In this paper, we propose a new semi-analytic approach based on the generalized Taylor series for solving nonlinear differential equations of fractional order. Assuming the solution is expanded as the generalized Taylor series, the coefficients of the series can be computed by solving the corresponding recursive relation of the coefficients which is generated by the given problem. This method is called the generalized differential transform method(GDTM). In several literatures the standard GDTM was applied in each sub-domain to obtain an accurate approximation. As noticed in [19], however, a direct application of the GDTM in each sub-domain loses a term of memory which causes an inaccurate approximation. In this work, we derive a new recursive relation of the coefficients that reflects an effect of memory. Several illustrative examples are demonstrated to show the effectiveness of the proposed method. It is shown that the proposed method is robust and accurate for solving nonlinear differential equations of fractional order.

The proper orthogonal decomposition (POD) method for time-dependent problems significantly reduces the computational time as it reduces the original problem to the lower dimensional space. Even a higher degree of reduction can be reached if the solution is smooth in space and time. However, if the solution is discontinuous and the discontinuity is parameterized e.g. with time, the POD approximations are not accurate in the reduced space due to the lack of ability to represent the discontinuous solution as a finite linear combination of smooth bases. In this paper, we propose to post-process the sample solutions and re-initialize the POD approximations to deal with discontinuous solutions and provide accurate approximations while the computational time is reduced. For the post-processing, we use the Gegenbauer reconstruction method. Then we regularize the Gegenbauer reconstruction for the construction of POD bases. With the constructed POD bases, we solve the given PDE in the reduced space. For the POD approximation, we re-initialize the POD solution so that the post-processed sample solution is used as the initial condition at each sampling time. As a proof-of-concept, we solve both one-dimensional linear and nonlinear hyperbolic problems. The numerical results show that the proposed method is efficient and accurate.

This paper examines individual and social strategies to form profitable cooperation networks. These two types of strategies measure network stability and efficiency that may not meet in a single network. We apply restrictions on knowledge flows (R&D spillovers) and links formation to integrate these benefits into structures that ensure high outcomes for both strategies. The results suggest that linking the spillovers to the firms' positions and restricting cooperation contribute to reducing the conflict between the individual and social strategies in the development of cooperative networks.

This article presents a new approach for conformal flattening with optimal cone singularity. The algorithm here takes an optimal control for selecting optimal cones and uses the Ricci flow to force the flattening. This work is considered as a modification to the work of Soliman et al. [1] in the sense that they make use of the Yamabe equation for the flattening, which is an approximation of the Ricci flow. We present a numerical algorithm based on the optimal control with the mathematical background. Several numerical results validate that our method is optimal in total cone angle and usage of the Ricci flow ensures the conformal flattening while selecting optimal cones.

A 1kW-class horizontal axis wind turbine (HAWT) rotor blade is taken into account to investigate elastic characteristics in 2-D. The elastic blade field is composed of symmetric cross-ply laminated composite material. Blade element momentum theory is applied to obtain the boundary conditions pressuring the blade, and the plane stress elasticity problem is formulated in terms of two displacement parameters with mixed boundary conditions. For the elastic characteristics a fair of differential equations are derived based on the elastic theory. The domain is divided by triangular and rectangular elements due to the complexity of the blade configuration, and a finite element method is developed for the governing equations to search approximate solutions. The results describe that the elastic behavior is deeply influenced by the layered angle of the middle laminate and the stability of the blade can be improved by controlling the layered angle of laminates, which can be evaluated by the mathematical approach.

In this paper, we are extending fractional partial differential equations to fuzzy fractional partial differential equation under Riemann-Liouville and Caputo fractional derivatives, namely Variational iteration methods, and this method have applied to the fuzzy fractional wave equation with initial conditions as in fuzzy. It is explained by one and two-dimensional wave equations with suitable fuzzy initial conditions.