Previously, Chang reported a new high-order Conservation Element Solution Element (CESE) method for solving nonlinear, scalar, hyperbolic partial differential equations in one dimensional space. Bilyeu et al. have extended Chang’s scheme for solving a onedimensional, coupled equations with an arbitrary order of accuracy. In the present paper, the one-dimensional, high-order CESE method is exten...

Generalized solution on Neumann problem of the fourth order ordinary differential equation in space $W^2_alpha(0,b)$ has been discussed, we obtain the condition on B.V.P when the solution is in classical form. Formulation of Quintic Spline Function has been derived and the consistency relations are given.Numerical method,based on Quintic spline approximation has been developed. Spline solution ...

In this paper, we use a deep mathematical result (namely, a minor modi cation of Kolmogorov's solution to Hilbert's 13th problem) to explain why fundamental physical equations are of second order. This same result explain why all these fundamental equations naturally lead to nonsmooth solutions like singularity. Formulation of the problem. Most physical phenomena are described in terms of parti...