International Journal of Partial Differential Equations and Applications
ISSN (Print): 2376-9548 ISSN (Online): 2376-9556 Website: http://www.sciepub.com/journal/ijpdea Editor-in-chief: Mahammad Nurmammadov
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International Journal of Partial Differential Equations and Applications. 2020, 8(1), 1-5
DOI: 10.12691/ijpdea-8-1-1
Open AccessArticle

Solving Advection-Diffusion Equations via Sobolev Space Notions

Atefeh Hasan-Zadeh1,

1Fouman Faculty of Engineering, College of Engineering, University of Tehran, Fouman, Iran

Pub. Date: September 16, 2020

Cite this paper:
Atefeh Hasan-Zadeh. Solving Advection-Diffusion Equations via Sobolev Space Notions. International Journal of Partial Differential Equations and Applications. 2020; 8(1):1-5. doi: 10.12691/ijpdea-8-1-1

Abstract

In this paper, the time-dependent advection-diffusion equation is studied. After introducing these equations in various engineering fields such as gas adsorption, solid dissolution, heat and mass transfer in falling film or pipe and other equations similar to transport phenomena, a new method has been proposed to find their solutions. Among the various works on solving these PDEs by numerical and somewhat analytical methods, a general analytical framework for solving these equations is presented. Using advanced components of Sobolev spaces, weak solutions and some important integral inequalities, an analytical method for the existence and uniqueness of the weak solution of these PDEs is presented, which is the best solution in the proposed structure. Then, with a reduced system of ODE, one can solve the problem of the general parabolic boundary value problem, which includes PDE transport phenomena. Besides, the new approach supports the infinite propagation speed of disturbances of (time-dependent) diffusion-time equations in semi-infinite media.

Keywords:
advection-diffusion equation Sobolev spaces weak solutions Harnack's inequality semi-infinite media

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