Acta Physica Hungarica 72. (1992)

1992 / 2-4. szám - Interdisciplinary - E. V. Krishnan: Travelling wave solutions of density dependent diffusion equations

Acta Physica Hungarica 72 (2-4), VP- 193-202 (1992) TRAVELLING WAVE SOLUTIONS OF DENSITY DEPENDENT DIFFUSION EQUATIONS E. V. Krishnan Department of Mathematics and Computing, Sultan Qaboos University Alkhod, Muscat, Sultanate of Oman (Received 4 August 1992) Travelling wave solutions for two nonlinear diffusion equations have been found by a direct method. The behaviour of solutions for these equations with c and the parameter a in the problem varying have been investigated numerically as a boundary value problem. The equilibrium solutions (c = 0) of these equations have been found in terms of Weierstrass elliptic functions. 1. Introduction The prototype for the spatial diffusion of biological populations in population dynamics is taken as [1] Щ = (u2)xx + F(u). (1.1) The first term on the right-hand side of Eq. (1.1) represents density dependent diffusion and the second term describes population supply due to births and deaths. The phenomena like flow of liquids in porous media, the transport of thermal energy in plasma etc. have also (1.1) as the governing equation. The exact solutions for Eq. (1.1) have been presented by Gurtin and MacCamy [2], Newman [3] and Hosono [4]. Gurtin and MacCamy considered the case F(u) = /ли and then by variable transformations, w = ■ w and r = (eMt — 1 )/p reduced (1.1) to wT = (iw2)**, (1.2) for which similarity solutions are known. Newman considered the case with F(u) = u(u — 1) and Hosono showed that the travelling wave solutions и — u(x — ct) of Eq. (1.1) with F(u) — u(u — l)(a — it) varies its profile with the sign of the velocity c. Satsuma [5] chose the same F(u) as that of Hosono and found an explicit expression for the travelling wave solution using Painlevé analysis. Ablowitz and Zeppetella [6] used the same method to obtain a travelling wave solution of Fisher’s equation Acta Physica Hungarica 72, 1992 Akadémiai Kiadó, Budapest Щ = uxx + u(l - u). (1.3)

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