This study introduces a finite element method using a higher-order INTERPOLATION FUNCTION for effective simulations of wave transformation. Finite element methods with a higher-order INTERPOLATION FUNCTION usually employ a Lagrangian INTERPOLATION FUNCTION that gives accurate solutions with a lesser number of elements compared to lower order INTERPOLATION FUNCTION. At the same time, it takes a lot of time to get a solution because the size of the local matrix increases resulting in the increase of band width of a global matrix as the order of the INTERPOLATION FUNCTION increases. Mass lumping can reduce computation time by making the local matrix a diagonal form. However, the efficiency is not satisfactory because it requires more elements to get results. In this study, the LEGENDRE CARDINAL INTERPOLATION FUNCTION, a modified Lagrangian INTERPOLATION FUNCTION, is used for efficient calculation. Diagonal matrix generation by applying direct numerical integration to the LEGENDRE CARDINAL INTERPOLATION FUNCTION like conducting mass lumping can reduce calculation time with favorable accuracy. Numerical simulations of regular, irregular and solitary waves using the Boussinesq equations through applying the INTERPOLATION approaches are carried out to compare the higher-order finite element models on wave transformation and examine the efficiency of calculation.

In this paper, we formulate a numerical method to approx-imate the solution of non-linear fractal-fractional Burgers equation. In this model, differential operators are defined in the Atangana-Riemann-Liouville sense with Mittag-Leffler kernel. We first expand the spatial derivatives using barycentric INTERPOLATION method, and then we derive an operational matrix (OM) of the fractal-fractional derivative for the LEGENDRE polynomials. To be more precise, two approximation tools are coupled to convert the fractal-fractional Burgers equation into a system of algebraic equations which is technically uncomplicated and can be solved using available mathematical software such as MATLAB. To investigate the agreement between exact and approximate solutions, several examples are examined.

We estimate a FUNCTION f with N independent observations by using Leg-endre wavelets operational matrices. The FUNCTION f is approximated with the solution of a special minimization problem. We introduce an explicit expression for the penalty term by LEGENDRE wavelets operational matrices. Also, we obtain a new upper bound on the approximation error of a differentiable FUNCTION f using the partial sums of the LEGENDRE wavelets. The validity and ability of these operational matrices are shown by several examples of real-world problems with some constraints. An accurate ap-proximation of the regression FUNCTION is obtained by the LEGENDRE wavelets estimator. Furthermore, the proposed estimation is compared with a non-parametric regression algorithm and the capability of this estimation is illustrated.

In this paper we have used Sen Welfare FUNCTION along with Generalized Sen Welfare FUNCTION for evaluating the welfare change in Iran. We have also used the substitution rate between efficiency and inequality, marginal substitution rate between social welfare and income, and elasticity of welfare with respect to inequality for the evaluation. Results of the finding show that social welfare has increased in 2002-2007, 1997-2001, and 1992-1996 periods compared to 1971-1976 period by the rates of 4.9, 3.1, and 2.7 percent respectively. This shows that the public policy decisions taken by the government during the 1997-2007 periods had an effective role in increasing the per capita income and also in reducing the income inequality. Results of the study also showed that the welfare change due to per capita income is greater than welfare change due to income inequality reduction in the period 1971-2007 (except for 1977-1986). The important finding of this study was that increase in per capita income did not result in increase in income inequality. Therefore it seems that there was not a tradeoff between equity and efficiency in that period in Iranian economy. So, pursuing the policies for increasing the efficiency in economy is a proper policy for increasing the social welfare in Iran.

A new method for structural optimization is presented for successive approximation of the objective FUNCTION and constraints in conjunction with Lagrange Multipliers Approach. The focus is on presenting the methodology with simple examples. The basis of the iterative algorithm is that after each iteration it brings the approximate location of the estimated minimum closer to the exact location gradually. In other words, instead of the linear or parabolic term used in Taylor expansion which works based on a short step length, an arc is used that has a constant curvature but a longer step length. Using this approximation, the equations of optimization involve the Lagrange multipliers as the only unknown variables. The equations which depend on the design variables are decoupled linearly as these variables are directly obtained. One mathematical example is solved to explain in details how the method works. Next the method is applied to the optimization of a simple truss structure to explain how the method can be used in structural optimization. The same problems have been solved by penalty method and compared. The results from both methods have been the same. However because of the long step length and reduction in the number of variables, the speed of convergence has been higher in the presented method.