scholarly journals Periodic solutions of the boundary value problem for the nonlinear heat equation

1984 ◽  
Vol 30 (1) ◽  
pp. 99-110 ◽  
Author(s):  
M. N. Nkashama ◽  
M. Willem
Keyword(s):  
Boundary Value Problem ◽  
Heat Equation ◽  
Periodic Solutions ◽  
Boundary Value ◽  
Coincidence Degree ◽  
Nonlinear Heat Equation

We prove the existence of generalized periodic solutions of the boundary value problem for the nonlinear heat equation. The proof is based on classical Leray-Schauder's techniques and coincidence degree.

On a nonlinear heat equation with a time delay

2002 ◽  
Vol 13 (3) ◽  
pp. 321-335 ◽  
Author(s):  
YUNKANG LIU
Keyword(s):  
Time Delay ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Heat Equation ◽  
Turbulent Shear ◽  
Heat And Mass Exchange ◽  
Initial Boundary Value

A nonlinear forward-backward heat equation with a regularization term was proposed by Barenblatt et al. [1, 2] to model the heat and mass exchange in stably stratified turbulent shear flow. It was proven to be well-posed in the case of given initial and Neumann boundary conditions. However, the solution was found to have an unphysical discontinuity with certain smooth initial functions. In this paper, a nonlinear heat equation with a time delay originally used by Barenblatt et al. [1, 2] to derive their model is investigated. The same type of initial-boundary value problem is shown to have a unique smooth global solution when the initial function is reasonably smooth. Numerical examples are used to demonstrate that its solution forms step-like profiles in finite times. A semi-discretization of the initial-boundary value problem is proved to have a unique asymptotically and globally stable equilibrium.

scholarly journals On a fractional-order p-Laplacian boundary value problem at resonance on the half-line with two dimensional kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
O. F. Imaga ◽  
S. A. Iyase
Keyword(s):  
Differential Operator ◽  
Boundary Value Problem ◽  
Fractional Order ◽  
Boundary Value ◽  
Coincidence Degree ◽  
Half Line ◽  
At Resonance ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Problem At Resonance

AbstractIn this work, we consider the solvability of a fractional-order p-Laplacian boundary value problem on the half-line where the fractional differential operator is nonlinear and has a kernel dimension equal to two. Due to the nonlinearity of the fractional differential operator, the Ge and Ren extension of Mawhin’s coincidence degree theory is applied to obtain existence results for the boundary value problem at resonance. Two examples are used to validate the established results.

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