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Mathematical Modelling

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Mathematical Geoscience

Part of the book series: Interdisciplinary Applied Mathematics ((IAM,volume 36))

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Abstract

Chapter 1 is a condensed overview of a raft of techniques which are used in applied mathematics in studying natural phenomena. The chapter focusses on phenomena which occur in continuous systems, modelled by differential equations. Non-dimensionalisation and scaling are treated. For ordinary differential equations, oscillations, hysteresis and resonance are discussed; for partial differential equations, topics covered are waves, nonlinear diffusion, blow-up and reaction–diffusion equations.

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Notes

  1. 1.

    We can understand why T follows the equilibrium curve as follows. We can write (1.26) in terms of suitable dimensionless variables as \(\dot{\Delta}=T_{0}-g(\Delta)\), where g(Δ) is a cubic-like curve similar to the function T 0(Δ) depicted in Fig. 1.9. if T 0 is slowly varying, then T 0=T 0(δt) where δ≪1, and putting τ=δt, we have δdΔ/dτ=T 0(τ)−g(Δ); thus on the slow time scale τ, Δ will tend rapidly to a (quasi-equilibrium) zero of the right hand side.

  2. 2.

    Note that as T 0→T c , (1.43) matches with (1.40).

  3. 3.

    We need \(u \mbox{ \raisebox{-.9ex}{$\stackrel{\textstyle<}{\sim}$} }O (\frac{1}{x^{2}} )\) in order that the second term in (1.81) be significant (otherwise we regain (1.82)), and in fact we need the two terms to be approximately equal, so that 0<u<1: hence (1.83).

  4. 4.

    Geomorphologists would call this surface convex; see Chap. 6.

  5. 5.

    Ice sheets and their marginal movement are discussed further in Chap. 10.

  6. 6.

    We use the summation convention, which implies summation over repeated suffixes.

  7. 7.

    Physicists call (1.212) the KPZ equation (after Kardar et al. 1986). The substitution \(u=\exp(\bar{\alpha}\psi)\) reduces it to the diffusion equation for u; this is the Hopf–Cole transformation (see Whitham 1974).

  8. 8.

    See Watson (1944, pp. 199 f.) for these results.

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Authors and Affiliations

  1. MACSI, Department of Mathematics & Statistics, University of Limerick, Limerick, Ireland

    Andrew Fowler

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  1. Andrew Fowler
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Correspondence to Andrew Fowler .

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Fowler, A. (2011). Mathematical Modelling. In: Mathematical Geoscience. Interdisciplinary Applied Mathematics, vol 36. Springer, London. https://doi.org/10.1007/978-0-85729-721-1_1

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