A New Method for Numerical Integration of Higher-Order Ordinary Differential Equations Without Losing the Periodic Responses

A new numerical method is presented for the solution of initial value problems described by systems of N linear ordinary differential equations (ODEs). Using the state-space representation, a differential equation of order n > 1 is transformed into a system of L = n×N first-order equations, thus the numerical method developed recently by Katsikadelis for first-order parabolic differential equations can be applied. The stability condition of the numerical scheme is derived and is investigated using several well-corroborated examples, which demonstrate also its convergence and accuracy. The method is simply implemented. It is accurate and has no numerical damping. The stability does not require symmetrical and positive definite coefficient matrices. This advantage is important because the scheme can find the solution of differential equations resulting from methods in which the space discretization does not result in symmetrical matrices, for example, the boundary element method. It captures the periodic behavior of the solution, where many of the standard numerical methods may fail or are highly inaccurate. The present method also solves equations having variable coefficients as well as non-linear ones. It performs well when motions of long duration are considered, and it can be employed for the integration of stiff differential equations as well as equations exhibiting softening where widely used methods may not be effective. The presented examples demonstrate the efficiency and accuracy of the method.

Sustainable Decision Making Using a Consensus Model for Consistent Hesitant Fuzzy Preference Relations—Water Allocation Management Case Study

This paper presents an improved consensus-based procedure to handle multi-person decision making (MPDM) using hesitant fuzzy preference relations (HFPRs) which are not in normal format. At the first level, we proposed a ukasiewicz transitivity (TL-transitivity) based scheme to get normalized hesitant fuzzy preference relations (NHFPRs), subject to which, a consensus-based model is established. Then, a transitive closure formula is defined to construct TL-consistent HFPRs and creates symmetrical matrices. Following this, consistency analysis is made to estimate the consistency degrees of the information provided by the decision-makers (DMs), and consequently, to assign the consistency weights to them. The final priority weights vector of DMs is calculated after the combination of consistency weights and predefined priority weights (if any). The consensus process concludes whether the aggregation of data and selection of the best alternative should be originated or not. The enhancement mechanism is indulged in improving the consensus measure among the DMs, after introducing an identifier used to locate the weak positions, in case of the poor consensus reached. In the end, a comparative example reflects the applicability and the efficiency of proposed scheme. The results show that the proposed method can offer useful comprehension into the MPDM process.

Application of Corrected Clarke’s Transformation to the Solution of the Modal Analysis of Un-Transposed Transmission Lines

Clark’s transformation can be used to conduct a modal analysis of perfectly balanced three-phase lines. For un-transposed transmission lines, the parameter matrices of line such as impedance matrix and admittance matrix are no longer symmetrical matrices, so Clark’s transformation can’t be used directly in modal analysis. Nevertheless, it still plays an important role in determining the exact phase-mode transformation matrix. This paper showed how this can be done by a series of matrix transformation starting with Clark’s transformation and then used the corrected Clark’s transformation in modal analysis.

ON SOME VARIATIONAL FORMULATIONS FOR MINIMUM SURFACE

An approach to resolving the minimum surface problem based on the total energy balance of the nodal system is represented in this work. The approach leads to design of two or three (for 2D or 3D problems respectively) independent systems of linear algebraic equations with symmetrical matrices in the banded form similar to global stiffness matrix of FEM assemblage. The efficiency of the approach is demonstrated by some 2D and 3D numerical tests. The formulation allows one to minimize the surface embodied into non-plane and plane, closed and non-closed contours. If a contour is closed and plane it allows to correct the triangle network distortion.