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We discuss the numerical methods and their error estimates of the linear and non-linear differential equations, the singular non-linear differential equation and the Timoshenko beam and the circular arch problems in chapter two, chapter three and chapter four, respectively, then we get the following conclusions:Theorem 2.1 For the problemwhere the error bound of the finite difference method is o（h）, and if [1], the error bound is O（h2）, where h = maxkhk-->0Theorem 2.2 For the second order non-linear differential equation with smooth coefficientthe error bound of the three-point difference scheme is O（h4） Theorem 3.1 Assume f（x） = f（x.y（x）） C2[0,l], exists and is continuous and . For the singular two-point boundary value problemwhere A is a real constant, w（x）,p（x）, f（x） = f（x,y（x）） : I = （0, 1） --> R is L integrabel, our new spline method provides uniformly convergent approximations s（x） over [0, 1] for the solution y（x） of the singular two-point value problem, that is. for sufficiently small h,where C = 4h2 + c（2 + 6uμ（π））.Theorem 4.1 For the Timoshenko beam problem, the solution of shooting method is stable for the parameter ε.Theorem 4.2 For the circular arch problem, the solution of shooting method is stable for the parameter ε
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