Lévy stable distribution and space-fractional Fokker-Planck type equation

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• 作者：Jun-Sheng Duan<sup>asup><sup> ; sup><sup>duanjs@sit.edu.cn" class="auth_mail" title="E-mail the corresponding authorsup> ; Temuer Chaolu<sup>bsup> ; Zhong Wang<sup>csup> ; Shou-Zhong Fu<sup>csup>
• 关键词：Fokker&ndash ; Planck equation ; Laplace operator ; Fourier transform ; Fox H function
• 刊名：Journal of King Saud University - Science
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：28
• 期：1
• 页码：17-20
• 全文大小：495 K

文摘

The space-fractional Fokker&ndash;Planck type equation <span id="mmlsi9" class="mathmlsrc">source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1018364715000336&_mathId=si9.gif&_user=111111111&_pii=S1018364715000336&_rdoc=1&_issn=10183647&md5=688339e688acb4c4d95ad430aa8abee2">ss="imgLazyJSB inlineImage" height="23" width="245" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1018364715000336-si9.gif">script>style="vertical-align:bottom" width="245" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1018364715000336-si9.gif">script><span class="mathContainer hidden"><span class="mathCode">scroll">∂p∂t+γ∂p∂x=-Dsup>stretchy="false">(-Δstretchy="false">)α/2sup>pspace width="0.35em">space>stretchy="false">(0<α⩽2stretchy="false">)span>span>span> subject to the initial condition <span id="mmlsi10" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1018364715000336&_mathId=si10.gif&_user=111111111&_pii=S1018364715000336&_rdoc=1&_issn=10183647&md5=0826bb84500fb53ce38af1358e4307df" title="Click to view the MathML source">p(x,0)=δ(x)span><span class="mathContainer hidden"><span class="mathCode">scroll">pstretchy="false">(x,0stretchy="false">)=δstretchy="false">(xstretchy="false">)span>span>span> is solved in terms of Fox H functions. The solution as <span id="mmlsi11" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1018364715000336&_mathId=si11.gif&_user=111111111&_pii=S1018364715000336&_rdoc=1&_issn=10183647&md5=8c2a75e3e8c3e45fdc0dbeadb08307f3" title="Click to view the MathML source">γ=0span><span class="mathContainer hidden"><span class="mathCode">scroll">γ=0span>span>span> expresses the Lévy stable distribution with the index <span id="mmlsi12" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1018364715000336&_mathId=si12.gif&_user=111111111&_pii=S1018364715000336&_rdoc=1&_issn=10183647&md5=4019575657237264054e64fe9ce77677" title="Click to view the MathML source">αspan><span class="mathContainer hidden"><span class="mathCode">scroll">αspan>span>span>. From the properties of Fox H functions, the series representation and asymptotic behavior for the solution are also obtained. Lévy stable distribution as <span id="mmlsi13" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1018364715000336&_mathId=si13.gif&_user=111111111&_pii=S1018364715000336&_rdoc=1&_issn=10183647&md5=d2caea7bca9f17ac1dd1395c45990ce4" title="Click to view the MathML source">0<α<2span><span class="mathContainer hidden"><span class="mathCode">scroll">0<α<2span>span>span> describes anomalous superdiffusion and its diffusion velocity is characterized by <span id="mmlsi14" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1018364715000336&_mathId=si14.gif&_user=111111111&_pii=S1018364715000336&_rdoc=1&_issn=10183647&md5=ee2b673399c46c39cef26c68b450119e" title="Click to view the MathML source">x<sub>dsub>∝(Dt)<sup>1/αsup>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>xdsub>∝sup>stretchy="false">(Dtstretchy="false">)1/αsup>span>span>span>.