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In this paper, by using the CESTAC method and the CADNA library a procedure is proposed to solve a fuzzy initial value problem based on RBF-meshless methods under generalized H-differentiability. So a reliable approach is presented to determine optimal shape parameter and number of points for RBF-meshless methods. The results reveal that the proposed method is very effective and simple. Also, the numerical accuracy of the method is shown in the tables and figures, and algorithms are given based on the stochastic arithmetic. The examples illustrate the efficiency and importance of using the stochastic arithmetic in place of the floating-point arithmetic. [ABSTRACT FROM AUTHOR]

The goal of this paper is to present a new scheme based on the stochastic arithmetic (SA) to find the optimal convergence control parameter, the optimal iteration and the optimal approximation of the homotopy analysis method (HAM). This scheme is called the CESTAC1 method. Also, the CADNA2 library is applied to implement the CESTAC method on the proposed algorithms. CADNA is able to present a new, robust and valid environment to implement the HAM and optimize the results. By using this method, not only the optimal auxiliary control parameter can be computed but also the unnecessary iterations can be neglected and optimal step of this method is achieved. The main theorems are presented to guarantee the validity and accuracy of the HAM. Different kinds of integral equations such as singular and first kind are considered to find the optimal results by applying the proposed algorithms. The numerical results show the importance and efficiency of the SA in comparison with the floating-point arithmetic (FPA). [ABSTRACT FROM AUTHOR]

In this work, an adaptive method with memory is developed such that all previous information are applied. The importance of the proposed method can be seen because of the optimization in important effecting factors, i.e., least number of iterations steps, least number of functional evaluations, least value of absolute error, and maximum efficiency index in final as well as in individual step as compared with the other methods. Indeed, it is proved that this adaptive method with memory has efficiency index 2 and competes all the existing methods without and with memory in the literature. The order of convergence is obtained by using two self-accelerating parameters, which is increased from 2 to 4 without any new function evaluation. It means that, the order of convergence can be improved until 100%. Numerical examples and the comparison with existing methods are included to demonstrate exceptional convergence speed of the proposed method and confirm theoretical results. [ABSTRACT FROM AUTHOR]