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The quasilinear parabolic kirchhoff equation

100%
Dawidowski Ł.
Open Mathematics
|
2017
|
tom 15
|
nr 1
382-392
EN
In this paper the existence of solution of a quasilinear generalized Kirchhoff equation with initial – boundary conditions of Dirichlet type will be studied using the Leray – Schauder principle.
2
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Spatially-dependent and nonlinear fluid transport: coupling framework

64%
Geiser J.
Open Mathematics
|
2012
|
tom 10
|
nr 1
116-136
EN
We introduce a solver method for spatially dependent and nonlinear fluid transport. The motivation is from transport processes in porous media (e.g., waste disposal and chemical deposition processes). We analyze the coupled transport-reaction equation with mobile and immobile areas. The main idea is to apply transformation methods to spatial and nonlinear terms to obtain linear or nonlinear ordinary differential equations. Such differential equations can be simply solved with Laplace transformation methods or nonlinear solver methods. The nonlinear methods are based on characteristic methods and can be generalized numerically to higher-order TVD methods [Harten A., High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 1983, 49(3), 357–393]. In this article we will focus on the derivation of some analytical solutions for spatially dependent and nonlinear problems which can be embedded into finite volume methods. The main contribution is to embed one-dimensional analytical solutions into multi-dimensional finite volume methods with the construction idea of mass transport [Geiser J., Mobile and immobile fluid transport: coupling framework, Internat. J. Numer. Methods Fluids, 2010, 65(8), 877–922]. At the end of the article we present some results of numerical experiments for different benchmark problems.
3
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The finite speed of propagation of solutions of the Neumann problem of a degenerate parabolic equation

64%
Pan J.
Open Mathematics
|
2011
|
tom 9
|
nr 3
673-685
EN
In this paper the finite speed of propagation of solutions and the continuous dependence on the nonlinearity of a degenerate parabolic partial differential equation are discussed. Our objective is to derive an explicit expression for the speed of propagation and the large time behavior of the solution and to show that the solution continuously depends on the nonlinearity of the equation.
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