Matrix Methods for Parabolic Partial Differential Equations

Many of the problems of physics and engineering that require numerical approximations are special cases of the following second-order linear parabolic differential equation: 8.1 $$ \begin{array}{*{20}c} {\phi \left( {\rm x} \right)u_t \left( {{\rm x};t} \right)} \hfill & { = \sum\limits_{i = 1}^n {\left( {K_i \left( {\rm x} \right)u_{x_i } } \right)_{x_i } } + \sum\limits_{i = 1}^n {G_i \left( {\rm x} \right)u_{x_i } } } \hfill \\ \, \hfill & {\begin{array}{*{20}c} { - \sigma \left( {\rm x} \right)u\left( {{\rm x};t} \right) + S\left( {{\rm x};t} \right),} \hfill & {{\rm x} \in R,\,t > 0,} \hfill \\ \end{array}} \hfill \\ \end{array} $$ where R is a given finite (connected) region in Euclidean n-dimentional space, with (external) boundary conditions 8.2 $$ \begin{array}{*{20}c} {\alpha \left( {\rm x} \right)u\left( {{\rm x};t} \right) + \beta \left( {\rm x} \right)\frac{{\partial u\left( {{\rm x};t} \right)}}{{\partial n}} = \gamma \left( {\rm x} \right),} \hfill & {{\rm x} \in \Gamma ,t > 0,} \hfill \\ \end{array}$$ where Г is the external boundary of R. Characteristic of such problems is the additional initial condition8.3 $$ \begin{array}{*{20}c} {u\left( {{\rm x};0} \right) = g\left( {\rm x} \right),} \hfill & {{\rm x} \in R.} \hfill \\ \end{array} $$