In the present paper, hybrid DIFFERENTIAL TRANSFORM and finite difference method (HDTFD) is applied to solve 2D transient nonlinear straight annular fin equation. For the case of linear heat transfer the results are verified with analytical solution. The effect of different parameters on in temperature distribution is investigated. Effect of TIME INTERVAL of DIFFERENTIAL TRANSFORM on the stability of results has been examined. Results show the excellent capability of HDTFD to solve different engineering problems and also indicate that appropriate selection of DIFFERENTIAL TRANSFORM TIME INTERVAL can solve the divergence problem of the method and lead to reduction in computational costs.

In this research, in order to investigate the effect of the piezoelectric patch which is used as a sensor or actuator in rotating flexible structures such as a helicopter blade, the free vibrations of the rotating rectangular sheet with and without the piezoelectric patch have been presented. First-order shear deformation theory is considered for plate displacement and piezoelectric field. Considering the effect of Coriolis acceleration, centrifugal acceleration and centrifugal in-plane forces, the equations of motion are derived from Hamilton's principle and the electromechanical couple equation is obtained from Maxwell's equation. For piezoelectric, two electrical conditions, open circuit and closed circuit, which are used in sensors and actuators, respectively, have been considered. The equations are discretized with the help of the numerical method of generalized DIFFERENTIAL squares and the matrices of inertia mass, eccentricity, Coriolis and stiffness matrix are obtained. Natural frequency values for beam and rotating plate have been compared in Abaqus software. Also, the values obtained from the numerical solution in MATLAB have been verified with articles and ABAQUS, which have high accuracy. The effect of parameters such as hub radius, rotation speed, sheet thickness, aspect ratio, piezoelectric patch thickness and applied voltage on the natural frequency of the system has also been investigated.

TIME series is a type of data with complex structure. Analysis of TIME series is used in sciences such as meteorology, economics, geology, marine science, medicine and engineering widely. So, Because of TIME series applications in various sciences, the interest to analyze these data has been increased. On the other hand by developing information gathering technologies such as mobile, GPS and sensors, and Access to large volumes of TIME series data, we always require methods to extract useful information from large datasets. Thus, data mining is an important method for solving this problem. Clustering analysis as the most commonly used function of data mining, has attracted many researchers in computer science. Clustering is a strong instrument for knowledge discovery and it provides useful information about existing patterns in datasets. In general, the purpose of clustering is representing large datasets by a fewer number of cluster centers. It simplifies large datasets and thus is an important step in the process of knowledge discovery and data mining. Fuzzy C-means (FCM) clustering is one of the most important classic clustering methods that have been used in many researches. The main disadvantage of this method is the high probability of getting trapped in local optima especially in facing high-dimensional data such as TIME series. Furthermore Euclidean distance is the most commonly used similarity measure in Fuzzy C-means but someTIMEs, its necessary to use another similarity/dissimilarity measures instead of Euclidean distance. In this paper in order to compensate the shortcomings of Fuzzy C-means algorithm, we used one of the existing evolutionary algorithms. Evolutionary algorithms has gained huge popularity in the field of pattern recognition and clustering recently. Among the existing evolutionary algorithms, the DIFFERENTIAL evolution algorithm as a strong, fast and efficient global search method has been attracted the attention of researchers. In this paper, we proposed a technique for clustering TIME series data using a combination of Fuzzy C-means and DIFFERENTIAL evolution (DE) approach and we considered dynamic TIME warping (DTW) as distance measures between TIME series. Also, in this method we used Discrete Cosine TRANSFORM (DCT) to TIME series dimension reduction. Finding all elements of cluster centers using DIFFERENTIAL evolution is TIME consuming and the large number of unknown parameters related to the cluster centers will reduce the efficiency and the speed of DIFFERENTIAL evolution algorithm. So, for reducing the search space, the most important Discrete Cosine TRANSFORM coefficients of the cluster centers were recognized as the main unknown clustering problem in the proposed method and DIFFERENTIAL evolution algorithm tries to determine the near optimal Discrete Cosine TRANSFORM coefficients of cluster centers by minimizing the Fuzzy C-means objective function. Experimental results over two popular data sets indicate the superiority of the proposed technique compared to fuzzy C-means and a clustering algorithm based on DIFFERENTIAL evolution without using dimension reduction techniques. Comparing the run TIME of the methods, the proposed method is slower than the Fuzzy C-means clustering algorithm, but due to the use of discrete cosine TRANSFORM method to reduce unknowns, it operates faster than DIFFERENTIAL evolution without using dimension reduction techniques.

Modelling phenomena with INTERVAL DIFFERENTIAL equations (IDEs) is an effective way to consider the uncertainties that are unavoidable when collecting data. Similarly to the theory of ordinary DIFFERENTIAL equations, IDEs have been parallelly investigated with the INTERVAL difference equations from the beginning. These two branches can be regarded as one when unifying continuous and discrete solution domains. A conspicuous advantage when merging these areas is that the proof of several analogous properties in both theories need not be repeated. The paper provides a common and efficient tool for studying IDEs not only with continuous or discrete solution domains but also with more general ones. We propose the diamond-$\alpha$ derivative for INTERVAL-valued functions (IVFs) on TIME scales with respect to the generalized Hukuhara difference. Differently from most of the studies on the derivatives of functions on TIME scales, using the language of epsilon-delta, the novel concept is naturally studied according to the limit of IVFs on TIME scales as in classical mathematics. A particular class of IDEs on TIME scales is then considered with respect to the diamond-$\alpha$ derivative. Numerical problems are elaborated to illustrate the necessity and efficiency of the latter.

We expand a new generalization of the two-dimensional DIFFERENTIAL trans-form method. The new generalization is based on the two-dimensional differ-ential TRANSFORM method, fractional power series expansions, and conformable fractional derivative. We use the new method for solving a nonlinear con-formable fractional partial DIFFERENTIAL equation and a system of conformable fractional partial DIFFERENTIAL equation. Finally, numerical examples are pre-sented to illustrate the preciseness and effectiveness of the new technique.

Introduction: The theory and application of linear and nonlinear delay DIFFERENTIAL equations is an important subject within physics and applied mathematics.There are several numerical approaches for solving linear and nonlinear delay DIFFERENTIAL equations.Aim: In this paper, DIFFERENTIAL TRANSFORM method (DTM) is applied to numerical solution of linear and nonlinear delay DIFFERENTIAL equations.Materials and Methods: In this work, we presented simple proofs of the DIFFERENTIAL TRANSFORM Theorems and then we applied them to solving linear and nonlinear delay DIFFERENTIAL equations.Results: Numerical results compared to exact solutions are reported and it is shown that DTM is a reliable tool for the solution of delay DIFFERENTIAL equations.Conclusion: The present method reduces the computational difficulties of the other traditional methods and all the calculations can be made simple manipulations. Several examples were tested by applying the DTM and the results have shown remarkable performance.