Systems of simultaneous second-order nonlinear ordinary differential equations with boundary conditions at two points are solved by a new numerical scheme. By adding fictitious accumulation terms, ordinary differential equations become parabolic partial differential equations. A stable numerical method based upon Saul'yev's technique is used to integrate the parabolic partial differential equations. At an early stage of integration only a few mesh points are required and later the number increases gradually as the steady-state is approached.
Since one is only interested in obtaining the steady-state solution to the partial differential equations, one can change the values of coefficients of accumulation terms and also use relatively large time increments. Two examples are given to illustrate this technique. The first problem is three simultaneous ordinary differential equations describing two reactions within porous catalyst particles under nonisothermal conditions. The second problem deals with Taylor diffusion in tubular reactors. Results are obtained within short computer times and one does not require a close initial guess to the steady-state solution.