Solvability of boundary problems for quasi-linear parabolic equations of higher order

1967 ◽  
pp. 1-40 ◽  
Author(s):  
M. I. Višik
Keyword(s):  
Parabolic Equations ◽  
Higher Order ◽  
Boundary Problems ◽  
Linear Parabolic Equations

scholarly journals Galerkin-Chebyshev Pseudo Spectral Method and a Split Step New Approach for a Class of Two dimensional Semi-linear Parabolic Equations of Second Order

2018 ◽  
Vol 3 (1) ◽  
pp. 255-264 ◽  
Author(s):  
F. Talay Akyildiz ◽  
K. Vajravelu
Keyword(s):  
Spectral Method ◽  
Parabolic Equations ◽  
Splitting Method ◽  
Higher Order ◽  
Two Dimensional ◽  
Order Accuracy ◽  
Special Cases ◽  
Linear Parabolic Equations ◽  
Pseudo Spectral Method ◽  
Pseudo Spectral

AbstractIn this paper, we use a time splitting method with higher-order accuracy for the solutions (in space variables) of a class of two-dimensional semi-linear parabolic equations. Galerkin-Chebyshev pseudo spectral method is used for discretization of the spatial derivatives, and implicit Euler method is used for temporal discretization. In addition, we use this novel method to solve the well-known semi-linear Poisson-Boltzmann (PB) model equation and obtain solutions with higher-order accuracy. Furthermore, we compare the results obtained by our method for the semi-linear parabolic equation with the available analytical results in the literature for some special cases, and found excellent agreement. Furthermore, our new technique is also applicable for three-dimensional problems.

scholarly journals Initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Liming Xiao ◽  
Mingkun Li
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Evolution Problem ◽  
Positive Energy ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations which do not have positive energy and come from the soil mechanics, the heat conduction, and the nonlinear optics. By the mountain pass theorem we first prove the existence of nonzero weak solution to the static problem, which is the important basis of evolution problem, then based on the method of potential well we prove the existence of global weak solution to the evolution problem.

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