This paper solves implicit differential equations involving Hilfer-Katugampola fractional derivatives with nonlocal, boundary, and impulsive conditions. In addition, some sufficient conditions are formulated for the existence and uniqueness of solutions to the given problem, and Hyers-Ulam stability results are also presented.

In this research paper, the authors studied some problems related to harmonic summability of double Fourier series on Nörlund summability method. These results constitute substantial extension and generalization of related work of Moricz [1] and Rhodes et al., [2]. We also constructed a new result on \((N,p^{(1)}_b,p^{(2)}_a)\) by regular N\”orlund method of summability.

The aim of this paper is to present an optimal control problem to reduce the MDR-TB (multidrug-resistant tuberculosis) and XDR-TB (extensively drug-resistant TB) cases, using controls in these compartments and controlling reinfection/reactivation of the bacteria. The model used studies the efficacy of the tuberculosis treatment taking into account the influence of HIV/AIDS and diabetes, and we prove the global stability of the disease-free equilibrium point based on the behavior of the basic reproduction number. Various control strategies are proposed with the combinations of controls. We show the existence of optimal control using Pontryagin’s maximum principle. We solve the optimality system numerically with an algorithm based on forward/backward Runge-Kutta method of the fourth-order. The numerical results indicate that the implementation of the strategy that activates all controls and of type I (starting with the highest controls) is the most cost-effective of the strategies studied. This strategy reduces significantly the number of MDR-TB and XDR-TB cases in all sub-populations, which is an important factor in combating tuberculosis and its resistant strains.

Repeated integration is a major topic of integral calculus. In this article, we study repeated integration. In particular, we study repeated integrals and recurrent integrals. For each of these integrals, we develop reduction formulae for both the definite as well as indefinite form. These reduction formulae express these repetitive integrals in terms of single integrals. We also derive a generalization of the fundamental theorem of calculus that expresses a definite integral in terms of an indefinite integral for repeated and recurrent integrals. From the recurrent integral formulae, we derive some partition identities. Then we provide an explicit formula for the \(n\)-th integral of \(x^m(\ln x)^{m’}\) in terms of a shifted multiple harmonic star sum. Additionally, we use this integral to derive new expressions for the harmonic sum and repeated harmonic sum.

In 2006, Konstantinova proposed the marginal entropy of a graph based on the Wiener index. In this paper, we obtain the marginal entropy of the complete multipartite graphs, firefly graphs, lollipop graphs, clique-chain graphs, Cartesian product and join of two graphs, which extends the results of ¸Sahin.

The concept of Lucky colorings of a graph is used to introduce the notion of the Lucky \(k\)-polynomials of null graphs. We then give the Lucky \(k\)-polynomials for complete split graphs and generalized star graphs. Finally, further problems of research related to this concept are discussed.

In the present communication, we introduce the theory of Type-II generalized Pythagorean bipolar fuzzy soft sets and define complementation, union, intersection, AND, and OR. The Type-II generalized Pythagorean bipolar fuzzy soft sets are presented as a generalization of soft sets. We showed De Morgan’s laws, associate laws, and distributive laws in Type-II generalized Pythagorean bipolar fuzzy soft set theory. Also, we advocate an algorithm to solve the decision-making problem based on a soft set model.

A \(k\)-plane tree is a tree drawn in the plane such that the vertices are labeled by integers in the set \(\{1,2,\ldots,k\}\), the children of all vertices are ordered, and if \((i,j)\) is an edge in the tree, where \(i\) and \(j\) are labels of adjacent vertices in the tree, then \(i+j\leq k+1\). In this paper, we construct bijections between these trees and the sets of \(k\)-noncrossing increasing trees, locally oriented \((k-1)\)-noncrossing trees, Dyck paths, and some restricted lattice paths.

TEMO = topological effect on molecular orbitals was discovered by Polansky and Zander in 1982, in connection with the eigenvalues of molecular graphs. Eventually, analogous regularities were established for a variety of other topological indices. We now show that a TEMO-type regularity also holds for the Sombor index (\(SO\)): For the graphs \(S\) and \(T\), constructed by connecting a pair of vertex-disjoint graphs by two edges, \(SO(S) < SO(T)\) holds. Analogous relations are verified for several other degree-based graph invariants.

This study focused on line integral and its applications. The study was designed to show the areas where line integral is applicable and point out the role of line integral in solving practical problems. The study found out that space curves, and the concepts of scalar and vector fields are basic concepts to deal line integral. Also, the study found out that line integral is used to calculate mass, center of mass and moments of inertia of a wire, work done by a force on an object moving in a vector field, magnetic field around a conduct, voltage generated in a loop, length of a curve, area of a region bounded by a closed curve, and volume of a solid formed by rotating a closed curve about the \(x-\)axis.