The double side approach method combines the method of weighted residuals (MWR) with mathematical programming to solve the differential equations. Once the differential equation is proved to satisfy the maximum principle, collocation method and mathematical programming are used to transfer the problem into a bilateral inequality. By utilizing Genetic Algorithms optimization method, the maximum and minimum solutions which satisfy the inequality can be found. Adopting this method, quite less computer memory and time are needed than those required for finite element method.
In this paper, the incompressible-Newtonian, fully-developed, steady-state laminar flow in equilateral triangular, rectangular, elliptical and super-elliptical ducts is studied. Based on the maximum principle of differential equations, the monotonicity of the Laplace operator can be proved and the double side approach method can be applied. Different kinds of trial functions are constructed to meet the no-slip boundary condition, and it was demonstrated that the results are in great agreement with the analytical solutions or the formerly presented works. The efficiency, accuracy, and simplicity of the double side approach method are fully illustrated in the present study to indicate that the method is powerful for solving boundary value problems.