Numerical Solution of the First Passage Problem

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 k_x/C, k_y/C and k_z/C are the diffusion coefficients; u_x/C, u_y/C and u_z/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.