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Fourth Order Diffusion Equations with Increasing Entropy
Department of Mathematics and Statistics, La Trobe University, Bundoora VIC 3086, Australia
* Author to whom correspondence should be addressed.
Received: 11 April 2012; in revised form: 19 June 2012 / Accepted: 19 June 2012 / Published: 25 June 2012
Abstract: The general quasi-linear autonomous fourth order diffusion equation ut = −[G(u)uxxx + h(u, ux, uxx)]x with positive variable diffusivity G(u) and lower-order flux component h is considered on the real line. A direct algorithm produces a general class of equations for which the Shannon entropy density obeys a reaction-diffusion equation with a positive irreducible source term. Such equations may have any positive twice-differentiable diffusivity function G(u). The forms of such equations are the indicators of more general conservation equations whose entropy equation may be expressed in an alternative reaction-diffusion form whose source term, although reducible, is positive.
Keywords: Shannon entropy; fourth order diffusion; irreversibility
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MDPI and ACS Style
Tehseen, N.; Broadbridge, P. Fourth Order Diffusion Equations with Increasing Entropy. Entropy 2012, 14, 1127-1139.
Tehseen N, Broadbridge P. Fourth Order Diffusion Equations with Increasing Entropy. Entropy. 2012; 14(7):1127-1139.
Tehseen, Naghmana; Broadbridge, Philip. 2012. "Fourth Order Diffusion Equations with Increasing Entropy." Entropy 14, no. 7: 1127-1139.