In this paper, we intend to introduce the Sturm-Liouville fractional problem and solve it using the collocation method based on CHEBYSHEV CARDINAL polynomials. To this end, we first provide an introduction to the Sturm-Liouville fractional equation. Then the CHEBYSHEV CARDINAL FUNCTIONs are introduced along with some of their properties and the operational matrices of the derivative, fractional integral, and Caputo fractional derivative are obtained for it. Here, for the first time, we solve the equation using the operational matrix of the fractional derivative without converting it to the corresponding integral equation. In addition to efficiency and accuracy, the proposed method is simple and applicable. The convergence of the method is investigated, and an example is presented to show its accuracy and efficiency.

In this manuscript, a numerical technique is presented for finding the eigenvalues of the regular Sturm-Liouville problems. The CHEBYSHEV CARDINAL FUNCTIONs are used to approximate the eigenvalues of a regular Sturm-Liouville problem with Dirichlet boundary conditions. These FUNCTIONs defined by the CHEBYSHEV FUNCTION of the first kind. By using the operational matrix of derivative the problem is reduced to a set of algebraic equation. Finally we use some numerical examples to show that this method include to demonstrate the validity and applicability of technique.

This study concentrated on the numerical solution of a nonlinear Volterra integral equation. The approach is accorded to a type of orthogonal wavelets named the CHEBYSHEV CARDINAL wavelets. The undetermined solution is extended concerning the CHEBYSHEV CARDINAL wavelets involving unknown coefficients. Hence, a system of nonlinear algebraic equations is drawn out by changing the introduced expansion to the predetermined problem, applying the generated operational matrix, and supposing the CARDINALity of the basis FUNCTIONs. Conclusively, the estimated solution is achieved by figuring out the mentioned system. Relatively, the convergence of the founded procedure process is reviewed in the Sobolev space. In addition, the results achieved from utilizing the method in some instances display the applicability and validity of the method.

In this paper, an effective direct method to determine the numerical solution of linear and nonlinear Fredholm and Volterra integral and integro-differential equations is proposed. The method is based on expanding the required approximate solution as the elements of CHEBYSHEV CARDINAL FUNCTIONs. The operational matrices for the integration and product of the CHEBYSHEV CARDINAL FUNCTIONs are described in detail. These matrices play the important role of reducing an integral equation to a system of algebraic equations. Illustrative examples are shown, which confirms the validity and applicability of the presented technique.

A new and effective direct method to determine the numerical solution of linear and nonlinear differential-algebraic equations (DAEs) is proposed. The method consists of expanding the required approximate solution as the elements of CHEBYSHEV CARDINAL FUNCTIONs. The operational matrices for the integration and product of the CHEBYSHEV CARDINAL FUNCTIONs are presented. A general procedure for forming these matrices is given. These matrices play an important role in modelling of problems. By using these operational matrices together, a differentialalgebraic equation can be transformed to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

In this study, an effective numerical method for solving fractional differential equations using CHEBYSHEV CARDINAL FUNCTIONs is presented. The fractional derivative is described in the Caputo sense. An operational matrix of fractional order integration is derived and is utilized to reduce the fractional differential equations to system of algebraic equations. In addition, illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method.

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.