Mathematics

In this paper, we introduce a Laplace-type integral transform called the Shehu transform which is a generalization of the Laplace and the Sumudu integral transforms for solving differential equations in the time domain. The proposed integral transform is successfully derived from the classical Fourier integral transform and is applied to both ordinary and partial differential equations to show its simplicity, efficiency, and the high accuracy.

We define the Pythagorean fuzzy parameterized soft set and investigate some properties of the new set. Further, we propose to the solution of decision-making application for the Pythagorean fuzzy parameterized soft set and other related concepts.

By a non-Gaussian integral we mean integral of the product of an arbitrary function and exponent of a polynomial. We develop a theory of such integrals, which generalizes and simplifies the theory of general hypergeometric functions in the sense of I. M. Gelfand et al.

The interval numbers is the set of compact intervals of R with addition and multiplication operation, which are very useful for solving calculations where there are intervals of error or uncertainty, however, it lacks an algebraic structure with an inverse element, both additive and multiplicative This fundamental disadvantage results in overestimation of solutions in an interval equation or also overestimation of the image of a function over square regions. In this article we will present a solution to this problem, through a morphism that preserves both the addiction and the multiplication between the space of the interval numbers to the space of square diagonal matrices.

We study inhomogeneous nonlinear second-order differential equations in one dimension. The inhomogeneities can be point sources or continuous source distributions. We consider second order differential equations of type ϕ ′′ (x)+V(ϕ(x))=Qδ(x) , where V(ϕ) is a continuous, differentiable, analytic function and Qδ(x) is a point source. In particular we study cubic functions of the form V(ϕ(x))=Aϕ(x)+B ϕ 3 (x) . We show that Green functions can be determined for modifications of such cubic equations, and that such Green's functions can be used to determine the solutions for cases where the point source is replaced by a continuous source distribution.

According to harmonic analysis (Fourier analysis), any function f(x) , periodic over the interval [−L,L] , which satisfies the Dirichlet conditions, can be developed into an infinite sum (known in the literature as the trigonometric series, and for which, for reasons which will become evident in the course of this work, we will use the name of sinusoidal series), consisting of the weighted components of a complete biortogonal base, formed of the unitary function 1, of the fundamental harmonics sin(πx/L) , even and cos(πx/L) , odd ( 2L -periodic functions) and of the secondary harmonics sin(nπx/L) and cos(nπx/L) (periodic functions, with period 2L/n , where n∈ Z + , positive integers). The coefficients of these expansions (Fourier coefficients) can be calculated using Euler formulas. We will generalize this statement and show that the function f(x) can also be developed into non-sinusoidal periodic series, formed from the sum of the weighted components of a complete, non-orthogonal base: the unit function 1, the fundamental quasi-harmonics g(x) , even and h(x) , odd ( 2L -periodic functions, with zero mean value over the definition interval) and the secondary quasi-harmonics g n (x) and h n (x) ( 2L/n -periodic functions), where n∈ Z + . The fundamental quasi-harmonics g(x) and h(x) are any functions which admit expansions in sinusoidal series (satisfy Dirichlet conditions, or belong to L 2 space). The coefficients of these expansions are obtained with the help of certain algebraic relationships between the Fourier coefficients of the expansions of the functions f(x) , g(x) and h(x) . In addition to their obvious theoretical importance, these types of expansions can have practical importance in the approximation of functions and in the numerical and analytical resolution of certain classes of differential equations.

The aim of this paper is to investigate the sufficient condition for the invariance of a normal curve on a smooth immersed surface under isometry. We also find the the deviations of the tangential and normal components of the curve with respect to the given isometry.

Catalan's constant and the lemniscate constants have been important mathematical constants of interest to the mathematical society, yet various properties are unknown. An important property of significant mathematical constants is whether they are normal numbers. This paper evaluates the normality of decimal and hexadecimal representations of current world record computations of digits for the Catalan's constant (600,000,000,100 decimal digits and 498,289,214,317 hexadecimal digits) and the arc length of a lemniscate with a=1 (600,000,000,000 decimal digits and 498,289,214,234 hexadecimal digits). All analyzed frequencies have been found to be persistent to the conjecture of Catalan's constant and the arc length of a lemniscate with a=1 being a normal number in bases 10 and 16.

Since the theory of rough sets was introduced by Zdzislaw Pawlak, several approaches have been proposed to combine rough set theory with fuzzy set theory. In this paper, we examine one of these approaches, namely fuzzy rough sets with crisp reference sets, from a lattice-theoretic point of view. We connect the lower and upper approximations of a fuzzy relation R to the approximations of the core and support of R . We also show that the lattice of fuzzy rough sets corresponding to a fuzzy equivalence relation R and the crisp subsets of its universe is isomorphic to the lattice of rough sets for the (crisp) equivalence relation E , where E is the core of R . We establish a connection between the exact (fuzzy) sets of R and the exact (crisp) sets of the support of R .

The question of a possible excitation and emergence of fractional type dynamics, as a more realistic framework for understanding emergence of complex systems, directly from a conventional integral order dynamics, in the form a continuous transition or deformation, is of significant interest. Although there have been a lot of activities in nonlinear, fractional or not, dynamical systems, the above question appears yet to be addressed systematically in the current literature. The present work may be considered to be a step forward in this direction. Based on a novel concept of asymptotic duality structure, we present here an extended analytical framework that would provide a scenario for realizing the above stated continuous deformation of integral order dynamics to a local fractional order dynamics on a fractal and fractional space. The related concepts of self dual and strictly dual asymptotics are introduced and there relevance in connection with smooth and nonsmooth deformation of the real line are pointed out. The relationship of the duality structure and renormalization group is examined. The ordinary derivation operator is shown to be invariant under this duality enabled renormalization group transformation, leading thereby to a {\em natural} realization of local fractional type derivative in a fractal space. As an application we discuss linear wave equation in one and two dimensions and show how the underlying integral order wave equation could be deformed and renormalized suitably to yield meaningful results for vibration of a fractal string or wave propagation in a region with fractal boundary.