  This paper points out that the differential quadrature (DQ) and differential
cubature (DC) methods due to their global domain property are more efficient
for nonlinear problems than the traditional numerical techniques such as finite
element and finite difference methods. By introducing the Hadamard product of
matrices, we obtain an explicit matrix formulation for the DQ and DC solutions
of nonlinear differential and integro-differential equations. Due to its
simplicity and flexibility, the present Hadamard product approach makes the DQ
and DC methods much easier to be used. Many studies on the Hadamard product can
be fully exploited for the DQ and DC nonlinear computations. Furthermore, we
first present SJT product of matrix and vector to compute accurately and
efficiently the Frechet derivative matrix in the Newton-Raphson method for the
solution of the nonlinear formulations. We also propose a simple approach to
simplify the DQ or DC formulations for some nonlinear differential operators
and thus the computational efficiency of these methods is improved
significantly. We give the matrix multiplication formulas to compute
efficiently the weighting coefficient matrices of the DC method. The spherical
harmonics are suggested as the test functions in the DC method to handle the
nonlinear differential equations occurring in global and hemispheric weather
forecasting problems. Some examples are analyzed to demonstrate the simplicity
and efficiency of the presented techniques. It is emphasized that innovations
presented are applicable to the nonlinear computations of the other numerical
methods as well.
