The goal of computational science is to develop models that predict phenomena observed in nature. However, these models are often based on parameters that are uncertain. In recent decades, main numerical methods for s...The goal of computational science is to develop models that predict phenomena observed in nature. However, these models are often based on parameters that are uncertain. In recent decades, main numerical methods for solving SPDEs have been used such as, finite difference and finite element schemes [1]-[5]. Also, some practical techniques like the method of lines for boundary value problems have been applied to the linear stochastic partial differential equations, and the outcomes of these approaches have been experimented numerically [7]. In [8]-[10], the author discussed mean square convergent finite difference method for solving some random partial differential equations. Random numerical techniques for both ordinary and partial random differential equations are treated in [4] [10]. As regards applications using explicit analytic solutions or numerical methods, a few results may be found in [5] [6] [11]. This article focuses on solving random heat equation by using Crank-Nicol- son technique under mean square sense and it is organized as follows. In Section 2, the mean square calculus preliminaries that will be required throughout the paper are presented. In Section 3, the Crank-Nicolson scheme for solving the random heat equation is presented. In Section 4, some case studies are showed. Short conclusions are cleared in the end section.展开更多
Pedestrian navigation has become an important theoretical and practical research topic in many disciplines such as cartography,geographical information science,global and indoor positioning,spatial behavior,psychology...Pedestrian navigation has become an important theoretical and practical research topic in many disciplines such as cartography,geographical information science,global and indoor positioning,spatial behavior,psychology,sociology,and neuroscience.Many research studies view pedestrian navigation using process-oriented and goal-directed approaches.However,this paper revisits people’s needs in pedestrian navigation and classifies their needs as three layers:physical sense layer,physiological safety layer,and mental satisfaction layer according to Maslow’s theory.This paper introduces a people-centric framework for pedestrian navigation theory based on these three layers and discusses theoretical challenges for meeting each layer of people’s needs.These challenging theories may represent promising and valuable research and promote usage of pedestrian navigation systems or devices in the future.展开更多
文摘The goal of computational science is to develop models that predict phenomena observed in nature. However, these models are often based on parameters that are uncertain. In recent decades, main numerical methods for solving SPDEs have been used such as, finite difference and finite element schemes [1]-[5]. Also, some practical techniques like the method of lines for boundary value problems have been applied to the linear stochastic partial differential equations, and the outcomes of these approaches have been experimented numerically [7]. In [8]-[10], the author discussed mean square convergent finite difference method for solving some random partial differential equations. Random numerical techniques for both ordinary and partial random differential equations are treated in [4] [10]. As regards applications using explicit analytic solutions or numerical methods, a few results may be found in [5] [6] [11]. This article focuses on solving random heat equation by using Crank-Nicol- son technique under mean square sense and it is organized as follows. In Section 2, the mean square calculus preliminaries that will be required throughout the paper are presented. In Section 3, the Crank-Nicolson scheme for solving the random heat equation is presented. In Section 4, some case studies are showed. Short conclusions are cleared in the end section.
基金This work was supported in part by the National Science Foundation of China[grant number 41371420],[grant number 41231171]the Shenzhen Dedicated Funding of Strategic Emerging Industry Development Program[grant number JCYJ20121019111128765]the Funding for Excellent Young Scholars in Wuhan University[grant number 2042015KF0167].
文摘Pedestrian navigation has become an important theoretical and practical research topic in many disciplines such as cartography,geographical information science,global and indoor positioning,spatial behavior,psychology,sociology,and neuroscience.Many research studies view pedestrian navigation using process-oriented and goal-directed approaches.However,this paper revisits people’s needs in pedestrian navigation and classifies their needs as three layers:physical sense layer,physiological safety layer,and mental satisfaction layer according to Maslow’s theory.This paper introduces a people-centric framework for pedestrian navigation theory based on these three layers and discusses theoretical challenges for meeting each layer of people’s needs.These challenging theories may represent promising and valuable research and promote usage of pedestrian navigation systems or devices in the future.
