solving partial differential equations the initial conditions are /ui xi / ( ( we can apply any method for systems of odes to solve ( )-( construction of test problem with known discrete solution at this pointit is tempting to implement real physical case and run it howeverpartial differential equations constitute non-trivial topic where mathematical and programming mistakes come easy better start is therefore to address carefully designed test example where we can check that the method works the most attractive examples for testing implementations are those without approximation errorsbecause we know exactly what numbers the program should produce it turns out that solutions xtthat are linear in time and in space can be exactly reproduced by most numerical methods for partial differential equations candidate solution might be xtd linserting this in the governing equation gives ld xtg xtd lwhat about the boundary conditionswe realize that @ =@ for lwhich breaks the assumption of @ =@ at in the formulation of the numerical method above moreoveru td /so we must set td tc and td finallythe initial condition dictates xd xl/but recall that we must have /and ui xi / it is important that starts out at the right value dictated by tin case is not equal this value first we need to generalize our method to handle @ =@ $ at we then have un un td un un xwhich inserted in ( gives dun un tc un tc gn tdv dt ( implementationforward euler method in particularwe may use the forward euler method as implemented in the general function ode_fe in the module ode_system_fe from sect the ode_fe function needs specification of the right-hand side of the ode system