85
Views
3
CrossRef citations to date
0
Altmetric
Articles

Finite-difference modeling à la Mickens of the distribution of the stopping time in a stochastic differential equation

&
Pages 799-820 | Received 15 Jul 2016, Accepted 16 Jan 2017, Published online: 27 Jan 2017

References

  • R.J. Allen, G.S. Booth, and T. Jutla, A review of fatigue crack growth characterisation by linear elastic fracture mechanics (LEFM). Part I-principles and methods of data generation, Fatigue Fracture Eng. Mater. Struct. 11 (1988), pp. 45–69.
  • R.J. Allen, G.S. Booth, and T. Jutla, A review of fatigue crack growth characterisation by linear elastic fracture mechanics (LEFM). Part II-Advisory documents and applications within national standards, Fatigue Fracture Eng. Mater. Struct. 11 (1988), pp. 71–108.
  • R. Anguelov, Y. Dumont, J.S. Lubuma, and M. Shillor, Dynamically consistent nonstandard finite difference schemes for epidemiological models, J. Comput. Appl. Math. 255 (2014), pp. 161–182.
  • S. Asmussen and H. Albrecher, Ruin Probabilities 14, World Scientific, New Jersey, NJ, 2010.
  • A.N. Borodin and P. Salminen, Handbook of Brownian Motion-facts and Formulae, 2nd ed., Birkhäuser, New York, 2012.
  • J. Dominguez, J. Zapatero, and B. Moreno, A statistical model for fatigue crack growth under random loads including retardation effects, Eng. Fracture Mech. 62 (1999), pp. 351–369.
  • R. Durrett, Stochastic Calculus: A Practical Introduction, 1st ed., Vol. 6, CRC Press, New York, 1996.
  • V.J. Ervin, J.E. Macías-Díaz, and J. Ruiz-Ramírez, A positive and bounded finite element approximation of the generalized Burgers--Huxley equation, J. Math. Anal. Appl. 424 (2015), pp. 1143–1160.
  • T. Fujimoto and R.R. Ranade, Two characterizations of inverse-positive matrices: The Hawkins--Simon condition and the Le Chatelier--Braun principle, Electron. J. Linear Algebra 11 (2004), pp. 59–65.
  • K. Itô and H.P. McKean, Diffusion Processes and their Sample Paths, 2nd ed., Springer-Verlag, New York, 1974.
  • M.F. Kaplan, Crack propagation and the fracture of concrete, in ACI Journal Proceedings, M.F. Kaplan, ed., Vol. 58, Springer, 1961, pp. 591–610.
  • I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, 1st ed., Vol. 113, Springer Science & Business Media, Berlin, 1991.
  • A. Kroshko, O. Sharomi, A.B. Gumel, and R.J. Spiteri, Design and analysis of NSFD methods for the diffusion-free Brusselator, in Second Order Elliptic Equations and Elliptic Systems: Contemporary Mathematics, A.B. Gumel, ed., Vol. 618, American Mathematical Society, Providence, RI, 2014, p. 163.
  • J.A. León, L.P. Hernández, and J. Villa-Morales, On the distribution of explosion time of stochastic differential equations, Bol. Soc. Mat. Mex. 19 (2013), pp. 125–138.
  • J.A. León and J. Villa, On the distributions of the sup and inf of the classical risk process with exponential claim, Commun. Stoch. Anal. 13 (2009), pp. 69–84.
  • J.A. León and J. Villa, An Osgood criterion for integral equations with applications to stochastic differential equations with an additive noise, Stat. Probab. Lett. 81 (2011), pp. 470–477.
  • K.J. Mach, B.B. Hale, M.W. Denny, and D.V. Nelson, Death by small forces: A fracture and fatigue analysis of wave-swept macroalgae, J. Exp. Biol. 210 (2007), pp. 2231–2243.
  • J. Macías-Díaz and J. Ruiz-Ramírez, A non-standard symmetry-preserving method to compute bounded solutions of a generalized Newell--Whitehead--Segel equation, Appl. Numer. Math. 61 (2011), pp. 630–640.
  • J.E. Macías-Díaz, On a boundedness-preserving semi-linear discretization of a two-dimensional nonlinear diffusion-reaction model, Int. J. Comput. Math. 89 (2012), pp. 1678–1688.
  • J.E. Macías-Díaz and A. Puri, An explicit positivity-preserving finite-difference scheme for the classical Fisher--Kolmogorov--Petrovsky--Piscounov equation, Appl. Math. Comput. 218 (2012), pp. 5829–5837.
  • J.E. Macías-Díaz and K.A. Rejniak, On a conditionally stable nonlinear method to approximate some monotone and bounded solutions of a generalized population model, Appl. Math. Comput. 229 (2014), pp. 273–282.
  • J.E. Macías-Díaz and A. Szafrańska, Existence and uniqueness of monotone and bounded solutions for a finite-difference discretization à la Mickens of the generalized Burgers--Huxley equation, J. Diff. Equ. Appl. 20 (2014), pp. 989–1004.
  • J. Maljaars, H.M.G.M. Steenbergen, and A.C.W.M. Vrouwenvelder, Probabilistic model for fatigue crack growth and fracture of welded joints in civil engineering structures, Int. J. Fatigue 38 (2012), pp. 108–117.
  • H.P. McKean, Stochastic Integrals, 1st ed., American Mathematical Society, New York, 1969.
  • G. Meral and M. Tezer-Sezgin, The comparison between the DRBEM and DQM solution of nonlinear reaction--diffusion equation, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), pp. 3990–4005.
  • R.E. Mickens, Nonstandard finite difference schemes for differential equations, J. Diff. Equ. Appl. 8 (2002), pp. 823–847.
  • R.E. Mickens, Dynamic consistency: A fundamental principle for constructing nonstandard finite difference schemes for differential equations, J. Differ. Equ. Appl. 11 (2005), pp. 645–653.
  • R.E. Mickens, A note on a non-standard finite difference scheme for the Reluga x – y – z model, J. Diff. Equ. Appl. 16 (2010), pp. 1501–1504.
  • R.E. Mickens, A SIR-model with square-root dynamics: An NSFD scheme, J. Diff. Equ. Appl. 16 (2010), pp. 209–216.
  • R.E. Mickens and T.M. Washington, A note on exact finite difference schemes for the differential equations satisfied by the Jacobi cosine and sine functions, J. Diff. Equ. Appl. 30 (2012), pp. 1–6.
  • P.C. Paris, M.P. Gomez, and W.E. Anderson, A rational analytic theory of fatigue, Trend Eng. 13 (1961), pp. 9–14.
  • N. Pugno, M. Ciavarella, P. Cornetti, and A. Carpinteri, A generalized Paris law for fatigue crack growth, J. Mech. Phys. Solids 54 (2006), pp. 1333–1349.
  • L.I.W. Roeger and R.E. Mickens, Exact finite difference and non-standard finite difference schemes for dx/dt = –λxα, J. Diff. Equ. Appl. 18 (2012), pp. 1511–1517.
  • J. Ruiz-Ramírez and J.E. Macías-Díaz, A finite-difference scheme to approximate non-negative and bounded solutions of a FitzHugh-Nagumo equation, Int. J. Comput. Math. 88 (2011), pp. 3186–3201.
  • M.A. Safi and A.B. Gumel, Qualitative study of a quarantine/isolation model with multiple disease stages, Appl. Math. Comput. 218 (2011), pp. 1941–1961.
  • P. Salminen and O. Wallin, Perpetual integral functionals of diffusions and their numerical computations, in Stochastic Analysis and Applications, F.E. Benth, G.D. Nunno, T. Lindstrøm, B. Øksendal, and T. Zhang, eds., Springer, 2007, pp. 569–594.
  • K. Sobczyk, Random Fatigue: From Data to Theory, Kluwer Academic Publishers, Norwell, MA, 1992.
  • A. Szafrańska and J.E. Macías-Díaz, On the convergence of a finite-difference discretization à la Mickens of the generalized Burgers--Huxley equation, J. Diff. Equ. Appl. 20 (2014), pp. 1444–1451.
  • S. Tomasiello, Stability and accuracy of the iterative differential quadrature method, Int. J. Numer. Methods Eng. 58 (2003), pp. 1277–1296.
  • S. Tomasiello, Numerical solutions of the Burgers-Huxley equation by the IDQ method, Int. J. Comput. Math. 87 (2010), pp. 129–140.
  • S. Tomasiello, Numerical stability of DQ solutions of wave problems, Numer. Algorithms 57 (2011), pp. 289–312.
  • S. Tomasiello, A new DQ-based method and its application to the stability of columns, Appl. Math. Comput. 226 (2014), pp. 145–156.
  • S. Tomasiello, A new least-squares DQ-based method for the buckling of structures with elastic supports, Appl. Math. Model. 39 (2015), pp. 2809–2814.
  • T. Wang and B. Guo, Analysis of some finite difference schemes for two-dimensional Ginzburg--Landau equation, Numer. Methods Partial Differ. Equ. 27 (2011), pp. 1340–1363.
  • R.P. Wei, Fracture mechanics approach to fatigue analysis in design, J. Eng. Mater. Technol. 100 (1978), pp. 113–120.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.