Existence of generalized solutions, increasing at infinity, of boundary-value problems for linear and quasilinear parabolic equations

1986 ◽  
Vol 37 (4) ◽  
pp. 378-385
Author(s):  
A. E. Shishkov
Keyword(s):  
Boundary Value Problems ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Generalized Solutions ◽  
Quasilinear Parabolic Equations ◽  
Quasilinear Parabolic

On Convergence of Difference Schemes for IBVP for Quasilinear Parabolic Equations with Generalized Solutions

2014 ◽  
Vol 14 (3) ◽  
pp. 361-371 ◽  
Author(s):  
Piotr Matus
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Parabolic Equations ◽  
Time Derivative ◽  
Initial Boundary ◽  
Generalized Solutions ◽  
Quasilinear Parabolic Equations ◽  
Differential Problem ◽  
The Maximum Principle ◽  
Quasilinear Parabolic

Abstract.We construct an usual linearized difference scheme for initial boundary value problems (IBVP) for one-dimensional quasilinear parabolic equations with generalized solutions. The uniform parabolicity condition $0<k_1\le k(u) \le k_2$ is assumed to be fulfilled for the sign alternating solution $u(x,t) \in \bar{D}(u)$ only in the domain of exact solution values (unbounded non-linearity). On the basis of new corollaries of the maximum principle, we establish not only two-sided estimates for the approximate solution y but its belonging to the domain of exact solution values. We assume that the solution is continuous and its first derivative $\frac{\partial u}{\partial x}$ has discontinuity of the first kind in the neighborhood of the finite number of discontinuity lines. An existence of time derivative in any sense is not assumed. We prove convergence of the approximate solution to the generalized solution of the differential problem in the grid norm L2.

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