## 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.

This is a preview of subscription content, access via your institution.

## Notes

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.

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.See Humphreys (2004, p. 64).

*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).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.

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

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

*q*,*p*are coordinates of phase space and*t*is time.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”.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.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).

## References

Babelon, O., Bernard, D., & Talon, M. (2003).

*Introduction to classical integrable systems*. Cambridge: Cambridge University Press.Barberousse, A., & Imbert, C. (2014). Recurring models and sensitivity to computational constraints.

*The Monist*,*97*(3), 259–279.Barberousse, A., Franceschelli, S., & Imbert, C. (2009). Computer simulations as experiments.

*Synthese*,*169*(3), 557–574.Batterman, R.W. (2007). On the specialness of special functions (The nonrandom effusions of the divine mathematician).

*The British Journal for the Philosophy of Science*,*58*(2), 263–286.Belendez, A., Pascual, C., Mendez, D.I., Belendez, T., & Neipp, C. (2007). Exact solution for the nonlinear pendulum.

*Revista Brasileira de Ensino de Fisica*,*29*(4), 645–648.Borwein, J., & Crandall, R. (2013). Closed forms: what they are and why we care.

*Notices of the American Mathematical Society*,*60*(1), 50–65.Corless, R.M., & Fillion, N. (2014).

*A graduate introduction to numerical methods*. Springer.Diacu, F. (1996). The solution of the N-boby problem.

*The Mathematical Intelligencer*,*18*(3), 66–70.Dutt, R. (1976). Application of Hamilton-Jacobi theory to the Lotka-Volterra oscillator.

*Bulletin of Mathematical Biology*,*38*, 459–465.Einstein, T.L. (2003). Applications of ideas from random matrix theory to step distributions on “misoriented” surfaces.

*Annales Henri Poincaré*,*4*(Suppl. 2), 811–824.Evans, C.M., & Findley, G.L. (1999). Analytic solutions to a family of Lotka–Volterra related differential equations.

*Journal of Mathematical Chemistry*,*25*, 181–189.Fillion, N., & Corless, R.M. (2014). On the epistemological analysis of modeling and computational error in the mathematical sciences.

*Synthese*,*191*, 1451–1467.Fillion, N., & Bangu, S. (2015). Numerical methods, complexity and epistemic hierarchies.

*Philosophy of Science*. forthcoming.Forsythe, G.E. (1970). Pitfalls in computation, or why a math book isn’t enough.

*The American Mathematical Monthly*,*77*(9), 931–956.French, A. P., & Taylor, E.F. (1998).

*An introduction to quantum physics*. Cheltenham: Stanley Thomas.Gallant, J. (2012).

*Doing physics with scientific notebook*. Wiley.Goldstein, H., Poole, C., & Safko, J. (2001). Classical mechanics.

Goriely, A. (2001).

*Integrability and nonintegrability of dynamical systems. Advanced series in nonlinear dynamics*Vol. 19. World Scientific.Hartmann, S. (1996). The world as a process - Simulations in the natural and social sciences. In R.U.M. Hegselmann, & K. Troitzsch (Eds.),

*Modelling and simulation in the social sciences from the philosophy of science point of view*(pp. 77–100). Dordrecht: Kluwer.Hairer, E., Nørsett, S.P., & Wanner, G. (1992).

*Solving ordinary differential equations I: Nonstiff problems*. Springer.Hiestand, J.W. (2009).

*Numerical methods with VBA programming*. John & Bartlett Publishers.Henkel, M. (2001). Sur la solution de Sundman du problme des trois corps.

*Philosophia Scientiae*,*5*(2), 161–184.Humphreys, P. (2004).

*Extending ourselves: Computational science, empiricism, and scientific method*. New-York: Oxford University Press.Humphreys, P. (2009). The philosophical novelty of computer simulation methods.

*Synthese*,*169*(3), 615–626.Masoliver, J., & Ros, A. (2011). Integrability and chaos: The classical uncertainty.

*European Journal of Physics*,*32*, 431–458.Murray, J. (2002).

*Mathematical biology. An Introduction*Vol. 1. Springer.Morrison, M. (2009). Models, measurement and computer simulation: The changing face of experimentation.

*Philosophical Studies*,*143*, 33–57.Ortega, J.M. (1992).

*Numerical analysis*. Ed. SIAM.Singer, M.F. (1990). Formal solutions of differential equations.

*Journal of Symbolic Computation*,*10*, 59–94.Stern, A., & Desbrun, M. (2008).

*Discrete geometric mechanics for variational time integrators*. Siggraph 2006 Course Notes, chap. 15.Stoer, J., & Bulisch, R. (2002).

*Introduction to numerical analysis, Texts in Applied Mathematics*Vol. 12. Springer.Süli, E., & Mayers, D.F. (2003).

*An introduction to numerical analysis*. Cambridge: Cambridge University Press.Sundman, K. (1907). Recherches sur le problème des trios corps.

*Acta Societatis Scientiarum Fennicae*,*34*(6).Sundman, K. (1909). Nouvelles recherches sur le problème des trois corps.

*Acta Societatis Scientiarum Fennicae*,*35*(9).Wang, Q.-D. (1991). The global solution of the n-body problem.

*Celestial Mechanics and Dynamics Astronomy*,*50*, 73–88.Zoladek, H. (1998). The extented monodromy group and liouvillian first intergrals.

*Journal of Dynamical and Control Systems*,*4*(1), 1–28.

## 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.

## Author information

### Authors and Affiliations

### Corresponding author

## Rights and permissions

## About this article

### Cite this article

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

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s13194-016-0152-2

### Keywords

- Applied mathematics
- Exactness
- Analytical solutions
- Numerical methods