solving partial differential equations to reduce the partial differential equation to ordinary differential equations one important technique for achieving thisis based on finite difference discretization of spatial derivatives reduction of pde to system of odes introduce spatial mesh in with mesh points xn the space between two mesh points xi and xi the interval oexi xi is call cell we shall herefor simplicityassume that each cell has the same length xi xi the partial differential equation is valid at all spatial points "but we may relax this condition and demand that it is fulfilled at the internal mesh points onlyx xn @ xi @ xi tc xi /dv @ @ ( nowat any point xi we can approximate the second-order derivative by finite difference@ xi tu xi xi tc xi ( @ it is common to introduce short notation ui tfor xi / approximated at some mesh point xi in space with this new notation we canafter inserting ( in ( )write an approximation to the partial differential equation at mesh point xi tas dui tui ui tc ui tc gi /dv dt ( note that we have adopted the notation gi tfor xi ttoo what is ( )this is nothing but system of ordinary differential equations in unknowns /un /in other wordswith aid of the finite difference approximation ( )we have reduced the single partial differential equation to system of odeswhich we know how to solve in the literaturethis strategy is called the method of lines we need to look into the initial and boundary conditions as well the initial condition xtranslates to an initial condition for every unknown function ui /ui xi / at the boundary we need an ode in our ode systemwhich must come from the boundary condition at this point the boundary condition reads td twe can derive an ode from this equation by differentiating both sidesu td tthe ode system above cannot be used for since that equation involves some quantity outside the domain insteadwe use the equation td tderived from the boundary condition for this particular equation we also need to make sure the initial condition is (otherwise nothing will happenwe get forever