Approximating the solution stochastic process of the random Cauchy one-dimensional heat model

[EN] This paper deals with the numerical solution of the random Cauchy one-dimensional heat model. We propose a random finite difference numerical scheme to construct numerical approximations to the solution stochastic process. We establish sufficient conditions in order to guarantee the consistency and stability of the proposed random numerical scheme.The theoretical results are illustrated by means of an example where reliable approximations of the mean and standard deviation to the solution stochastic process are given.This work has been partially supported by the Ministerio de Economía y Competitividad Grant MTM2013-41765-P. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València. M. A. Sohaly is also indebted to Egypt Ministry of Higher Education, Cultural Affairs, for its financial support [mohe-casem (2016)].Navarro Quiles, A.; Romero, J.; Roselló, M.; Sohaly, M. (2016). Approximating the solution stochastic process of the random Cauchy one-dimensional heat model. Abstract and Applied Analysis. 2016:1-7. https://doi.org/10.1155/2016/5391368S172016Logan, J. D. (2004). Partial Differential Equations on Bounded Domains. Undergraduate Texts in Mathematics, 121-171. doi:10.1007/978-1-4419-8879-9_4Wang, J. (1994). A Model of Competitive Stock Trading Volume. Journal of Political Economy, 102(1), 127-168. doi:10.1086/261924Tsynkov, S. V. (1998). Numerical solution of problems on unbounded domains. A review. Applied Numerical Mathematics, 27(4), 465-532. doi:10.1016/s0168-9274(98)00025-7Koleva, M. N. (2006). Numerical Solution of the Heat Equation in Unbounded Domains Using Quasi-uniform Grids. Lecture Notes in Computer Science, 509-517. doi:10.1007/11666806_58Han, 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)00329-7Wu, X., & Sun, Z.-Z. (2004). Convergence of difference scheme for heat equation in unbounded domains using artificial boundary conditions. Applied Numerical Mathematics, 50(2), 261-277. doi:10.1016/j.apnum.2004.01.001Cortés, J. C., Sevilla-Peris, P., & Jódar, L. (2005). Analytic-numerical approximating processes of diffusion equation with data uncertainty. Computers & Mathematics with Applications, 49(7-8), 1255-1266. doi:10.1016/j.camwa.2004.05.015Casabán, M.-C., Cortés, J.-C., García-Mora, B., & Jódar, L. (2013). Analytic-Numerical Solution of Random Boundary Value Heat Problems in a Semi-Infinite Bar. Abstract and Applied Analysis, 2013, 1-9. doi:10.1155/2013/676372Casabán, M.-C., Company, R., Cortés, J.-C., & Jódar, L. (2014). Solving the random diffusion model in an infinite medium: A mean square approach. Applied Mathematical Modelling, 38(24), 5922-5933. doi:10.1016/j.apm.2014.04.063Villafuerte, 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.2009.08.061Øksendal, B. (2003). Stochastic Differential Equations. Universitext. doi:10.1007/978-3-642-14394-6Kloeden, P. E., & Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. doi:10.1007/978-3-662-12616-5Holden, H., Øksendal, B., Ubøe, J., & Zhang, T. (2010). Stochastic Partial Differential Equations. doi:10.1007/978-0-387-89488-

On The Foundations of Digital Games

Computers have lead to a revolution in the games we play, and, following this, an interest for computer-based games has been sparked in research communities. However, this easily leads to the perception of a one-way direction of influence between that the field of game research and computer science. This historical investigation points towards a deep and intertwined relationship between research on games and the development of computers, giving a richer picture of both fields. While doing so, an overview of early game research is presented and an argument made that the distinction between digital games and non-digital games may be counter-productive to game research as a whole

How Ordinary Elimination Became Gaussian Elimination

Newton, in notes that he would rather not have seen published, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method that Euler did not recommend, that Legendre called "ordinary," and that Gauss called "common" - is now named after Gauss: "Gaussian" elimination. Gauss's name became associated with elimination through the adoption, by professional computers, of a specialized notation that Gauss devised for his own least squares calculations. The notation allowed elimination to be viewed as a sequence of arithmetic operations that were repeatedly optimized for hand computing and eventually were described by matrices.Comment: 56 pages, 21 figures, 1 tabl