Fractional-parabolic equations and systems. Cauchy problem

2019 ◽  
pp. 145-158
Author(s):  
Anatoly N. Kochubei
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equations

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 Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

2020 ◽  
Vol 10 (1) ◽  
pp. 353-370 ◽  
Author(s):  
Hans-Christoph Grunau ◽  
Nobuhito Miyake ◽  
Shinya Okabe
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Fourth Order ◽  
Parabolic Equations ◽  
Sufficient Conditions ◽  
Nonlinear Parabolic Equations ◽  
Heat Equations ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Positivity Of Solutions

Abstract This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

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

scholarly journals On polynomials of Sheffer type arising from a Cauchy problem

2003 ◽  
Vol 2003 (33) ◽  
pp. 2119-2137
Author(s):  
D. G. Meredith
Keyword(s):  
Cauchy Problem ◽  
Heat Conduction ◽  
Parabolic Equations ◽  
Analytic Solution ◽  
Heat Conduction Problem ◽  
Laguerre Polynomials ◽  
Inverse Heat Conduction Problem ◽  
Inverse Heat Conduction ◽  
Operator Calculus ◽  
Classical Polynomials

A new sequence of eigenfunctions is developed and studied in depth. These theta polynomials are derived from a recent analytic solution of the canonical Cauchy problem for parabolic equations, namely, the inverse heat conduction problem. By appealing to the methods of the operator calculus, it is possible to categorize the new functions as polynomials of binomial and Sheffer types. The connection of the new set with the classical polynomials of Laguerre is carefully examined. Some integral relations involving the Laguerre polynomials and the theta polynomials are presented along with a number of binomial identities. The inverse heat conduction problem is revisited and an analytic solution depending on the generalized theta polynomials is presented.

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