This monograph deals with Burgers' equation and its generalisations. Such equations describe a wide variety of nonlinear diffusive phenomena, for instance, in nonlinear acoustics, laser physics, plasmas and atmospheric physics. The Burgers equation also has mathematical interest as a canonical nonlinear parabolic differential equation that can be exactly linearised. It is closely related to equations that display soliton behaviour and its study has helped elucidate other such nonlinear behaviour. The approach adopted here is applied mathematical. The author discusses fully the mathematical properties of standard nonlinear diffusion equations, and contrasts them with those of Burgers' equation. Of particular mathematical interest is the treatment of self-similar solutions as intermediate asymptotics for a large class of initial value problems whose solutions evolve into self-similar forms. This is achieved both analytically and numerically.

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The Falkner-Skan equation, subject to appropriate physical boundary conditions arising from boundary layer theory, is exactly solved. The results obtained from this solution are compared with the numerical solution. The Blasius equation, subject to the same boundary conditions, is also solved exactly; the solution is compared with the earlier work on this equation. The analytic solution presented here agrees closely with the corresponding numerical results.

Articles you may be interested in Stationary axisymmetric solutions involving a third order equation irreducible to Painlevé transcendentsThe problem ( 1.5) and ( 1.6) also occurs in the treatment of spherical and cylindrical nonlinear sound waves, in which g(x) has the particularformsg(x) = ~ and x, respectively. Scott studied, in particular, strictly self-similar solutions of the form u = net Ix). He considered the intermediate asymptotic behavior of three kinds of self-similar solutions 2397

An exact representation of N-wave solutions for the non-planar Burgers equation u t + uu x + ½ju/t = ½δu xx , j = m/n, m < 2n , where m and n are positive integers with no common factors, is given. This solution is asymptotic to the inviscid solution for | x | < √(2 Q 0 t ), where Q 0 is a function of the initial lobe area, as lobe Reynolds number tends to infinity, and is also asymptotic to the old age linear solution, as t tends to infinity; the formulae for the lobe Reynolds numbers are shown to have the correct behaviour in these limits. The general results apply to all j = m/n, m < 2n , and are rather involved; explicit results are written out for j = 0, 1, ½, 1/3 and 1/4. The case of spherical symmetry j = 2 is found to be ‘singular’ and the general approach set forth here does not work; an alternative approach for this case gives the large time behaviour in two different time regimes. The results of this study are com pared with those of Crighton & Scott (1979).