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Numerical Solution of the First Passage Problem
Abstract Consider the general form of the three-dimensional convection-diffusion equation given by $${\partial \over {\partial x}}{k_x}{{\partial \psi } \over {\partial x}} + {\partial \over {\partial y}}{k_y}{{\partial \psi } \over {\partial y}} + {\partial \over {\partial z}} - {u_x}{{\partial \psi } \over {\partial z}} - {u_y}{{\partial \psi } \over {\partial y}} - {u_z}{{\partial \psi } \over {\partial z}} + c{{\partial \psi } \over {\partial t}} + Q = 0$$ (3.1) where kx/C, ky/C and kz/C are the diffusion coefficients; ux/C, uy/C and uz/C are the velocities; Q/C is the source term; and the boundary conditions are appropriately defined. The nonlinear partial differential equations developed in Chapter II which govern the first passage behavior of the simple hysteretic oscillator are degenerate forms of Equation 3.1. The nature of Equation 3.1 for many engineering problems gives little hope for analytical solution; thus numerical methods must be adopted to solve the problem.
Numerical Solution of the First Passage Problem
Abstract Consider the general form of the three-dimensional convection-diffusion equation given by $${\partial \over {\partial x}}{k_x}{{\partial \psi } \over {\partial x}} + {\partial \over {\partial y}}{k_y}{{\partial \psi } \over {\partial y}} + {\partial \over {\partial z}} - {u_x}{{\partial \psi } \over {\partial z}} - {u_y}{{\partial \psi } \over {\partial y}} - {u_z}{{\partial \psi } \over {\partial z}} + c{{\partial \psi } \over {\partial t}} + Q = 0$$ (3.1) where kx/C, ky/C and kz/C are the diffusion coefficients; ux/C, uy/C and uz/C are the velocities; Q/C is the source term; and the boundary conditions are appropriately defined. The nonlinear partial differential equations developed in Chapter II which govern the first passage behavior of the simple hysteretic oscillator are degenerate forms of Equation 3.1. The nature of Equation 3.1 for many engineering problems gives little hope for analytical solution; thus numerical methods must be adopted to solve the problem.
Numerical Solution of the First Passage Problem
Spencer, B. F. Jr. (author)
1986-01-01
42 pages
Article/Chapter (Book)
Electronic Resource
English
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