J. D. Achenbach, Wave Propagation in Elastic Solids, Journal of Applied Mechanics, vol.41, issue.2, 1973.
DOI : 10.1115/1.3423344

G. E. Andrews, R. Askey, and R. Roy, Special functions, 1999.

R. Askey and G. Gasper, Certain Rational Functions Whose Power Series Have Positive Coefficients, The American Mathematical Monthly, vol.79, issue.4, pp.327-341, 1972.
DOI : 10.2307/2978081

J. Avigad and K. Donnelly, A Decision Procedure for Linear ???Big O??? Equations, Journal of Automated Reasoning, vol.5, issue.4:4, pp.353-373, 2007.
DOI : 10.1007/s10817-007-9066-1

P. Baudin, P. Cuoq, J. Filliâtre, C. Marché, B. Monate et al., ACSL: ANSI/ISO C Specification Language, version 1.5, 2009.

Y. Bertot and P. Castéran, Interactive Theorem Proving and Program Development. Coq'Art: The Calculus of Inductive Constructions. Texts in Theoretical Computer Science, 2004.
DOI : 10.1007/978-3-662-07964-5
URL : https://hal.archives-ouvertes.fr/hal-00344237

Y. Bertot, G. Gonthier, S. Ould-biha, and I. Pasca, Canonical Big Operators, 21st International Conference on Theorem Proving in Higher Order Logics, pp.86-101, 2008.
DOI : 10.1007/3-540-44659-1_29
URL : https://hal.archives-ouvertes.fr/inria-00331193

F. Bobot, S. Conchon, E. Contejean, and M. Iguernelala, Stéphane Lescuyer , and Alain Mebsout. The Alt-Ergo automated theorem prover, 2008.

S. Boldo, Preuves formelles en arithmétiquesarithmétiquesà virgule flottante, 2004.

S. Boldo, Floats and Ropes: A Case Study for Formal Numerical Program Verification, 36th International Colloquium on Automata, Languages and Programming, pp.91-102, 2009.
DOI : 10.1007/978-3-642-02930-1_8

S. Boldo, F. Clément, J. Filliâtre, M. Mayero, G. Melquiond et al., Formal Proof of a Wave Equation Resolution Scheme: The Method Error, 1st Interactive Theorem Proving Conference (ITP), pp.147-162, 2010.
DOI : 10.1007/978-3-642-14052-5_12
URL : https://hal.archives-ouvertes.fr/inria-00450789

S. Boldo and J. Filliâtre, Formal Verification of Floating-Point Programs, 18th IEEE Symposium on Computer Arithmetic (ARITH '07), pp.187-194, 2007.
DOI : 10.1109/ARITH.2007.20
URL : https://hal.archives-ouvertes.fr/hal-01174892

S. Boldo, J. Filliâtre, and G. Melquiond, Combining Coq and Gappa for Certifying Floating-Point Programs, 16th Calculemus Symposium, pp.59-74, 2009.
DOI : 10.1109/TC.2008.200
URL : https://hal.archives-ouvertes.fr/inria-00432726

S. Boldo and G. Melquiond, Flocq: A Unified Library for Proving Floating-Point Algorithms in Coq, 2011 IEEE 20th Symposium on Computer Arithmetic, pp.243-252, 2011.
DOI : 10.1109/ARITH.2011.40
URL : https://hal.archives-ouvertes.fr/inria-00534854

S. Boldo, T. M. , and T. Nguyen, Proofs of numerical programs when the compiler optimizes, Innovations in Systems and Software Engineering, pp.1-10, 2011.
DOI : 10.1007/s11334-011-0151-6
URL : https://hal.archives-ouvertes.fr/hal-00777639

L. M. Brekhovskikh and V. Goncharov, Mechanics of Continua and Wave Dynamics, 1994.

E. Sylvain-conchon, J. Contejean, S. Kanig, and . Lescuyer, CC(X): Semantic Combination of Congruence Closure with Solvable Theories, Post-proceedings of the 5th International Workshop on Satisfiability Modulo Theories, pp.51-69, 2007.
DOI : 10.1016/j.entcs.2008.04.080

T. Coquand and C. Paulin-mohring, Inductively defined types, Colog'88, 1990.
DOI : 10.1007/3-540-52335-9_47

S. Boldo, F. Clément, J. Filliâtre, M. Mayero, G. Melquiond et al., On the partial difference equations of mathematical physics, IBM Journal of Research and Development, vol.11, issue.2, pp.215-234, 1967.

P. Cousot, R. Cousot, J. Feret, and L. Mauborgne, Antoine Miné, David Monniaux, and Xavier Rival. The ASTRÉEASTR´ASTRÉE analyzer, ESOP, number 3444 in LNCS, pp.21-30, 2005.

L. Cruz-filipe, A Constructive Formalization of the Fundamental Theorem of Calculus, 2nd International Workshop on Types for Proofs and Programs, 2002.
DOI : 10.1007/3-540-39185-1_7

M. Daumas and G. Melquiond, Certification of bounds on expressions involving rounded operators, ACM Transactions on Mathematical Software, vol.37, issue.1, pp.1-20, 2010.
DOI : 10.1145/1644001.1644003
URL : https://hal.archives-ouvertes.fr/hal-00127769

M. Daumas, L. Rideau, and L. Théry, A Generic Library for Floating-Point Numbers and Its Application to Exact Computing, TPHOLs, pp.169-184, 2001.
DOI : 10.1007/3-540-44755-5_13
URL : https://hal.archives-ouvertes.fr/hal-00157285

C. Florent-de-dinechin, G. Lauter, and . Melquiond, Certifying the Floating-Point Implementation of an Elementary Function Using Gappa, IEEE Transactions on Computers, vol.60, issue.2, pp.242-253, 2011.
DOI : 10.1109/TC.2010.128

L. De, M. , and N. Bjørner, Z3, an efficient SMT solver, TACAS, pp.337-340, 2008.

D. Delmas, E. Goubault, S. Putot, J. Souyris, K. Tekkal et al., Towards an Industrial Use of FLUCTUAT on Safety-Critical Avionics Software, FMICS, pp.53-69, 2009.
DOI : 10.1007/978-3-642-04570-7_6

B. Dutertre, Elements of mathematical analysis in PVS, 9th International Conference on Theorem Proving in Higher Order Logics (TPHOLs'96), pp.141-156, 1996.
DOI : 10.1007/BFb0105402

J. Filliâtre and C. Marché, The Why/Krakatoa/Caduceus Platform for Deductive Program Verification, 19th International Conference on Computer Aided Verification, pp.173-177, 2007.
DOI : 10.1007/978-3-540-73368-3_21

J. D. Fleuriot, On the Mechanization of Real Analysis in Isabelle/HOL, 13th International Conference on Theorem Proving and Higher- Order Logic (TPHOLs'00), volume 1869 of LNCS, pp.145-161, 2000.
DOI : 10.1007/3-540-44659-1_10

R. Gamboa and M. Kaufmann, Nonstandard analysis in ACL2, Journal of Automated Reasoning, vol.27, issue.4, pp.323-351, 2001.
DOI : 10.1023/A:1011908113514

H. Geuvers and M. Niqui, Constructive Reals in Coq: Axioms and Categoricity, 1st International Workshop on Types for Proofs and Programs, pp.79-95, 2000.
DOI : 10.1007/3-540-45842-5_6

J. Harrison, Theorem Proving with the Real Numbers, 1998.
DOI : 10.1007/978-1-4471-1591-5

J. Harrison, A HOL Theory of Euclidean Space, 18th International Conference on Theorem Proving and Higher-Order Logic (TPHOLs'05), pp.114-129, 2005.
DOI : 10.1007/11541868_8

F. John, Partial Differential Equations, 1986.

R. Krebbers and B. Spitters, Type classes for efficient exact real arithmetic in Coq, Logical Methods in Computer Science, vol.9, issue.1, 2011.
DOI : 10.2168/LMCS-9(1:1)2013

J. Le-rond and D. , Recherches sur la courbe que forme une corde tendue mise en vibrations, Histoire de l'Académie Royale des Sciences et Belles Lettres, pp.214-249, 1747.

G. Lee and B. Werner, Proof-irrelevant model of CC with predicative induction and judgmental equality, Logical Methods in Computer Science, vol.7, issue.4, 2011.
DOI : 10.2168/LMCS-7(4:5)2011

C. Lelay and G. Melquiond, Différentiabilité et intégrabilité en Coq Applicationàtionà la formule de d'Alembert, 23èmes Journées Francophones des Langages Applicatifs, pp.119-133, 2012.

P. Letouzey, A New Extraction for Coq, 2nd International Workshop on Types for Proofs and Programs, 2002.
DOI : 10.1007/3-540-39185-1_12
URL : https://hal.archives-ouvertes.fr/hal-00150914

C. Marché, Jessie, Proceedings of the 2007 workshop on Programming languages meets program verification , PLPV '07, pp.1-2, 2007.
DOI : 10.1145/1292597.1292598

M. Mayero, Formalisation et automatisation de preuves en analyses réelle et numérique, 2001.

M. Mayero, Using theorem proving for numerical analysis (correctness proof of an automatic differentiation algorithm) In Victor Carreño, César Muñoz, andSofì ene Tahar, 15th International Conference on Theorem Proving and Higher-Order Logic, pp.246-262, 2002.

I. Newton, Axiomata, sive Leges Motus, Philosophiae Naturalis Principia Mathematica, p.1687

O. Russell and . Connor, Certified exact transcendental real number computation in Coq, 21st International Conference on Theorem Proving in Higher Order Logics, pp.246-261, 2008.

O. Russell, B. Connor, and . Spitters, A computer-verified monadic functional implementation of the integral, Theoretical Computer Science, vol.411, issue.37, pp.3386-3402, 2010.

E. Elemer and . Rosinger, Propagation of round-off errors and the role of stability in numerical methods for linear and nonlinear PDEs, Applied Mathematical Modelling, vol.9, issue.5, pp.331-336, 1985.

E. Elemer and . Rosinger, L-convergence paradox in numerical methods for PDEs, Applied Mathematical Modelling, vol.15, issue.3, pp.158-163, 1991.

J. Christopher, W. L. Roy, and . Oberkampf, A comprehensive framework for verification, validation, and uncertainty quantification in scientific computing, Computer Methods in Applied Mechanics and Engineering, vol.200, pp.25-282131, 2011.

P. Rudnicki, An overview of the MIZAR project, Types for Proofs and Programs, pp.311-332, 1992.

B. Szyszka, An interval method for solving the one-dimensional wave equation, 7th EUROMECH Solid Mechanics Conference (ESMC2009), 2009.

R. Zach, Hilbert's " Verunglueckter Beweis, " the first epsilon theorem, and consistency proofs
DOI : 10.1080/01445340310001606930
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.9.4767

D. Zwillinger, Handbook of Differential Equations INRIA Centre de recherche INRIA Saclay ? Île-de, des Vignes 4, rue Jacques Monod -91893 Orsay Cedex, 1998.