Quasilinearization Methods for Nonlinear Parabolic Equations with Functional Dependence

2002 ◽  
Vol 9 (3) ◽  
pp. 431-448
Author(s):  
A. Bychowska
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equations ◽  
Unknown Function ◽  
Functional Dependence ◽  
Nonlinear Parabolic Equations ◽  
Quasilinearization Method ◽  
Convergence Theorems ◽  
Nonlinear Parabolic ◽  
Quasilinearization Methods ◽  
Derivatives Of

Abstract We consider a Cauchy problem for nonlinear parabolic equations with functional dependence. We prove convergence theorems for a general quasilinearization method in two cases: (i) the Hale functional acting only on the unknown function, (ii) including partial derivatives of the unknown function.

scholarly journals On Weak Solution of Cauchy Problem for Nonlinear Parabolic Equations

2019 ◽  
Vol 1341 ◽  
pp. 062038
Author(s):  
Naimah Aris ◽  
Muh. Rifki Nisardi ◽  
Kasbawati ◽  
Jeffry Kusuma ◽  
Budi Nurwahyu ◽  
...  
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Parabolic Equations ◽  
Nonlinear Parabolic Equations ◽  
Nonlinear Parabolic

The Cauchy Problem for Nonlinear Parabolic Equations with Lévy Laplacian

2006 ◽  
Vol 24 (2) ◽  
pp. 125-136 ◽  
Author(s):  
S. Albeverio ◽  
Ya. Belopolskaya ◽  
M. Feller
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equations ◽  
Nonlinear Parabolic Equations ◽  
Nonlinear Parabolic ◽  
Lévy Laplacian ◽  
The Cauchy Problem

GENERALIZED SOLUTIONS OF THE CAUCHY PROBLEM FOR SYSTEMS OF NONLINEAR PARABOLIC EQUATIONS AND DIFFUSION PROCESSES

2012 ◽  
Vol 12 (01) ◽  
pp. 1150001 ◽  
Author(s):  
YANA BELOPOLSKAYA ◽  
WOJBOR A. WOYCZYNSKI
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equations ◽  
Diffusion Processes ◽  
Stochastic Approach ◽  
Generalized Solution ◽  
Nonlinear Parabolic Equations ◽  
Kolmogorov Equation ◽  
Generalized Solutions ◽  
Nonlinear Parabolic ◽  
The Cauchy Problem

The purpose of this paper is to construct both strong and weak solutions (in certain functional classes) of the Cauchy problem for a class of systems of nonlinear parabolic equations via a unified stochastic approach. To this end we give a stochastic interpretation of such a system, treating it as a version of the backward Kolmogorov equation for a two-component Markov process with coefficients depending on the distribution of its first component. To extend this approach and apply it to the construction of a generalized solution of a system of nonlinear parabolic equations, we use results from Kunita's theory of stochastic flows.

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