Mean Values Connected with the Sawtooth Curve of Figure 5

J. M. Burgers
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J. M. Burgers: University of Maryland, Institute for Fluid Dynamics and Applied Mathematics

Chapter Chapter VII in The Nonlinear Diffusion Equation, 1974, pp 124-131 from Springer

Abstract: Abstract For large values of t the solution of the nonlinear Equation (1.1), with v→0, asymptotically approaches to a ‘sawtooth’ curve, consisting of upward sloping segments, having a slope equal to 1/t, separated by steep vertical descents or ‘shocks’, as indicated in Figure 5 and in more detail in the lower part of Figure 13 . The location of a ‘shock’ is determined by the position of the axis of the doubly contacting parabola at the instant t under consideration and is denoted by x k . The endpoints of the parabolic arc are at x k + x ′ k $${x_k} + {x'_k}$$ and x k + x ′ ″ k $${x_k} + {x'''_k}$$ so that x k ′ and x k ″ are relative coordinates with reference to the axis of the parabola They correspond to x 1, x 2, respectively, occurring in the integrals of the preceding chapter.

Date: 1974
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DOI: 10.1007/978-94-010-1745-9_8

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