Application of ( q , τ )‐Bernoulli Interpolation to the Spectral Solution of Quantum Differential Equations

Global structure and asymptotic behavior of weak solutions to flood wave equations

Modal Analysis of Fluid Flows: An Overview

We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class is in bijection with a subset of negative definite functions and we name it the class of generalized Laguerre semigroups. Our approach, which goes beyond the framework of perturbation theory, is based on an in-depth and original analysis of an intertwining relation that we establish between this class and a self-adjoint Markov semigroup, whose spectral expansion is expressed in terms of the classical Laguerre polynomials. As a by-product, we derive smoothness properties for the solution to the associated Cauchy problem as well as for the heat kernel. Our methodology also reveals a variety of possible decays, including the hypocoercivity type phenomena, for the speed of convergence to equilibrium for this class and enables us to provide an interpretation of these in terms of the rate of growth of the weighted Hilbert space norms of the spectral projections. Depending on the analytic properties of the aforementioned negative definite functions, we are led to implement several strategies, which require new developments in a variety of contexts, to derive precise upper bounds for these norms.

Structural behaviour of an open car park under real fire scenarios

Realizing the behavior of the solution in the asymptotic situations is essential for repetitive applications in the control theory and modeling of the real-world systems. This study discusses a robust and definitive attitude to find the interval approximate asymptotic solutions of fractional differential equations (FDEs) with the Atangana-Baleanu (A-B) derivative. In fact, such critical tasks require to observe precisely the behavior of the noninterval case at first. In this regard, we initially shed light on the noninterval cases and analyze the behavior of the approximate asymptotic solutions, and then, we introduce the A-B derivative for FDEs under interval arithmetic and develop a new and reliable approximation approach for fractional interval differential equations with the interval A-B derivative to get the interval approximate asymptotic solutions. We exploit Laplace transforms to get the asymptotic approximate solution based on the interval asymptotic A-B fractional derivatives under interval arithmetic. The techniques developed here provide essential tools for finding interval approximation asymptotic solutions under interval fractional derivatives with nonsingular Mittag-Leffler kernels. Two cases arising in the real-world systems are modeled under interval notion and given to interpret the behavior of the interval approximate asymptotic solutions under different conditions as well as to validate this new approach. This study highlights the importance of the asymptotic solutions for FDEs regardless of interval or noninterval parameters.

Fractional calculus, in the understanding of its theoretical and real-world presentations in numerous regulations, for example, astronomy and manufacturing problems, is discovered to be accomplished of pronouncing phenomena owning long range memory special effects that are challenging to handle through traditional integer-order calculus. Nearby an increasing concentration has been in the modification of fractional calculus as a successful modelling instrument for complicated systems, contributing to innovative viewpoints in their dynamical investigation and regulator. This improvement in the methodical knowledge is established by an enormous quantity of evens developing on the subject, manuscripts, and presentations in the past years. Nevertheless, countless singularities still pose significant confronts to the apprehensive population and fractional calculus appears to be plausibly contestant to incorporate larger exemplars through detaching graceful dependent on the explanation of involvedness. This special issue contains papers about recent theoretical development and methods and applications results on the topics in almost all branches of sciences and engineering. We have received 56 papers during the submission period. Five were withdrawn; 34 were rejected including the papers submitted to the member of our editorial board. Only 17 good papers were accepted for publication. The papers of this special issue cover some new algorithms and procedures designed to explore conventional, fractional, and time-scales differential equations of general interest. New understandings of existences and uniqueness theorems of some differential equations were also offered. In the following we give the brief summary of the content of the special issue. The existence and uniqueness theorems for impulsive fractional equations with the two-point and integral boundary conditions and sufficient condition on the fractional integral for the convergence of a function were presented. Besides the stability, boundedness, and Lagrange stability of fractional differential equation with initial time difference, stability of nonlinear Dirichlet BVPs governed by fractional Laplacian was proposed. A novel study on the singular perturbations fractional equations, analysis of a fractional-order couple model with acceleration in feelings, q-Sumudu transforms of q-analogues of Bessel functions, certain fractional integral formulas involving the product of generalized Bessel functions, and an expansion formula with higher-order derivatives for fractional operators of variable order were investigated in detail. A novel study underpinning construction of solution for fractional differential equations such as decomposition method for time fractional reactional-diffusion equation and high-order compact difference scheme for numerical solution of time fractional heat equation, a procedure to construct exact solutions of nonlinear fractional differential equations, were presented. An investigation on impulsive multiterm fractional differential equations and multiple positive solutions for nonlinear fractional boundary value problems were undertaken. The editorial board trust that the set of nominated papers will offer readers an opportune renovate of significant investigation subjects and may also operate as a policy for encouraging additional contribution in this fast evolving ground.

Potassium resources are abundant in the brine of chloride-type salt lakes. The main challenge in the efficient separation and extraction of potassium from salt lakes lies in the insufficient understanding of the structure and crystallization behavior of brine solutions and their correlation. In the present work, X-ray scattering (XRS) and computational simulation methods were used to study the microstructure of KCl and MgCl2 mixed solutions, including the hydration and association structures of ions in the solutions. Furthermore, infrared (IR) spectroscopy was used to further study the crystallization behavior of solution droplets. The results indicate that the hydrogen bond network structure is disrupted as the mass fraction of MgCl2 increases. The addition of MgCl2 causes Mg2+ to compete with K+ for Cl- in solutions, hindering K+-Cl- association and forming contact K+-Cl--Mg2+ clusters, which results in a slower precipitation and crystallization rate of mixed solutions compared with that of aqueous KCl solutions. This study is expected to provide theoretical guidance for the efficient separation and extraction process of potassium resources in salt lake brine.

Numerical solution of two-dimensional fractional-order partial differential equations using hybrid functions

Fractional calculus is the integration and differentiation to an arbitrary or fractional order. The techniques of fractional calculus are not commonly taught in engineering curricula since physical laws are expressed in integer order notation. Dr. Richard Magin (2006) notes how engineers occasionally encounter dynamic systems in which the integer order methods do not properly model the physical characteristics and lead to numerous mathematical operations. In the following study, the application of fractional order calculus to approximate the angular position of the disk oscillating in a Newtonian fluid was experimentally validated. The proposed experimental study was conducted to model the nonlinear response of an oscillating system using fractional order calculus. The integer and fractional order mathematical models solved the differential equation of motion specific to the experiment. The experimental results were compared to the integer order and the fractional order analytical solutions. The fractional order mathematical model in this study approximated the nonlinear response of the designed system by using the Bagley and Torvik fractional derivative. The analytical results of the experiment indicate that either the integer or fractional order methods can be used to approximate the angular position of the disk oscillating in the homogeneous solution. The following research was in collaboration with Dr. Richard Mark French, Dr. Garcia Bravo, and Rajarshi Choudhuri, and the experimental design was derived from the previous experiments conducted in 2018.

Using the recently developed effective interaction potentials between polyelectrolyte stars, we examine the structure and phase behavior of solutions of the same. The effective interaction is ultrasoft and density dependent, owing to the integration of the counterionic degrees of freedom. The latter contribute extensive volume terms that must be taken into account in drawing the phase diagram of the system. The structural behavior of the uniform fluid is characterized by anomalous structure factors, akin to those found previously for solutions of uncharged star polymers. The phase diagram of the system is very rich, featuring a fluid phase at low arm numbers of the stars, two reentrant melting regions, as well as a variety of crystal structures with unusual symmetry. The physical origin of these features can be traced back to the ultrasoft nature of the effective interaction potential.

Generalized fractional power series solutions for fractional differential equations

Post-buckling behaviour of slender structures with a bi-linear bending moment–curvature relationship

A new approach for space-time fractional partial differential equations by residual power series method

In this paper, a numerical approach for solving systems of nonlinear fractional differential equations (FDEs) is presented Using the Euler wavelets technique and associated operational matrices for fractional integration, we try to solve those systems of FDEs. The method’s major objective is to transform the nonlinear FDE into a nonlinear system of algebraic equations that is straightforward to solve with matrix techniques. The Euler wavelets are constructed using Euler polynomials, which have fewer terms than most other polynomials used to construct other types of wavelets, therefore, using Euler wavelets for the numerical approach provides sparse operational matrices. Thanks to the sparsity of those operational matrices, the proposed numerical approach requires less computation and takes less time to evaluate. The approach described here is also applicable to systems of fractional differential equations with variable orders. To illustrate the strength and performance of the method, four numerical examples are provided.

This paper presents an algorithm to obtain numerically stable differentiation matrices for approximating the left- and right-sided Caputo-fractional derivatives. The proposed differentiation matrices named fractional Chebyshev differentiation matrices are obtained using stable recurrence relations at the Chebyshev–Gauss–Lobatto points. These stable recurrence relations overcome previous limitations of the conventional methods such as the size of fractional differentiation matrices due to the exponential growth of round-off errors. Fractional Chebyshev collocation method as a framework for solving fractional differential equations with multi-order Caputo derivatives is also presented. The numerical stability of spectral methods for linear fractional-order differential equations (FDEs) is studied by using the proposed framework. Furthermore, the proposed fractional Chebyshev differentiation matrices obtain the fractional-order derivative of a function with spectral convergence. Therefore, they can be used in various spectral collocation methods to solve a system of linear or nonlinear multi-ordered FDEs. To illustrate the true advantages of the proposed fractional Chebyshev differentiation matrices, the numerical solutions of a linear FDE with a highly oscillatory solution, a stiff nonlinear FDE, and a fractional chaotic system are given. In the first, second, and forth examples, a comparison is made with the solution obtained by the proposed method and the one obtained by the Adams–Bashforth–Moulton method. It is shown the proposed fractional differentiation matrices are highly efficient in solving all the aforementioned examples.