The idea of the construction of Boehmians was initiated by the concept of regular operators. The construction of Boehmians is similar to the construction of the field of quotients and, in some cases, it just gives the field of quotients. In this article we consider two spaces of Boehmians. The strong space of Boehmians is continuously viewed in the space of general Boehmians. The Hartley transform is extended and obtained as a welldefined continuous mapping with respect to the ı convergence for which certain theorems have been proved. The article is ended up in defining the inverse Hartley transform and discussing some of its properties in detail.
Nonclassical quantum mechanics along with dispersive interactions of free particles, long-range boson stars, hydrodynamics, harmonic oscillator, shallow-water waves, and quantum condensates can be modeled via the nonlinear fractional Schrödinger equation. In this paper, various types of optical soliton wave solutions are investigated for perturbed, conformable space-time fractional Schrödinger model competed with a weakly nonlocal term. The fractional derivatives are described by means of conformable space-time fractional sense. Two different types of nonlinearity are discussed based on Kerr and dual power laws for the proposed fractional complex system. The method employed for solving the nonlinear fractional resonant Schrödinger model is the hyperbolic function method utilizing some fractional complex transformations. Several types of exact analytical solutions are obtained, including bright, dark, singular dual-power-type soliton and singular Kerr-type soliton solutions. Moreover, some graphical simulations of those solutions are provided for understanding the physical phenomena.
The space of Boehmians is constructed using an algebraic approach that utilizes convolution and approximate identities or delta sequences. A proper subspace can be identified with the space of distributions. In this paper, we first construct a suitable Boehmian space on which the Sumudu transform can be defined and the function space S can be embedded. In addition to this, our definition extends the Sumudu transform to more general spaces and the definition remains consistent for S elements. We also discuss the operational properties of the Sumudu transform on Boehmians and finally end with certain theorems for continuity conditions of the extended Sumudu transform and its inverse with respect to δ-and -convergence.MSC: Primary 54C40; 14E20; secondary 46E25; 20C20
In this article, we propose a new method that determines an efficient numerical procedure for solving second-order fuzzy Volterra integro-differential equations in a Hilbert space. This method illustrates the ability of the reproducing kernel concept of the Hilbert space to approximate the solutions of second-order fuzzy Volterra integro-differential equations. Additionally, we discuss and derive the exact and approximate solutions in the form of Fourier series with effortlessly computable terms in the reproducing kernel Hilbert space W 3 2 [a, b] ⊕ W .3 2 [a, b]. The convergence of the method is proven and its exactness is illustrated by three numerical examples.
This article aims to discuss a class of quaternion Fourier integral operators on certain set of generalized functions, leading to a method of discussing various integral operators on various spaces of generalized functions. By employing a quaternion Fourier integral operator on points closed to the origin, we introduce convolutions and approximating identities associated with the Fourier convolution product and derive classical and generalized convolution theorems. Working on such identities, we establish quaternion and ultraquaternion spaces of generalized functions, known as Boehmians, which are more general than those existed on literature. Further, we obtain some characteristics of the quaternion Fourier integral in a quaternion sense. Moreover, we derive continuous embeddings between the classical and generalized quaternion spaces and discuss some inversion formula as well. KEYWORDSBoehmian space, generalized quaterion space, quaternion, quaternion fourier PRELIMINARIESQuaternions were invented by William Hamilton as an extension to complex numbers. Soon after quaternions were discovered, Hamilton has derived connections between quaternion algebra and spatial rotations. Compared with the calculus of vectors, quaternions have slipped into realm of obscurity and are an example of a larger class of clifford's geometric algebras, complex encoding vector spaces and all their subspace elements.Quaternions are used in rotations, physical laws, and quantum mechanics as well. The ultimate reason for such awareness of the theory of quaternions can be referred to the fact that multiplying quaternions turns the sphere into a group, by the relation v → qvq −1 , and substitutions of matrices speeds up numerical computations and some operations.The field of generalized functions has been developed along the requirements of its applications in linear and nonlinear partial differential equations, geometry, mathematical physics, stochastic analysis, and in harmonic analysis, both in theoretical and numerical aspects. Generalized functions being a type of linear functionals are very important in making discontinuous functions more likely smooth that lead to an ample use in physics and engineering problems. The space of Boehmians is the space of generalized functions that recently defined by an algebraic construction alike to the field of quotients. 1-10 When the construction of Boehmians is employed on various function spaces and the multiplication is explained as a convolution, the construction yields different sets of generalized functions. However, the idea of delta sequences of supports shrinking to the origin is very useful in the construction of the Boehmian spaces, leading to a uniqueness theorem which can be interpreted for Boehmian spaces as an uncertainty principle.Results of this article are spread as follows. In Section 2, we give definitions and establish various theorems to generate a quaternion spaces whose elements are Boehmians and ultraBoehmians. Further, we derive an embedding between Math Meth Appl Sci. 201...
In this paper, we discuss Hartley-Hilbert and Fourier-Hilbert transforms on a certain class of generalized functions. The extended transforms considered in this article are shown to be well-defined, one-to-one, linear and continuous mappings with respect to δ and convergence. Certain theorems are also established.MSC: Primary 54C40; 14E20; secondary 46E25; 20C20
Our aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential equation based on a reproducing kernel algorithm with model endowed with a Caputo–Fabrizio fractional derivative. By means of such an approach, we utilize the Gram–Schmidt orthogonalization process to create an orthonormal set of bases that leads to an appropriate solution in the Hilbert space $\mathcal{H}^{2}[a,b]$ H 2 [ a , b ] . We investigate and discuss stability and convergence of the proposed method. The n-term series solution converges uniformly to the analytic solution. We present several numerical examples of potential interests to illustrate the reliability, efficacy, and performance of the method under the influence of the Caputo–Fabrizio derivative. The gained results have shown superiority of the reproducing kernel algorithm and its infinite accuracy with a least time and efforts in solving the fractional Abel-type model. Therefore, in this direction, the proposed algorithm is an alternative and systematic tool for analyzing the behavior of many nonlinear temporal fractional differential equations emerging in the fields of engineering, physics, and sciences.
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