Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line

We consider the accelerated propagation of solutions to equations with a nonlocal linear dispersion on the real line and monostable nonlinearities (both local or nonlocal, however, not degenerated at $0$), in the case when either of the dispersion kernel or the initial condition has regularly heavy tails at both $\pm\infty$, perhaps different. We show that, in such case, the propagation in the right direction is fully determined by the right tails of either the kernel or the initial condition. We describe both cases of integrable and monotone initial conditions which may give different orders of the acceleration. Our approach is based, in particular, on the extension of the theory of sub-exponential distributions, which we introduced early in [D. Finkelshtein, P. Tkachov. 'Kesten's bound for sub-exponential densities on the real line and its multi-dimensional analogues', Advances in Applied Probability, 2018, 50(2), 373-395]

Solving random boundary heat model using the finite difference method under mean square convergence

"This is the peer reviewed version of the following article: Cortés, J. C., Romero, J. V., Roselló, M. D., Sohaly, MA. Solving random boundary heat model using the finite difference method under mean square convergence. Comp and Math Methods. 2019; 1:e1026. https://doi.org/10.1002/cmm4.1026 , which has been published in final form at https://doi.org/10.1002/cmm4.1026. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving."[EN] This contribution is devoted to construct numerical approximations to the solution of the one-dimensional boundary value problem for the heat model with uncertainty in the diffusion coefficient. Approximations are constructed via random numerical schemes. This approach permits discussing the effect of the random diffusion coefficient, which is assumed a random variable. We establish results about the consistency and stability of the random difference scheme using mean square convergence. Finally, an illustrative example is presented.Spanish Ministerio de Economía y Competitividad. Grant Number: MTM2017-89664-PCortés, J.; Romero, J.; Roselló, M.; Sohaly, M. (2019). Solving random boundary heat model using the finite difference method under mean square convergence. Computational and Mathematical Methods. 1(3):1-15. https://doi.org/10.1002/cmm4.1026S11513Han, X., & Kloeden, P. E. (2017). Random Ordinary Differential Equations and Their Numerical Solution. Probability Theory and Stochastic Modelling. doi:10.1007/978-981-10-6265-0Villafuerte, L., Braumann, C. A., Cortés, J.-C., & Jódar, L. (2010). Random differential operational calculus: Theory and applications. Computers & Mathematics with Applications, 59(1), 115-125. doi:10.1016/j.camwa.2009.08.061Logan, J. D. (2004). Partial Differential Equations on Bounded Domains. Undergraduate Texts in Mathematics, 121-171. doi:10.1007/978-1-4419-8879-9_4Cannon, J. R. (1964). A Cauchy problem for the heat equation. Annali di Matematica Pura ed Applicata, 66(1), 155-165. doi:10.1007/bf02412441LinPPY.On The Numerical Solution of The Heat Equation in Unbounded Domains[PhD thesis].New York NY:New York University;1993.Li, J.-R., & Greengard, L. (2007). On the numerical solution of the heat equation I: Fast solvers in free space. Journal of Computational Physics, 226(2), 1891-1901. doi:10.1016/j.jcp.2007.06.021Han, H., & Huang, Z. (2002). Exact and approximating boundary conditions for the parabolic problems on unbounded domains. Computers & Mathematics with Applications, 44(5-6), 655-666. doi:10.1016/s0898-1221(02)00180-3Han, H., & Huang, Z. (2002). A class of artificial boundary conditions for heat equation in unbounded domains. Computers & Mathematics with Applications, 43(6-7), 889-900. doi:10.1016/s0898-1221(01)00329-7Strikwerda, J. C. (2004). Finite Difference Schemes and Partial Differential Equations, Second Edition. doi:10.1137/1.9780898717938Kloeden, P. E., & Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. doi:10.1007/978-3-662-12616-5Øksendal, B. (2003). Stochastic Differential Equations. Universitext. doi:10.1007/978-3-642-14394-6Holden, H., Øksendal, B., Ubøe, J., & Zhang, T. (2010). Stochastic Partial Differential Equations. doi:10.1007/978-0-387-89488-1El-Tawil, M. A., & Sohaly, M. A. (2012). Mean square convergent three points finite difference scheme for random partial differential equations. Journal of the Egyptian Mathematical Society, 20(3), 188-204. doi:10.1016/j.joems.2012.08.017Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Sohaly, M. A. (2018). Solving the random Cauchy one-dimensional advection–diffusion equation: Numerical analysis and computing. Journal of Computational and Applied Mathematics, 330, 920-936. doi:10.1016/j.cam.2017.02.001Cortés, J. C., Jódar, L., Villafuerte, L., & Villanueva, R. J. (2007). Computing mean square approximations of random diffusion models with source term. Mathematics and Computers in Simulation, 76(1-3), 44-48. doi:10.1016/j.matcom.2007.01.020Cortés, J. C., Jódar, L., & Villafuerte, L. (2009). Random linear-quadratic mathematical models: Computing explicit solutions and applications. Mathematics and Computers in Simulation, 79(7), 2076-2090. doi:10.1016/j.matcom.2008.11.008Henderson, D., & Plaschko, P. (2006). Stochastic Differential Equations in Science and Engineering. doi:10.1142/580

Some results on blow up for semilinear parabolic problems

The authors describe the asymptotic behavior of blow-up for the semilinear heat equation ut=uxx+f(u) in R×(0,T), with initial data u0(x)>0 in R, where f(u)=up, p>1, or f(u)=eu. A complete description of the types of blow-up patterns and of the corresponding blow-up final-time profiles is given. In the rescaled variables, both are governed by the structure of the Hermite polynomials H2m(y). The H2-behavior is shown to be stable and generic. The existence of H4-behavior is proved. A nontrivial blow-up pattern with a blow-up set of nonzero measure is constructed. Similar results for the absorption equation ut=uxx−up, 0<p<1, are discussed