Theory of Approximation and Applications
Volume:9 Issue: 2, Summer and Autumn 2014

Although homotopy-based methods, namely homotopy analysis method and homotopy perturbation method, have largely been used to solve functional equations, there are still serious questions on the convergence issue of these methods. Some authors have tried to prove convergence of these methods, but the researchers in this article indicate that some of those discussions are faulty. Here, after criticizing previous works, a sucient condition for convergence of homotopy methods is presented. Finally, examples are given to show that even if the homotopy method leads to a convergent series, it may not converge to the exact solution of the equation under consideration.

Envelopment Analysis (DEA) is a very e ective method to evaluate the relative eciency of decision- making units (DMUs). DEA models divided all DMUs in two categories: ecient and inecient DMUs, and dont able to discriminant between ecient DMUs. On the other hand, the observed values of the input and output data in real-life problems are sometimes imprecise or vague, such as interval data, ordinal data and fuzzy data. This paper develops a new ranking system under the condition of constant returns to scale (CRS) in the presence of imprecise data, In other words, in this paper, we reformulate the conventional ranking method by ideal point as an imprecise data envelopment analysis (DEA) problem, and propose a novel method for ranking the DMUs when the inputs and outputs are fuzzy and/or ordinal or vary in intervals. For this purpose we convert all data into interval data. In order to convert each fuzzy number into interval data we use the nearest weighted interval approximation of fuzzy numbers by applying the weighting function and also we convert each ordinal data into interval one. By this manner we could convert all data into intervaldata. The numerical example illustrates the process of ranking all the DMUs in the presence of fuzzy, ordinal and interval data.

An ecient method, based on the Legendre wavelets, is proposed to solve the second kind Fredholm and Volterra integral equations of Hammerstein type. The properties of Legendre wavelet family are utilized to reduce a nonlinear integral equation to a system of nonlinear algebraic equations, which is easily handled with the well-known Newton's method. Examples assuring eciency of the method and its superiority are presented.

In this paper, a new and ecient approach based on operational matrices with respect to the gener- alized Laguerre polynomials for numerical approximation of the linear ordinary di erential equations (ODEs) with variable coecients is introduced. Explicit formulae which express the generalized La- guerre expansion coecients for the moments of the derivatives of any di erentiable function in terms of the original expansion coecients of the function itself are given in the matrix form. The main importance of this scheme is that using this approach reduces solving the linear di erential equations to solve a system of linear algebraic equations, thus greatly simplify the problem. In addition, several numerical experiments are given to demonstrate the validity and applicability of the method.

The linear system of equations Ax = b where A = [aij ] in Cn.n is a crisp singular matrix and the right-hand side is a fuzzy vector is called a singular fuzzy linear system of equations. In this paper, solving singular fuzzy linear systems of equations using generalized inverses such as Drazin inverse and pseudo-inverse are investigated.

In this paper we derive convergence theorems for an -nonexpansive mapping of a nonempty closed and convex subset of a complete CAT(0) space for SP- iterative process and Thianwan's iterative process.

In this paper, a fuzzy Hunter-Saxton equation is solved by using the Adomian's decomposition method (ADM) and homotopy analysis method (HAM). The approximation solution of this equation is calculated in the form of series which its components are computed by applying a recursive relation. The existence and uniqueness of the solution and the convergence of the proposed methods are proved. A numerical example is studied to demonstrate the accuracy of the presented methods. In this paper, a fuzzy Hunter-Saxton equation is solved by using the Adomian's decomposition method (ADM) and homotopy analysis method (HAM). The approximation solution of this equation is calculated in the form of series which its components are computed by applying a recursive relation. The existence and uniqueness of the solution and the convergence of the proposed methods are proved. A numerical example is studied to demonstrate the accuracy of the presented methods.

In this work, we present a computational method for solving second kind nonlinear Fredholm Volterra integral equations which is based on the use of Haar wavelets. These functions together with the collocation method are then utilized to reduce the Fredholm Volterra integral equations to the solution of algebraic equations. Finally, we also give some numerical examples that shows validity and applicability of the technique.

It is proved that by using bounds of eigenvalues of an interval matrix, some conditions for checking positive de niteness and stability of interval matrices can be presented. These conditions have been proved previously with various methods and now we provide some new proofs for them with a unity method. Furthermore we introduce a new necessary and sucient condition for checking stability of interval matrices.

The purpose of this letter is to revisit the nonlinear reaction-di usion model in porous catalysts when reaction term is fractional function of the concen- tration distribution of the reactant. This model, which originates also in uid and solute transport in soft tissues and microvessels, has been recently given analytical solution in terms of Taylors series for di erent family of reaction terms. We apply the method so-called predictor homotopy analysis method (PHAM) which has been recently proposed to predict multiplicity of solutions of nonlinear BVPs. Consequently, it is indicated that the problem for some values of the parameter admits multiple solutions. Also, error analysis of these solutions are given graphically.