solving partial differential equations we should also mention that the diffusion equation may appear after simplifying more complicated partial differential equations for exampleflow of viscous fluid between two flat and parallel plates is described by one-dimensional diffusion equationwhere then is the fluid velocity partial differential equation is solved in some domain in space and for time interval oe the solution of the equation is not unique unless we also prescribe initial and boundary conditions the type and number of such conditions depend on the type of equation for the diffusion equationwe need one initial conditionu /stating what is when the process starts in additionthe diffusion equation needs one boundary condition at each point of the boundary @of this condition can either be that is known or that we know the normal derivativeru @ =@ ( denotes an outward unit normal to @"let us look at specific application and how the diffusion equation with initial and boundary conditions then appears we consider the evolution of temperature in one-dimensional mediummore precisely long rodwhere the surface of the rod is covered by an insulating material the heat can then not escape from the surfacewhich means that the temperature distribution will only depend on coordinate along the rodxand time at one end of the rodx lwe also assume that the surface is insulatedbut at the other endx we assume that we have some device for controlling the temperature of the medium herea function ttells what the temperature is in time we therefore have boundary condition td tat the other insulated endx lheat cannot escapewhich is expressed by the boundary condition @ lt/=@ the surface along the rod is also insulated and hence subject to the same boundary condition (here generalized to @ =@ at the curved surfacehoweversince we have reduced the problem to one dimensionwe do not need this physical boundary condition in our mathematical model in one dimensionwe can set oe lto summarizethe partial differential equation with initial and boundary conditions reads @ xt@ xtc xt/dv @ @ td / ltd @ / ; ( ( ( oe ( mathematicallywe assume that at the initial condition ( holds and that the partial differential equation ( comes into play for similarlyat the end pointsthe boundary conditions ( and ( govern and the equation therefore is valid for lboundary and initial conditions are neededthe initial and boundary conditions are extremely important without themthe solution is not uniqueand no numerical method will work unfortunatelymany physical applications have one or more initial or boundary conditions as unknowns such situations can be dealt with if we have measurements of ubut the mathematical framework is much more complicated