Self‐organized criticality of cellular automata model; absorbtion in finite‐time of supercritical region into the critical one. (15th February 2013)

• Record Type: Journal Article
• Authors: Barbu, Viorel
• Is Part Of: Mathematical methods in the applied sciences. Volume 36:Number 13(2013:Sep. 15)
• Page Start: 1726
• Page End: 1733
• Publication Date: 2013-02-15
• Subjects: Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.2718
• Languages: English
• ISSNs: 0170-4214

Abstract

<abstract abstract-type="main" id="mma2718-abs-0001"> <title>Abstract</title> <p id="mma2718-para-0001">In this work, it is studied the evolution and time behavior of solutions to nonlinear diffusion equation <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg277n2107" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:01704214:media:mma2718:mma2718-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfrac><mml:mrow><mml:mi>∂ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>∂t</mml:mi></mml:mrow></mml:mfrac><mml:mo class="MathClass-bin">−</mml:mo><mml:mi>ΔH</mml:mi><mml:mrow><mml:mo class="MathClass-open">(</mml:mo><mml:mrow><mml:mi>ρ</mml:mi><mml:mo class="MathClass-bin">−</mml:mo><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo class="MathClass-close">)</mml:mo></mml:mrow><mml:mo class="MathClass-rel">∋</mml:mo><mml:mn>0</mml:mn></mml:math></alternatives> in <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg277n218n" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:01704214:media:mma2718:mma2718-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo class="MathClass-open">(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo class="MathClass-punc">, </mml:mo><mml:mo class="MathClass-rel">∞</mml:mo></mml:mrow><mml:mo class="MathClass-close">)</mml:mo></mml:mrow><mml:mo class="MathClass-bin">×</mml:mo><mml:mi mathvariant="script">O</mml:mi></mml:math></alternatives> where <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg277n213w" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:01704214:media:mma2718:mma2718-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">O</mml:mi><mml:mo class="MathClass-rel">⊂</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:math></alternatives>, <italic>d</italic> ≥ 1, and <italic>H</italic> is the Heaviside function. For <italic>d</italic> = 1, 2, 3, this equation describes the dynamics of self‐organizing sandpile process with critical state <italic>ρ</italic><sub><italic>c</italic></sub>. The main conclusion is that the supercritical region <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg277n21hh" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:01704214:media:mma2718:mma2718-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo class="MathClass-open">{</mml:mo><mml:mrow><mml:mi>ξ</mml:mi><mml:mo class="MathClass-rel">∈</mml:mo><mml:mi mathvariant="script">O</mml:mi><mml:mo class="MathClass-punc">;</mml:mo><mml:mspace width="1em" class="nbsp" /><mml:mi>ρ</mml:mi><mml:mrow><mml:mo class="MathClass-open">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo class="MathClass-punc">, </mml:mo><mml:mi>ξ</mml:mi></mml:mrow><mml:mo class="MathClass-close">)</mml:mo></mml:mrow><mml:mo class="MathClass-rel">></mml:mo><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo class="MathClass-open">(</mml:mo><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mo class="MathClass-close">)</mml:mo></mml:mrow></mml:mrow><mml:mo class="MathClass-close">}</mml:mo></mml:mrow></mml:math></alternatives> is absorbed in a finite‐time in the critical region <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg277n21gz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:01704214:media:mma2718:mma2718-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo class="MathClass-open">{</mml:mo><mml:mrow><mml:mi>ξ</mml:mi><mml:mo class="MathClass-rel">∈</mml:mo><mml:mi mathvariant="script">O</mml:mi><mml:mo class="MathClass-punc">;</mml:mo><mml:mspace width="1em" class="nbsp" /><mml:mi>ρ</mml:mi><mml:mrow><mml:mo class="MathClass-open">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo class="MathClass-punc">, </mml:mo><mml:mi>ξ</mml:mi></mml:mrow><mml:mo class="MathClass-close">)</mml:mo></mml:mrow><mml:mo class="MathClass-rel">=</mml:mo><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo class="MathClass-open">(</mml:mo><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mo class="MathClass-close">)</mml:mo></mml:mrow></mml:mrow><mml:mo class="MathClass-close">}</mml:mo></mml:mrow></mml:math></alternatives>. Copyright © 2013 John Wiley & Sons, Ltd.</p> </abstract>