A generalization of the Bernoulli’s method applied to brachistochrone-like problems

Re-derivation of Young’s Equation, Wenzel Equation, and Cassie-Baxter Equation Based on Energy Minimization

“…With the variational method, we can solve many problems involving the determination of maxima or minima of functionals, such as the shortest smooth curve joining two distinct points in the plane [21], the shape of solid of revolution moving in a flow of gas with least resistance [22], the plane curve down that a particle will slide without friction from one point to the other point in the shortest time [23], the curve passing through two given points to have minimum surface area by the rotation of the curve [21], and the shape of the flexible cable of given length suspended between two poles [24].…”

The Study of Heat Transfer Phenomena Using PM for Approximate Solution with Dirichlet and Mixed Boundary Conditions

“…2 show a noticeable fact, that (19) and (36) turn out to be a good approximation of (3) and (20) respectively, although perturbation parameters 1 ε = and 0.30 ε = cannot be considered small. Finally, our approximate solutions (19) and (36) do not depend of any adjustment parameter, for which, are in principle, general expressions for proposed problems.…”

“…Although the PM method provides in general, better results for small perturbation parameters 1 ε << , we will see that our approximations, besides to be handy, have a good accuracy, even for relatively large values of the perturbation parameter [39,41]. This paper is organized as follows.…”

“…The assumption that the nonlinear part is small compared to the linear is considered as a disadvantage of the method. There are other modern alternatives to find approximate solutions to the differential equations that describe some nonlinear problems such as those based on: variational approaches [5][6][7]29], tanh method [8], exp-function [9, 10], Adomian's decomposition method [11][12][13][14][15][16], parameter expansion [17], homotopy perturbation method [3,4,16,[18][19][20][21][22][23][24][25][26][27][28][29][31][32][33][34][35][36]38,42], homotopy analysis method [30], Homotopy asymptotic method [1], and perturbation method [39,41] among many others. Also, a few exact solutions to nonlinear differential equations have been reported occasionally [40].…”

A Method of Moments Approach for Laminar Boundary Layer Flows

“…The pioneering work of Howarth has proved to be the first attempt in solving the laminar boundary layer equations by Runge Kutta method [6]. An approximate solution using the homotopy perturbation method was preferred by Filobello Nino et al [7] while Paranda et al has suggested to use the sinc collocation method [8]. Coupled method combining the iteration and the perturbation methods was used by He to obtain high accuracy of the equation [9].…”