scholarly journals IRREVERSIBILITY AND HYSTERESIS FOR A FORWARD–BACKWARD DIFFUSION EQUATION

2004 ◽  
Vol 14 (11) ◽  
pp. 1599-1620 ◽  
Author(s):  
L. C. EVANS ◽  
M. PORTILHEIRO
Keyword(s):  
Diffusion Equation ◽  
Nonlinear Diffusion ◽  
Cubic Nonlinearity ◽  
Important Work ◽  
Ill Posed ◽  
Entropy Flux ◽  
Formal Limit

Our intention in this paper is to publicize and extend somewhat important work of Plotnikov7 on the asymptotic limits of solutions of viscous regularizations of a nonlinear diffusion PDE with a cubic nonlinearity. Since the formal limit PDE is in general ill-posed, we expect that the limit solves instead a corresponding diffusion equation with hysteresis effects. We employ entropy/entropy flux pairs to prove various assertions consistent with this expectation.

scholarly journals Application of He's Homotopy Perturbation Method for Cauchy Problem of Ill-Posed Nonlinear Diffusion Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Ali Zakeri ◽  
Azim Aminataei ◽  
Qodsiyeh Jannati
Keyword(s):  
Cauchy Problem ◽  
Diffusion Equation ◽  
Perturbation Method ◽  
Nonlinear Diffusion ◽  
Homotopy Perturbation Method ◽  
Finite Interval ◽  
Nonlinear Diffusion Equation ◽  
Third Order ◽  
Homotopy Perturbation ◽  
Ill Posed

We consider a Cauchy problem of unidimensional nonlinear diffusion equation on finite interval. This problem is ill-posed and its approximate solution is unstable. We apply the He's homotopy perturbation method (HPM) and obtain the third-order asymptotic expansion. We show that if the conductivity term in diffusion equation has a specified condition, the above solution can be estimated. Finally, a numerical experiment is provided to illustrate the method.

NONLINEAR DIFFUSION EQUATION WITH A PERTURBED CONVECTIONTERM: POTENTIAL SYMMETRIES WITH RESPECT TO THE SECOND CONSERVATION LAW

2021 ◽  
Vol 10 (5) ◽  
pp. 2611-2624
Author(s):  
O.K. Narain ◽  
F.M. Mahomed
Keyword(s):  
Diffusion Equation ◽  
Conservation Law ◽  
Nonlinear Diffusion ◽  
Nonlinear Diffusion Equation ◽  
The Other ◽  
Linear Case ◽  
Exact Equation ◽  
Convection Term ◽  
Potential Symmetries ◽  
Perturbed Equation

We consider the nonlinear diffusion equation with a perturbed convection term. The potential symmetries for the exact equation with respect to the second conservation law are classified. It is found that these exist only in the linear case. It is further shown that no nontrivial approximate potential symmetries of order one exists for the perturbed equation with respect to the other conservation law.

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