In this study, we present a new analytical numerical technique for solving a class of time Fractional Differential Equations (FDEs) with variable coefficients based on the generalized Taylor series formula in the Caputo sense. This method provided the solution in the form of a rapidly convergent power series under a multiple fractional differentiability with easily computable components. An efficacious experiment is given to guarantee the procedure, to illustrate the theoretical statements of the present technique and to show its potentiality, generality and superiority for solving wide range of FDEs. The results reveal that the method is easy to implement, very effective, fully compatible with the complexity of such problems, straightforward and simple.

In this paper, an application of reproducing kernel Hilbert space (RKHS) method is applied to solve second-order integrodifferential equation of Volterra type. The analytical solution is represented in the form of series in the reproducing kernel space. The n−truncation approximation u n (x) is obtained and proved to converge to the analytical solution u(x). Moreover, the presented method has an advantages that it is possible to pick any point in the interval domain and as well the approximate solution and its derivatives will be applicable Numerical experiments are displayed to illustrate the validity, accuracy, efficiency and applicability of the proposed method. Results indicates that our technique is simple, straightforward and effective.

No abstract

The rough set principle was proposed as a methodology to cope with vagueness or uncertainty of data in the information systems. Day by day, this theory has proven its efficiency in handling and modeling many real-life problems. To contribute to this area, we present new topological approaches as a generalization of Pawlak’s theory by using j-adhesion neighborhoods and elucidate the relationship between them and some other types of approximations with the aid of examples. Topologically, we give another generalized rough approximation using near open sets. Also, we generate generalized approximations created from the topological models of j-adhesion approximations. Eventually, we compare the approaches given herein with previous ones to obtain a more affirmative solution for decision-making problems.

<abstract><p>This paper is devoted to study the concepts of compactness, Lindelöfness and connectedness via the class of soft somewhat open sets which represents one of the generalizations of soft open sets. Beside investigation the main properties of these concepts, it is demonstrated, with the help of examples, that some properties of their counterparts via soft open sets are invalid. Also, the relationships between these concepts and their counterparts defined in classical topology (which is studied herein under the name of parametric topology) are discussed in detail. Moreover, we provide the sufficient conditions that guarantee the equivalence between them. In this regard, it is proved that all introduced types of soft compact and Lindelöf spaces are transmitted to all parametric topologies without imposing any conditions, whereas the converse holds true under the conditions of a full soft topology and a finite (countable) set of parameters. These characterizations represent a unique behavior of these spaces compared to the other types defined by celebrated generalizations of soft open sets. Also, there is no relationship associating soft $ sw $-connectedness with its counterparts via parametric topologies. We successfully describe soft $ sw $-disconnectedness using soft open sets instead of soft $ sw $-open sets and consequently prove that the concepts of soft $ sw $-connected and soft hyperconnected spaces are identical. In conclusion, the obtained results show that the framework given in this manuscript enriches and generalizes the previous works, and has a good application prospect.</p></abstract>

Soft topological spaces (STSs) have received a lot of attention recently, and numerous soft topological ideas have been created from differing viewpoints. Herein, we put forth a new class of generalizations of soft open sets called “weakly soft semi-open subsets” following an approach inspired by the components of a soft set. This approach opens the door to reformulating the existing soft topological concepts and examining their behaviors. First, we deliberate the main structural properties of this class and detect its relationships with the previous generalizations with the assistance of suitable counterexamples. In addition, we probe some features that are obtained under some specific stipulations and elucidate the properties of the forgoing generalizations that are missing in this class. Next, we initiate the interior and closure operators with respect to the classes of weakly soft semi-open and weakly soft semi-closed subsets and look at some of their fundamental characteristics. Ultimately, we pursue the concept of weakly soft semi-continuity and furnish some of its descriptions. By a counterexample, we elaborate that some characterizations of soft continuous functions are invalid for weakly soft semi-continuous functions.

Rough set philosophy is a significant methodology in the knowledge discovery of databases. In the present paper, we suggest new sorts of rough set approximations using a multi-knowledge base; that is, a family of the finite number of general binary relations via different methods. The proposed methods depend basically on a new neighborhood (called basic-neighborhood). Generalized rough approximations (so-called, basic-approximations) represent a generalization to Pawlak’s rough sets and some of their extensions as confirming in the present paper. We prove that the accuracy of the suggested approximations is the best. Many comparisons between these approaches and the previous methods are introduced. The main goal of the suggested techniques was to study the multi-information systems in order to extend the application field of rough set models. Thus, two important real-life applications are discussed to illustrate the importance of these methods. We applied the introduced approximations in a set-valued ordered information system in order to be accurate tools for decision-making. To illustrate our methods, we applied them to find the key foods that are healthy in nutrition modeling, as well as in the medical field to make a good decision regarding the heart attacks problem.

Abstract:In this article, the reproducing kernel Hilbert space 4 2 W [0, 1] is employed for solving a class of third-order periodic boundary value problem by using fitted reproducing kernel algorithm. The reproducing kernel function is built to get fast accurately and efficiently series solutions with easily computable coefficients throughout evolution the algorithm under constraint periodic conditions within required grid points. The analytic solution is formulated in a finite series form whilst the truncated series solution is given to converge uniformly to analytic solution. The reproducing kernel procedure is based upon generating orthonormal basis system over a compact dense interval in sobolev space to construct a suitable analytical-numerical solution. Furthermore, experiments results of some numerical examples are presented to illustrate the good performance of the presented algorithm. The results indicate that the reproducing kernel procedure is powerful tool for solving other problems of ordinary and partial differential equations arising in physics, computer and engineering fields.