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On the presumed superiority of analytical solutions over numerical methods

Abstract

An important task in mathematical sciences is to make quantitative predictions, which is often done via the solution of differential equations. In this paper, we investigate why, to perform this task, scientists sometimes choose to use numerical methods instead of analytical solutions. Via several examples, we argue that the choice for numerical methods can be explained by the fact that, while making quantitative predictions seems at first glance to be facilitated by analytical solutions, this is actually often much easier with numerical methods. Thus we challenge the widely presumed superiority of analytical solutions over numerical methods.

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Notes

  1. Quantitative predictions should be understood in a broad sense. They are not limited to predictions about the future states of the target systems. They also include numerical results of the variables that describe a behavior or a property of the target systems.

  2. This is justified by the fact that y(t) is given by the first terms of Taylor series of the exponential function at t=0.

  3. See Humphreys (2004, p. 64).

  4. Algebraic functions are built from rational functions with the following operations: addition, subtraction, multiplication, division, and exponentiation with integral and fractional exponents. Elementary functions are built in admitting in addition the operations of exponentiation in general and derivation. Liouvillian functions admit in addition the operation of integration (Goriely 2001, pp. 38-40 and Zoladek 1998, pp 2-3). Closed-form functions admit in addition special functions like the non-liouvillian Airy and Bessel functions. Special functions are functions that are purposely defined as the solutions of some differential equations. For a detailed discussion on the role of special functions in physics, see (Batterman 2007). We borrow this list from Fillion and Bangu (2015, p. 4) to which we add liouvillian functions (Singer 1990, p. 66). Liouvillian functions play indeed an important role in classical mechanics: they are the solutions of integrable systems like the Kepler problem or the one-dimensional simple pendulum (Babelon et al. 2003, chap. 2).

  5. We emphasize that these algorithms are not doomed to be implemented on a computer since numerical methods (e.g Euler methods) have been used long before the development of computers. Numerical calculations were made by hand. However, in this paper, we are interested in the current use of numerical methods in mathematical sciences, which is based on computers.

  6. Additional errors are produced all along numerical computations though, as we will see in Section 3.

  7. This phenomenon is an example of a catastrophic cancellation. This series is indeed an alternating series in which differences between big numbers with finite significant figures are used to evaluate small numbers. For details, see Corless and Fillion (2014, p. 15).

  8. We are in debt to one of our anonymous reviewers for making this clear.

  9. q, p are coordinates of phase space and t is time.

  10. Action-angles variables are conjugate variables (I, 𝜃) – sometimes written (J, ω) – such as the action variables I (J) are constant with time (Babelon et al. 2003, p. 10; Goldstein et al. 2001, p. 452). We point out that the N-body problem with N=2 is an exception though, as, in this case, there are enough conserved quantities like energy, linear momentum and angular momentum to make this change of variables, and give a liouvillian solution. In this case, the system is said to be “integrable”.

  11. First, in the absence of any predation, the prey grows unboundedly in a Malthusian way; this is the a N term. Second, the effect of the predation is to reduce the prey growth rate by the term b and by the number of predators; this is the −b N P term. Third, the prey contribute to the predators’ growth; this is the d N P term. Fourth, in the absence of any prey to eat, the predators die following an exponential decay, that is, the −c P term.

  12. In Evans and Findley (1999, p. 181-182), the equations of the Lotka-Volterra model are slightly different and written as a function of the variables x 1(t) and x 2(t). We get the equations (7) with the mathematical transformations N(t)=(b/d)x 1(t) and P(t) = x 2(t).

  13. We emphasize how close are the conditions that guarantee that the numerical Euler method is convergent, and the conditions that guarantee that there exists a unique solution to a differential equation d x/d t = f(x,t) with an initial condition (Picard-Lindelöf theorem).

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Acknowledgments

We thank the anonymous reviewers for their helpful comments, and their contribution to enhancing the quality of this paper. This work was supported by French State funds managed by the National Research Agency on the behalf of Idex Sorbonne Universités within the Investissements d’Avenir Programme under reference ANR-11-IDEX-0004-02. One of the authors is currently a beneficiary of a “MOVE-IN Louvain” Incoming Post-doctoral Fellowship, co-funded by the Marie Curie Actions of the European Commission.

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Ardourel, V., Jebeile, J. On the presumed superiority of analytical solutions over numerical methods. Euro Jnl Phil Sci 7, 201–220 (2017). https://doi.org/10.1007/s13194-016-0152-2

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Keywords

  • Applied mathematics
  • Exactness
  • Analytical solutions
  • Numerical methods