Q-homotopy analysis method for time-fractional Newell–Whitehead equation and time-fractional generalized Hirota–Satsuma coupled KdV system

In this paper,two types of fractional nonlinear equations in Caputo sense,time-fractional Newell–Whitehead equation(FNWE)and time-fractional generalized Hirota–Satsuma coupled KdV system(HS-cKdVS),are investigated by means of the q-homotopy analysis method(q-HAM).The approximate solutions of the proposed equations are constructed in the form of a convergent series and are compared with the corresponding exact solutions.Due to the presence of the auxiliary parameter h in this method,just a few terms of the series solution are required in order to obtain better approximation.For the sake of visualization,the numerical results obtained in this paper are graphically displayed with the help of Maple.

APPLICATION OF MULTI-DIMENSIONAL OF CONFORMABLE SUMUDU DECOMPOSITION METHOD FOR SOLVING CONFORMABLE SINGULAR FRACTIONAL COUPLED BURGER'S EQUATION

In this article,several theorems of fractional conformable derivatives and triple Sumudu transform are given and proved.Based on these theorems,a new conformable triple Sumudu decomposition method(CTSDM)is intrduced for the solution of singular two-dimensional conformable functional Burger's equation.This method is a combination of the decomposition method(DM)and Conformable triple Sumudu transform.The exact and approximation solutions obtained by using the suggested method in the sense of conformable.Particular examples are given to clarify the possible application of the achieved results and the exact and approximate solution are sketched by using Matlab software.

New approximate solution for time-fractional coupled KdV equations by generalised differential transform method

In this paper, the genera]ised two-dimensiona] differentia] transform method （DTM） of solving the time-fractiona] coupled KdV equations is proposed. The fractional derivative is described in the Caputo sense. The presented method is a numerical method based on the generalised Taylor series expansion which constructs an analytical solution in the form of a polynomial. An illustrative example shows that the genera]ised two-dimensional DTM is effective for the coupled equations.

Approximate Analytical Solutions of Fractional Coupled mKdV Equation by Homotopy Analysis Method

In this paper, the approximate analytical solutions of the fractional coupled mKdV equation are obtained by homotopy analysis method (HAM). The method includes an auxiliary parameter which provides a convenient way of adjusting and controlling the convergence region of the series solution. The suitable value of auxiliary parameter is determined and the obtained results are presented graphically.

The fast method and convergence analysis of the fractional magnetohydrodynamic coupled flow and heat transfer model for the generalized second-grade fluid

In this paper,we first establish a new fractional magnetohydrodynamic(MHD)coupled flow and heat transfer model for a generalized second-grade fluid.This coupled model consists of a fractional momentum equation and a heat conduction equation with a generalized form of Fourier law.The second-order fractional backward difference formula is applied to the temporal discretization and the Legendre spectral method is used for the spatial discretization.The fully discrete scheme is proved to be stable and convergent with an accuracy of O(τ^(2)+N-r),whereτis the time step-size and N is the polynomial degree.To reduce the memory requirements and computational cost,a fast method is developed,which is based on a globally uniform approximation of the trapezoidal rule for integrals on the real line.The strict convergence of the numerical scheme with this fast method is proved.We present the results of several numerical experiments to verify the effectiveness of the proposed method.Finally,we simulate the unsteady fractional MHD flow and heat transfer of the generalized second-grade fluid through a porous medium.The effects of the relevant parameters on the velocity and temperature are presented and analyzed in detail.

The Quantum Microverse: A Prime Number Framework for Understanding the Universe

This study aims to demonstrate a proof of concept for a novel theory of the universe based on the Fine Structure Constant (α), derived from n-dimensional prime number property sets, specifically α = 137 and α = 139. The FSC Model introduces a new perspective on the fundamental nature of our universe, showing that α = 137.036 can be calculated from these prime property sets. The Fine Structure Constant, a cornerstone in Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD), implies an underlying structure. This study identifies this mathematical framework and demonstrates how the FSC model theory aligns with our current understanding of physics and cosmology. The results unveil a hierarchy of α values for twin prime pairs U{3/2} through U{199/197}. These values, represented by their fraction parts α♊ (e.g., 0.036), define the relative electromagnetic forces driving quantum energy systems. The lower twin prime pairs, such as U{3/2}, exhibit higher EM forces that decrease as the twin pairs increase, turning dark when they drop below the α♊ for light. The results provide classical definitions for Baryonic Matter/Energy, Dark Matter, Dark Energy, and Antimatter but mostly illustrate how the combined α♊ values for three adjacent twin primes, U{7/5/3/2} mirrors the strong nuclear force of gluons holding quarks together.

GENERALIZED FRACTIONAL TRACE VARIATIONAL IDENTITY AND A NEW FRACTIONAL INTEGRABLE COUPLINGS OF SOLITON HIERARCHY

Based on fractional isospectral problems and general bilinear forms, the gener-alized fractional trace identity is presented. Then, a new explicit Lie algebra is introduced for which the new fractional integrable couplings of a fractional soliton hierarchy are derived from a fractional zero-curvature equation. Finally, we obtain the fractional Hamiltonian structures of the fractional integrable couplings of the soliton hierarchy.

IMPULSIVE EXPONENTIAL SYNCHRONIZATION OF FRACTIONAL-ORDER COMPLEX DYNAMICAL NETWORKS WITH DERIVATIVE COUPLINGS VIA FEEDBACK CONTROL BASED ON DISCRETE TIME STATE OBSERVATIONS

This article aims to address the global exponential synchronization problem for fractional-order complex dynamical networks(FCDNs)with derivative couplings and impulse effects via designing an appropriate feedback control based on discrete time state observations.In contrast to the existing works on integer-order derivative couplings,fractional derivative couplings are introduced into FCDNs.First,a useful lemma with respect to the relationship between the discrete time observations term and a continuous term is developed.Second,by utilizing an inequality technique and auxiliary functions,the rigorous global exponential synchronization analysis is given and synchronization criterions are achieved in terms of linear matrix inequalities(LMIs).Finally,two examples are provided to illustrate the correctness of the obtained results.

New exact solutions for the time fractional coupled Boussinesq-Burger equation and approximate long water wave equation in shallow water

The aim of the article is to construct exact solutions for the time fractional coupled Boussinesq-Burger and approximate long water wave equations by using the generalized Kudryashov method.The fractional differential equation is converted into ordinary differential equations with the help of fractional complex transform and the modified Riemann-Liouville derivative sense.Applying the generalized Kudryashov method through with symbolic computer maple package,numerous new exact solutions are successfully obtained.All calculations in this study have been established and verified back with the aid of the Maple package program.The executed method is powerful,effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions with the integer and fractional order.

Symmetric and antisymmetric vector solitons for the fractional quadric-cubic coupled nonlinear Schrodinger equation

The fractional quadric-cubic coupled nonlinear Schrodinger equation is concerned,and vector symmetric and antisymmetric soliton solutions are obtained by the square operator method.The relationship between the Lévy index and the amplitudes of vector symmetric and antisymmetric solitons is investigated.Two components of vector symmetric and antisymmetric solitons show a positive and negative trend with the Lévy index,respectively.The stability intervals of these solitons and the propagation constants corresponding to the maximum and minimum instability growth rates are studied.Results indicate that vector symmetric solitons are more stable and have better interference resistance than vector antisymmetric solitons.

Generalized solitary wave solutions to the time fractional generalized Hirota-Satsuma coupled KdV via new definition for wave transformation

In this paper,the(G′/G)-expansion method has been applied to explore new solitary wave solutions for the time fractional generalized Hirota-Satsuma coupled KdV(FGHSC KdV)system.The fractional derivative is described with the use of conformable derivative.The results show that this method is a very useful and effective mathematical tool for solving nonlinear conformable fractional equations arising in mathematical physics.As a result,this method can also be applied to other nonlinear conformable fractional differential equations.©2019 Shanghai Jiaotong University.Published by Elsevier B.V.

Evaluating the soil evaporation loss rate in a gravel-sand mulching environment based on stable isotopes data

In order to cope with drought and water shortages,the working people in the arid areas of Northwest China have developed a drought-resistant planting method,namely,gravel-sand mulching,after long-term agricultural practices.To understand the effects of gravel-sand mulching on soil water evaporation,we selected Baifeng peach(Amygdalus persica L.)orchards in Northwest China as the experimental field in 2021.Based on continuously collected soil water stable isotopes data,we evaluated the soil evaporation loss rate in a gravel-sand mulching environment using the line-conditioned excess(lc-excess)coupled Rayleigh fractionation model and Craig-Gordon model.The results show that the average soil water content in the plots with gravel-sand mulching is 1.86%higher than that without gravel-sand mulching.The monthly variation of the soil water content is smaller in the plots with gravel-sand mulching than that without gravel-sand mulching.Moreover,the average lc-excess value in the plots without gravel-sand mulching is smaller.In addition,the soil evaporation loss rate in the plots with gravel-sand mulching is lower than that in the plots without gravel-sand mulching.The lc-excess value was negative for both the plots with and without gravel-sand mulching,and it has good correlation with relative humidity,average temperature,input water content,and soil water content.The effect of gravel-sand mulching on soil evaporation is most prominent in August.Compared with the evaporation data of similar environments in the literature,the lc-excess coupled Rayleigh fractionation model is better.Stable isotopes evidence shows that gravel-sand mulching can effectively reduce soil water evaporation,which provides a theoretical basis for agricultural water management and optimization of water-saving methods in arid areas.

Kinematics of faults in Bengal Basin:Constraints from GPS measurements

The crustal deformation of Bengal Basin is constrained by motion on faults within the basin and those along the basin boundaries on all sides except the southern margin.Eocene Hinge Zone(EHZ),the most prominent structure within Bengal Basin,running through Kolkata-Ranaghat-Mymensingh,shows present day reactivation by hosting 8 earthquakes of magnitude 5.3 to 7.2 on or near its vicinity since 1906.Analysis of GPS derived velocities of stations around the hinge belt reveals a change in crustal kinematics across the belt.The predicted fault model that best fits the observed GPS station velocities on the surface shows that EHZ is a low angle east dipping fault which is locked to a depth of 10 km below the alluvium cover with a maximum slip rate deficit of 1.6±0.7 mm/yr in its northern part at a shallow depth from surface while the deficit in the southern part of EHZ is 1.0±0.6 mm/yr.The Garh-Moyna Khandaghosh fault running almost parallel to the western basin-shield margin does not show significant slip rate deficit.The basin-bounding faults are more active than the faults within and therefore they are greater sources of seismic hazard in the basin.The model results reveal that the Sylhet trough region in the northeastern corner is most vulnerable to future earthquakes as it undergoes deformation due to a maximum slip rate deficit of 9.5±2.4 mm/yr on the shallowly dipping locked megathrust along the eastern margin of the basin plus that of 7.4±2.6 mm/yr on the over thrusting Dapsi thrust and Dauki fault in the north.

THE FRACTIONAL QUADRATIC-FORM IDENTITY AND BI-HAMILTONIAN STRUCTURES OF AN INTEGRABLE COUPLING OF THE FRACTIONAL COUPLED BURGERS HIERARCHY

Based on a general isospectral problem of fractional order and the fractional quadratic-form identity by Yue and Xia, the new integrable coupling of fractional coupled Burgers hierarchy and its fractional bi-Hamiltonian structures are obtained.

Theory and application of numerical simulation method of capillary force enhanced oil production

A kind of second-order implicit upwind fractional step finite difference methods are presented for the numerical simulation of coupled systems for enhanced （chemical） oil production with capillary force in the porous media. Some techniques, e.g., the calculus of variations, the energy analysis method, the commutativity of the products of difference operators, the decomposition of high-order difference operators, and the theory of a priori estimate, are introduced. An optimal order error estimate in the l2 norm is derived. The method is successfully used in the numerical simulation of the enhanced oil production in actual oilfields. The simulation results are satisfactory and interesting.