Abstract:In nature, the mechanical properties of geological bodies are very complex, and their various mechanical parameters are vague, incomplete, imprecise, and indeterminate. However, we cannot express them by the crisp values in classical probability and statistics. In geotechnical engineering, we need to try our best to approximate exact values in indeterminate environments because determining the joint roughness coefficient (JRC) effectively is a key parameter in the shear strength between rock joint surfaces. In this original study, we first propose neutrosophic interval probability (NIP) and define the confidence degree based on the cosine measure between NIP and the ideal NIP. Then, we propose a new neutrosophic interval statistical number (NISN) by combining the neutrosophic number with the confidence degree to express indeterminate statistical information. Finally, we apply NISNs to express JRC under indeterminate (imprecise, incomplete, and uncertain, etc.) environments. By an actual case, the results demonstrate that NISNs are suitable and effective for JRC expressions and have the objective advantage.

Abstract:In rock mechanics, the study of shear strength on the structural surface is crucial to evaluating the stability of engineering rock mass. In order to determine the shear strength, a key parameter is the joint roughness coefficient (JRC). To express and analyze JRC values, Ye et al. have proposed JRC neutrosophic numbers (JRC-NNs) and fitting functions of JRC-NNs, which are obtained by the classical statistics and curve fitting in the current method. Although the JRC-NNs and JRC-NN functions contain much more information (partial determinate and partial indeterminate information) than the crisp JRC values and functions in classical methods, the JRC functions and the JRC-NN functions may also lose some useful information in the fitting process and result in the function distortion of JRC values. Sometimes, some complex fitting functions may also result in the difficulty of their expressions and analyses in actual applications. To solve these issues, we can combine the neutrosophic numbers with neutrosophic statistics to realize the neutrosophic statistical analysis of JRC-NNs for easily analyzing the characteristics (scale effect and anisotropy) of JRC values. In this study, by means of the neutrosophic average values and standard deviations of JRC-NNs, rather than fitting functions, we directly analyze the scale effect and anisotropy characteristics of JRC values based on an actual case. The analysis results of the case demonstrate the feasibility and effectiveness of the proposed neutrosophic statistical analysis of JRC-NNs and can overcome the insufficiencies of the classical statistics and fitting functions. The main advantages of this study are that the proposed neutrosophic statistical analysis method not only avoids information loss but also shows its simplicity and effectiveness in the characteristic analysis of JRC.

Abstract:The Dombi operations of T-norm and T-conorm introduced by Dombi can have the advantage of good flexibility with the operational parameter. In existing studies, however, the Dombi operations have so far not yet been used for neutrosophic sets. To propose new aggregation operators for neutrosophic sets by the extension of the Dombi operations, this paper firstly presents the Dombi operations of single-valued neutrosophic numbers (SVNNs) based on the operations of the Dombi T-norm and T-conorm, and then proposes the single-valued neutrosophic Dombi weighted arithmetic average (SVNDWAA) operator and the single-valued neutrosophic Dombi weighted geometric average (SVNDWGA) operator to deal with the aggregation of SVNNs and investigates their properties. Because the SVNDWAA and SVNDWGA operators have the advantage of good flexibility with the operational parameter, we develop a multiple attribute decision-making (MADM) method based on the SVNWAA or SVNWGA operator under a SVNN environment. Finally, an illustrative example about the selection problem of investment alternatives is given to demonstrate the application and feasibility of the developed approach.Keywords: single-valued neutrosophic number; Dombi operation; single-valued neutrosophic Dombi weighted arithmetic average (SVNDWAA) operator; single-valued neutrosophic Dombi weighted geometric average (SVNDWGA) operator; multiple attribute decision-making

Abstract:In nature, the mechanical properties of geological bodies are very complex, and its various mechanical parameters are vague, incomplete, imprecise, and indeterminate. In these cases, we cannot always compute or provide exact/crisp values for the joint roughness coefficient (JRC), which is a quite crucial parameter for determining the shear strength in rock mechanics, but we need to approximate them. Hence, we need to investigate the anisotropy and scale effect of indeterminate JRC values by neutrosophic number (NN) functions, because the NN is composed of its determinate part and the indeterminate part and is very suitable for the expression of JRC data with determinate and/or indeterminate information. In this study, the lower limit of JRC data is chosen as the determinate information, and the difference between the lower and upper limits is chosen as the indeterminate information. In this case, the NN functions of the anisotropic ellipse and logarithmic equation of JRC are developed to reflect the anisotropy and scale effect of JRC values. Additionally, the NN parameter ψ is defined to quantify the anisotropy of JRC values. Then, a two-variable NN function is introduced based on the factors of both the sample size and measurement orientation. Further, the changing rates in various sample sizes and/or measurement orientations are investigated by their derivative and partial derivative NN functions. However, an actual case study shows that the proposed NN functions are effective and reasonable in the expression and analysis of the indeterminate values of JRC. Obviously, NN functions provide a new, effective way for passing from the classical crisp expression and analyses to the neutrosophic ones.

Abstract:A refined single-valued/interval neutrosophic set is very suitable for the expression and application of decision-making problems with both attributes and sub-attributes since it is described by its refined truth, indeterminacy, and falsity degrees. However, existing refined single-valued/interval neutrosophic similarity measures and their decision-making methods are scarcely studied in existing literature and cannot deal with this decision-making problem with the weights of both attributes and sub-attributes in a refined interval and/or single-valued neutrosophic setting. To solve the issue, this paper firstly introduces a refined simplified neutrosophic set (RSNS), which contains the refined single-valued neutrosophic set (RSVNS) and refined interval neutrosophic set (RINS), and then proposes vector similarity measures of RSNSs based on the Jaccard, Dice, and cosine measures of simplified neutrosophic sets in vector space, and the weighted Jaccard, Dice, and cosine measures of RSNSs by considering weights of both basic elements and sub-elements in RSNS. Further, a decision-making method with the weights of both attributes and sub-attributes is developed based on the weighted Jaccard, Dice, and cosine measures of RSNSs under RSNS (RINS and/or RSVNS) environments. The ranking order of all the alternatives and the best one can be determined by one of weighted vector similarity measures between each alternative and the ideal solution (ideal alternative). Finally, an actual example on the selecting problem of construction projects illustrates the application and effectiveness of the proposed method.

In this paper, we study some semi-closed 1-set-contractive operators A and investigate the boundary conditions under which the topological degrees of 1-set contractive fields, deg  are equal to 1. Correspondingly, we can obtain some new fixed point theorems for 1-set-contractive operators which extend and improve many famous theorems such as the Leray-Schauder theorem, and operator equation, etc. Lemma 2.1 generalizes the famous theorem. The calculation of topological degrees and index are important things, which combine the existence of solution of for integration and differential equation and or approximation by iteration technique. So, we apply the effective modification of He's variation iteration method to solve some nonlinear and linear equations are proceed to examine some a class of integral-differential equations, to illustrate the effectiveness and convenience of this method.

In this paper, the authors study the blow-up of solution for a class of nonlinear Schrodinger equation for some initial boundary problem. On the other hand, the authors give out some analyses and that new conclusion by Eigen-function method. In last section, the authors check the nonlinear parameter for light rule power by using of parameter method to get ground state and excite state correspond case, and discuss the global attractor of some fraction order case, and combine numerical test. To illustrate this physics meaning in dimension d = 1, 2 case. So, by numerable solution to give out these wave expression.

Abstract:In this paper, the author consider conditions of that initial boundary value to study the dynamic behavior for Generalized Burgers equation, and give these system for that new conditions of blow-up, and Extinction Phenomenon of these equation with in rich using engineering. Such as, these theories and method can be used in other similar systems, they have a wide range of applications. Finale, some numerical simulations are carried out to support these new results. The authors extend dynamic behavior and the critical value to continuum more previous work [3, 9, 13] for in-depth achieve for apply value.