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The notion of dual uniformity is introduced on UC(Y,Z)UC(Y,Z), the uniform space of uniformly continuous mappings between YY and ZZ, where (Y,V)(Y,{\mathcal{V}}) and (Z,U)(Z,{\mathcal{U}}) are two uniform spaces. It is shown that a function space uniformity on UC(Y,Z)UC(Y,Z) is admissible (resp. splitting) if and only if its dual uniformity on UZ(Y)={f2−1(U)∣f∈UC(Y,Z),U∈U}{{\mathcal{U}}}_{Z}(Y)=\{{f}_{2}^{-1}(U)\hspace{0.33em}| \hspace{0.33em}f\in UC(Y,Z),U\in {\mathcal{U}}\} is admissible (resp. splitting). It is also shown that a uniformity on UZ(Y){{\mathcal{U}}}_{Z}(Y) is admissible (resp. splitting) if and only if its dual uniformity on UC(Y,Z)UC(Y,Z) is admissible (resp. splitting). Using duality theorems, it is also proved that the greatest splitting uniformity and the greatest splitting family open uniformity exist on UZ(Y){{\mathcal{U}}}_{Z}(Y) and UC(Y,Z)UC(Y,Z), respectively, and these two uniformities are mutually dual splitting uniformities.

The aim of this work is to solve a numerical computation of the neutral fractional functional integro-differential equation based on a new approach to the Legendre wavelet method. The concept of fractional derivatives was examined in the sense of Caputo. The properties of the Legendre wavelet and function approximation were employed to determine the approximate solution of a given dynamical system. Moreover, the error estimations and convergence analysis of the truncated Legendre wavelet expansion for the proposed problem were discussed. The validity and applicability of this proposed technique to numerical computation were shown by illustrative examples. Eventually, the results of this technique demonstrate its great effectiveness and reliability. [ABSTRACT FROM AUTHOR]

In this paper, we discussed some qualitative properties of solutions to a thermostat system in the framework of a novel mathematical model designed by the new (p; q)-derivatives in fractional post-quantum calculus. We transformed the existing standard model into a new control thermostat system with the help of the Caputo-like (p; q)-derivatives. By the properties of the (p; q)-gamma function and applying the fractional Riemann-Liouville-like (p; q)-integral, we obtained the equivalent (p; q)-integral equation corresponding to the given Caputo-like post-quantum boundary value problem ((p; q)-BOVP) of the thermostat system. To conduct an analysis on the existence of solutions to this (p; q)-system, some theorems were proved based on the fixed point methods and the stability analysis was done from the Ulam-Hyers point of view. In the applied examples, we used numerical data to simulate solutions of the Caputo-like (p; q)-BOVPs of the thermostat system with respect to different parameters. The effects of given parameters in the model will show the performance of the thermostat system. [ABSTRACT FROM AUTHOR]

We investigate the bounded travelling wave solutions of the Biswas-Arshed model (BAM) including the low group velocity dispersion and excluding the self-phase modulation. We integrate the nonlinear structure of the model to obtain bounded optical solitons which pass through the optical fibers in the non-Kerr media. The bifurcation technique of the dynamical system is used to achieve the parameter bifurcation sets and split the parameter space into various areas which correspond to different phase portraits. All bounded optical solitons and bounded periodic wave solutions are identified and derived conforming to each region of these phase portraits. We also apply the extended sinh-Gordon equation expansion and the generalized Kudryashov integral schemes to obtain additional bounded optical soliton solutions of the BAM nonlinearity. We present more bounded optical shock waves, the bright-dark solitary wave, and optical rogue waves for the structure model via these schemes in different aspects. [ABSTRACT FROM AUTHOR]