In the preceding chapter, the numerical solution of ODEs, which involve a single independent variable, was discussed. However, for a wide variety of problems in science and engineering, the dependent variables are functions of two or more independent variables, such as time and the spatial coordinate distances. Consequently, the differential equations that govern such problems involve partial derivatives and are known as partial differential equations (PDEs). These equations arise in almost all areas of engineering, for instance, in ¬uid mechanics, elasticity, heat transfer, energy systems, environmental ¬ows, hydraulics, neutron diffusion in nuclear reactors, and structural analysis. The numerical solution of PDEs is generally more involved than that of ODEs because of the presence of several independent variables, each with its own initial and boundary conditions. Therefore, effort is often made, whenever possible, by the use of simplifying approximations and transformations, to reduce the governing PDE to an ODE. However, this simpli˜cation is possible in only a limited number of cases. Because of the complicated nature of PDEs, analytical solutions are rarely obtained, and numerical methods are necessary for most problems of practical interest.