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Jacobi Equation ◽  
Threshold Value ◽  
Monotonicity Property ◽  
Fast Diffusion ◽  
Extinction Time ◽  
One Step ◽  
Hamilton Jacobi Equation

A classification of the behaviour of the solutions f(·, a) to the ordinary differential equation (|f′|p-2f′)′ + f - |f′|p-1 = 0 in (0,∞) with initial condition f(0, a) = a and f′(0, a) = 0 is provided, according to the value of the parameter a > 0 when the exponent p takes values in (1, 2). There is a threshold value a* that separates different behaviours of f(·, a): if a > a*, then f(·, a) vanishes at least once in (0,∞) and takes negative values, while f(·, a) is positive in (0,∞) and decays algebraically to zero as r→∞ if a ∊ (0, a*). At the threshold value, f(·, a*) is also positive in (0,∞) but decays exponentially fast to zero as r→∞. The proof of these results relies on a transformation to a first-order ordinary differential equation and a monotonicity property with respect to a > 0. This classification is one step in the description of the dynamics near the extinction time of a diffusive Hamilton–Jacobi equation with critical gradient absorption and fast diffusion.

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