Continuity of generalized solutions of the cauchy problem for the porous medium equation

1974 ◽  
pp. 363-367 ◽  
Author(s):  
B. H. Gilding
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Porous Medium Equation ◽  
Generalized Solutions ◽  
Medium Equation ◽  
The Cauchy Problem

scholarly journals A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2843
Author(s):  
Ángel García ◽  
Mihaela Negreanu ◽  
Francisco Ureña ◽  
Antonio M. Vargas
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Meshless Method ◽  
Existence And Uniqueness ◽  
Fractional Laplacian ◽  
Porous Medium Equation ◽  
Medium Equation ◽  
The Cauchy Problem ◽  
Irregular Meshes ◽  
Complicated Geometry

The existence and uniqueness of the discrete solutions of a porous medium equation with diffusion are demonstrated. The Cauchy problem contains a fractional Laplacian and it is equivalent to the extension formulation in the sense of trace and harmonic extension operators. By using the generalized finite difference method, we obtain the convergence of the numerical solution to the classical/theoretical solution of the equation for nonnegative initial data sufficiently smooth and bounded. This procedure allows us to use meshes with complicated geometry (more realistic) or with an irregular distribution of nodes (providing more accurate solutions where needed). Some numerical results are presented in arbitrary irregular meshes to illustrate the potential of the method.

scholarly journals The Cauchy problem for the inhomogeneous porous medium equation

2006 ◽  
Vol 1 (2) ◽  
pp. 337-351 ◽  
Author(s):  
Guillermo Reyes ◽  
◽  
Juan-Luis Vázquez ◽  
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Porous Medium Equation ◽  
Medium Equation ◽  
The Cauchy Problem ◽  
Inhomogeneous Porous Medium

Large time estimates for solutions to the porous medium equation with nonintegrable data via comparison

1985 ◽  
Vol 100 (1-2) ◽  
pp. 1-10
Author(s):  
Nicholas D. Alikakos ◽  
Rouben Rostamian
Keyword(s):  
Porous Medium ◽  
Initial Data ◽  
Original Problem ◽  
Large Time ◽  
Porous Medium Equation ◽  
Comparison Method ◽  
Upper And Lower Bounds ◽  
Medium Equation ◽  
Time Estimates ◽  
The Cauchy Problem

SynopsisWe consider the Cauchy problem for the porous medium equation in one space dimension, with initial data which are locally integrable. We measure the asymptotic behaviour of the initial data near infinity in an integral sense and relate this to the pointwise rate of growth or decay of solution for large time. The emphasis is on a novel comparison method wherein the initial data are rearranged on the ×-axis to form a sequence of Dirac δ-masses. By using the explicit solution in the latter case, we derive upper and lower bounds for the solution to the original problem by comparisons.

Maximal viscosity solutions of the modified porous medium equation and their asymptotic behaviour

1996 ◽  
Vol 7 (5) ◽  
pp. 453-471 ◽  
Author(s):  
Josephus Hulshof ◽  
Juan Luis Vazquez
Keyword(s):  
Porous Medium ◽  
Asymptotic Behaviour ◽  
Viscosity Solutions ◽  
Propagation Speed ◽  
Porous Medium Equation ◽  
Main Theme ◽  
Optimal Regularity ◽  
Medium Equation ◽  
The Cauchy Problem ◽  
Similar Solutions

We construct a theory for maximal viscosity solutions of the Cauchy problem for the modified porous medium equation ut + γ|ut| = Δ(um) with γ∈(−1, 1) and m > 1. We investigate the existence, uniqueness, finite propagation speed and optimal regularity of these solutions. As a second main theme, we prove that the asymptotic behaviour is given by a certain family of self-similar solutions of the so-called second kind with anomalous similarity exponents.

Homotopy Perturbation Method for General Form of Porous Medium Equation

2009 ◽  
Vol 12 (11) ◽  
pp. 1121-1127 ◽  
Author(s):  
Jafar Biazar ◽  
Zainab Ayati ◽  
Hamideh Ebrahimi
Keyword(s):  
Porous Medium ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Porous Medium Equation ◽  
Homotopy Perturbation ◽  
Medium Equation
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