In this paper, a new set of spline functions called \Flat End FUZZY Spline" is de ned to interpolate given FUZZY data. Some important theorems on these splines together with their existence and unique-ness properties are discussed. Then numerical examples are presented to illustrate the di erences between of using our spline and other INTERPOLATIONs that have been studied before.

FUZZY rule INTERPOLATION (FRI) predicts an accountable outcome of a possible course of action in sparse FUZZY rule base system (FRBS). The geometry based linear FUZZY rule INTERPOLATION (GLFRI) is extended for multi-dimensional FUZZY rule base INTERPOLATION. Expansion/contraction (EC) of triangular, trapezoidal and complex polygonal FUZZY sets has been also proposed which enables the proposed FRI method to incorporate with FUZZY rules which include triangular, trapezoidal, hexagonal or complex FUZZY sets. The study further extends to introduce the process of backward rule base INTERPOLATION.It has been shown that the scale and move transformation-based FRI method can yield a non-convex FUZZY consequent which can be avoided by using the proposed method. The proposed method performs better without any risk of obtaining non-convex FUZZY consequent. The efficiency of proposed forward and backward FRI methods is projected with several numerical examples. A detailed comparison of EC transformation with scale and move transformation is also presented here.

Hidden Markov Model is a popular statisical method that is used in continious and discrete speech recognition. The probability density function of observation vectors in each state is estimated with discrete density or continious density modeling. The performance (in correct word recognition rate) of continious density is higher than discrete density HMM, but its computation complexity is very high, especially in very large discrete utterance recognition problems. For real time implementation of very large discrete utterance recognition, we must use discrete density HMM (DDHMM). To increase the performance of DDHMM, one usual solution is FUZZY INTERPOLATION. In this study, we present a new method named Gaussian INTERPOLATION. We implemented and compared the performance of two types of INTERPOLATION methods for 1500 Persian speech command words. Results show that precision and flexibility of Gaussian INTERPOLATION is better thanthose of the FUZZY INTERPOLATION.

FUZZY quasi-INTERPOLATIONs help to reduce the complexity of solving a linear system of equations compared with FUZZY INTERPOLATIONs. Almost all FUZZY quasi-INTERPOLATIONs are focused on the form of e f : R → F(R) or e f : F(R) → R. In this paper, we intend to offer a novel FUZZY radial basis function by the concept of source distance. Then, we will construct a FUZZY linear combination of such basis functions in order to introduce a fully FUZZY quasi-INTERPOLATION in the form of e f : F(R) → F(R). Also the error estimation of the proposed method is proved in terms of the fully FUZZY modulus of continuity which will be introduced in this paper. Finally some examples have been given to emphasize the acceptable accuracy of our method.

In this paper,  rstly, we review approximation of FUZZY functions by FUZZY bicubic splines interpo-lation and present a new approach based on the two-dimensional FUZZY splines INTERPOLATION and iterative method to approximate the solution of two-dimensional linear FUZZY Fredholm integral equa-tion (2DLFFIE). Also, we prove convergence analysis and numerical stability analysis for the proposed numerical algorithm. Finally, by an example, we show the e ciency of the proposed method.

introduction: In this paper, we propose a numerical method for approximating the solution of the FUZZY functional integral equations of Volterra and Fredholm type by using Lagrange INTERPOLATION. For this purpose, we convert the FUZZY Fredholm and Volterra integral equations to the crisp systems of integral equations. The proposed method is illustrated by various FUZZY numerical examples.Aim: In almost, the integral equations cannot be solved analytically or simply.Therefore, we propose the numerical methods for solving the integral equations.Material and Method: For solving FUZZY functional integral equations, at first, we introduce the parametric form of them and then we obtained (2n+2) x (2n+2) systems via Lagrange INTERPOLATION. In the following, this system convert to two (n+1) x (n+I) systems.By solving them, we have the support points which can be approximate the exact solution.Results: In this work, for solving Fredholm or Volterra FUZZY functional integral equation, we replaced each of this FUZZY integral equation by two crisp integral equations. For numerical solution of these equations, we applied the Lagrange INTERPOLATION with different r_cuts that is between zero and one. At last, by substituting x in the crisp equations by Xj for j = 0,1,...,n we obtained a linear system of equations with 2n + 2 equations and 2n + 2 unknown. By solving two (n+l) x (n+l) systems, y(xj.;r) and ȳ (xj;r) for j=0,1,...,n and 0£r£1 are calculated. Consequently, the approximation for exact solution is given by putting y(xj .;r) and . ȳ (xj;r) for j = 0, 1,...,n in Lagrange INTERPOLATION function. The advantage of this method in comparison with the other methods is that the solution of the integral equation is approximated by having the supported points. Also, the proposed method in comparison with Freidman et al.'s method converges to the exact solution with less number of the iterations and node points.Conclusion: This By the proposed method, we can numerically solve the FUZZY functional integral equations of Volterra and Fredholm of the second kind, Also, the proposed method in comparison with Freidman et al.' s method converges to the exact solution with less number of the iterations and node points.

In this paper FUZZY piecewise cubic INTERPOLATION is constructed for FUZZY data based on B-spline basis functions. We add two new additional conditions which guarantee uniqueness of FUZZY B-spline INTERPOLATION.Other conditions are imposed on the INTERPOLATION data to guarantee that the INTERPOLATION function to be a well-defined FUZZY function. Finally some examples are given to illustrate the proposed method.