Assessment of errors in approximate solutions of differential equations

The term assessment is applied to any process which enables us to set rigid bounds to the error or to estimate its value. It is shown that upper and lower bounds can be assigned whenever the Green's function of the problem is one- signed; this is true in many important problems. Another method is applicable to step by step -solutions of ordinary differential equations, linear or non-linear, and depends on use of the "index" of the process of integration. Lastly, the error in a linear problem can be estimated when an approximation to the Green's function is known.