In this paper, variational iteration method and homotopy perturbation method are applied to different forms of diffusion equation. The diffusion equations have found wide applications in heat transfer problems, theory of consolidation and many other problems in engineering. The methods proposed to solve the diffusion equations herein have been applied to a variety of problems in the recent past, and have proved to yield highly accurate solutions. Comparison is made between the exact solutions and the results of the variational iteration method (VIM) and homotopy perturbation method (HPM) in order to verify the accuracy of the results, revealing the fact that these methods are very effective and simple.

In this study, fin efficiency, temperature distribution and effectiveness of conductive, convective and radiative straight fins with temperature dependent thermal conductivity are solved using Galerkin Method (GM). The concept of galerkin method is briefly introduced, and then it is employed to derive solution of nonlinear governing equation of fin with a highly nonlinear term because of existing radiation in this study. The obtained results from GM are compared with numerical Boundary Value problem Method (BVP) to verify the accuracy of the proposed methods. The effects of some physical appropriate parameters in this problem such as thermo-geometric fin parameter and thermal parameters are analyzed.

In this study, the Homotopy Perturbation Method (HPM) is used to investigate nonlinear vibration behavior of a buckled beam subjected to an axial load. The motion equation has been solved due to vibration of a beam. Comparison between HPM results with time marching approach results demonstrates high accuracy of this method. This method can easily extend to solve other nonlinear vibration equations of the beams, plates and shells in the future.

Selection of cutting patterns in order to minimize production waste is an important issue in operations research, which has attracted many researchers. Solving this problem is very important in the industries that have priorities in minimizing the waste in the Pages section. In this paper, the two-dimensional cutting stock problem has been studied to reduce the cutting waste, with focus on men’s clothing (male pants size 42). In this method, regular and irregular shapes are enclosed by rectangles the goal is to minimize waste in total fabric. As solving these problems optimally are infeasible with current optimization algorithms because of the large solution space, the metaheuristic algorithm of simulated annealing (SA) was used. One of the main objectives of this research is to calculate the optimum length of fabric rolls, such that when several of such rolls are put together, the amount of cutting waste is minimized. To achieve this goal, we initially consider an unlimited length for each roll, and then obtain the optimum length of each roll. This research shows that the amount of required stock can be reduced in fabric cutting by using the SA algorithm. Moreover, if the length of pieces is not fixed, incontrollable stock can be changed into controllable ones; e.g., stock could be concentrated in a form which is usable in future consumption.

In this paper the classical Lie group formalism is applied to deduce new classes of solutions of a less studied nonlinear partial differential equation (nPDE) of the third-order. The nPDE under consideration is closely related to motions of plane curves.

Up to now no carefully performed symmetry analysis is available. Therefore we determine the classical Lie point symmetries including algebraic properties. Similarity solutions are given as well as nonlinear transformations could derived and periodic wave trains are obtained.

Since algebraic solution techniques fail symmetry analysis justifies the application yielding a deeper insight into the solution-manifold.

In addition, we shall see that the nPDE admits a new symmetry, the so called potential symmetry.

For some nonlinear ordinary differential equations (nODEs) created by similarity reductions the Painlevé property is discussed.

Nowadays, more and more ring footings are used in practice special for axi-symmetric structures. In this paper, a numerical analysis was performed using PLAXIS software for calculating bearing capacity and settlement of ring footing. The parameters used in this analysis are the results of geotechnical studies of Kazeroon cooling tower. The analysis was carried out using Mohr-Coulomb’s criterion for soil. The Bearing capacity was calculated for smooth and rough ring footing and then the bearing capacity factors were calculated. The analysis indicated that the bearing capacity of rough ring footing is obviously higher than the bearing capacity of smooth footing. Finally, the results were compared with those available in the literature.

In this study, three supported beam was studied. Longitudinal and transverse vibrations were investigated. Natural frequencies were calculated using Hamilton, Rayleigh’s and Rayleigh-Ritz Methods. For calculations, functions providing all boundary conditions of the problem were suggested. As a result natural frequencies of the system without using the equations of motion were calculated. Obtained results were compared with the real values.