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This paper proposes a novel exponential rational function method, a modification of the well-known exp-function method, to find exact solutions of the time fractional Cahn-Allen equation and the time fractional Phi-4 equation. The solution procedure is reduced to solve a system of algebraic equations, which is then solved by Wu?s method. The results show that the present method is effective, and can be applied to other fractional differential equations.

The objective of this study is to develop an analytical model for 1D gas‐water flow with evaporation of water into the mobile gas phase. The evaporation rate is proportional to the fluid‐fluid interfacial area, which is a function of the water saturation. Introduction of saturation‐dependent potential allows reducing the 2 × 2 non‐linear system of PDEs to one hyperbolic equation. The saturation distribution is obtained by non‐linear method of characteristics. The first integral allows for explicit expression of the vapor concentration versus saturation. It was shown that typical properties of rock‐CO2‐water system, the evaporation time has order of magnitude of millions of pore volumes injected. Close match was observed between a series of three corefloods and the analytical model, and the tuned model coefficients belong to their common intervals. This validates the developed model for 1D gas‐water flow with water evaporation into the injected gas. Plain Language Summary During injection of carbon dioxide or other gases into porous media containing small fractions of water, the water phase will slowly evaporate into the injected gas. This process is important as the gradual drying of the porous media near the injection site can lead to formation damage processes such as fines migration and salt precipitation, resulting in a significant decline in the ability to inject more gas. The evaporation process is dependent on the interfacial area between the gas and water. As the evaporation process progresses, this area gets smaller, and the evaporation rate slows down. In this work, we account for this effect directly, and solve the resulting differential equations, providing simple equations that can be used to predict the rate of water evaporation during gas injection. Key Points Exact solution of gas transport in porous media with non‐equilibrium evaporation kinetics is obtained The new solution relies on introducing a new potential function, which reduces the system of non‐linear PDEs to one equation The resulting exact solution shows good agreement with laboratory data and can predict accurately the evaporation time

This article considers multi-manned assembly line balancing problems with walking workers. The objective of the problem is the minimization of number of workers and workstations simultaneously. Several exact-solution algorithms based on Benders’ decomposition are proposed to solve the problem optimally. In one of the algorithms a constructive heuristic that generates effective task-worker assignments and some problem-specific symmetry breaking constraints are used. Moreover, the solutions obtained by meta-heuristic in the literature are used as starting points to increase the performance of proposed decomposition methods. A benchmark set of 99 instances are used to analyze the performance of the proposed exact methods, contribution of the developed heuristic and the ability of Benders’ decomposition on improving the starting solutions. Our results indicate a significiant improvement in the optimal solvability of the problem for larger-sized instances. Suggested methods also improve the results of the meta-heuristic method for significant number of instances. Consequntly, proposed methods solved most of instances optimally and they are able to find the optimal solutions of 17 instances that cannot be solved optimally with previous methods.

Upscaling of 1D two‐phase flows in heterogeneous porous media is important in interpretation of laboratory coreflood data, streamline quasi 3D modeling, and numerical reservoir simulation. In 1D heterogeneous media with properties varying along the flow direction, phase permeabilities are coordinate‐dependent. This yields the Buckley‐Leverett equation with coordinate‐dependent fractional flow f = f(s, x), which reflects the heterogeneity. So, an x‐dependency is considered to reflect microscale heterogeneity and averaging over x—upscaling. This work aims to average or upscale the heterogeneous system to obtain the homogenized media with such fractional flow function F(S) that provides the same water‐cut history at the reservoir outlet, x = 1. Thus, F(S) is an equivalent property of the medium. So far, the exact upscaling for 1D micro heterogeneous systems has not been derived. With the x‐dependency of fractional flow, the Riemann invariant is flux f, which yields exact integration of 1D flow problems. The novel exact solutions are derived for flows with continuous saturation profile, transition of shock into continuous wave, transition of continuous wave into shock, and transport in heterogeneous piecewise‐uniform rocks. The exact procedure of upscaling from f = f(s, x) to F(S) is as follows: the inverse function to the upscaled F(S) is equal to the averaged saturation over x of the inverse microscale function s = f −1(f, x). It was found that the Welge's method as applied to heterogeneous cores provides the upscaled F(S). For characteristic finite‐difference scheme, the fluxes for microscale and upscaled‐numerical‐cell systems, coincide in all grid nodes. Plain Language Summary Natural or industrial two‐phase flow of CO2, water and other fluids occurs in highly heterogeneous porous media. The flow dynamics is modeled by transport equations with highly oscillating coefficients, which require extensive computational resources. This study aims to develop an upscaling technique for the coordinate‐dependent flux resulting from the heterogeneity of the porous medium. The upscaling technique relies on the flux history (water‐cut) and saturation profiles across the porous medium. It is determined that, for a given sequence or configuration of numerical cells or laboratory core samples, the upscaling of the coordinate‐dependent flux is achievable by saturation averaging. It is observed that results from water‐cut based upscaling depend on length L while those of saturation profile‐based upscaling depend on the time interval T. Key Points Exact solutions for 1D two phase flow in heterogeneous porous media Implicit formula for upscaling of x‐dependent fractional flow function Flux‐history and saturation‐profile based upscaling yield the same equation for the upper‐scale fractional flow

A new method named bilinear neural network is introduced in this paper, and the corresponding tensor formula is proposed to obtain the exact analytical solutions of nonlinear partial differential equations (PDEs). This is the first time that the neural network model is used to find the exact analytical solution, and this method covers almost all methods of constructing a function after bilinearization to solve nonlinear PDEs. Furthermore, this method is most likely a universal method to obtain the exact analytical solutions of nonlinear PDEs. Abundant arbitrary functions solutions of the reduced p-gBKP equation are obtained by using this method. Various beautiful plots of the presented solutions, which show diversity of exact solutions to PDEs, are made. By choosing appropriate values and functions, the fractal solitons waves are obtained and the self-similar characteristics of these waves are observed by reducing the observation range and magnifying local images. Via various three-dimensional plots, the evolution characteristics of these waves are exhibited.

In this paper, our attention is given to the semidiscrete Gardner equation, which is a discrete analogue of the continuous Gardner equation describing the long wave propagation in a two-layer fluid. Firstly, a generalized (n,N-n)-fold Darboux transformation (DT) for this semi-discrete Gardner equation is constructed based on its recognized Lax pair. Secondly, the resulting DT is used to obtain exact solutions including ordinary soliton, rational soliton (RS) and their hybrid solutions within the non-zero seed background, and then analyze their asymptotic states as well as physical characteristics. Numerical simulations are also carried out to exhibit the dynamic characteristics of certain exact solutions. Thirdly, the soliton surface corresponding to this semi-discrete equation is investigated. Finally, employing continuum limit theory, we map the semi-discrete equation to the continuous equation, and obtain corresponding continuum limit for its Lax pair and DT. The findings given in this paper are conducive to a more profound understanding of the physical properties depicted by this equation.

The Ising model is famous in condensed matter and statistical physics. In this work we present a free-fermion formulation of the two-dimensional classical Ising models on honeycomb, triangular and Kagomé lattices. Each Ising model is studied in the cases of a zero field and of an imaginary field i(π/2)kBT. We employ the decorated lattice technique, star-triangle transformation, and weak-graph expansion method to exactly map each Ising model in both cases into an eight-vertex model on the square lattice. The resulting vertex weights are shown to satisfy the free-fermion condition. In the zero-field case, each Ising model is an even free-fermion model. In the case of the imaginary field, the Ising model on the honeycomb lattice is an even free-fermion model, while the models on the triangular and Kagomé lattices are odd free-fermion models. We obtain the exact solution of the Kagomé lattice Ising model under the imaginary field i(π/2)kBT, a result not previously reported in the literature. We also show that the frustrated Ising models on the triangular and Kagomé lattices in the imaginary field still exhibit a non-zero residual entropy.

This study investigates the exact solutions of laminar velocity and thermal boundary layers of Al 2 O 3 - CNT / water hybrid nanofluid flow over a permeable stretching sheet. The sheet with a nonlinear temperature distribution, placed through a porous medium under a vertical magnetic field, is considered as a general problem. The solutions of momentum and energy equations are derived for all the conditions under which there are analytical answers, based on comprehensive boundary conditions and Whittaker’s functions. The hybrid nanofluid flow performance is comprehensively investigated based on the velocity and temperature distributions as well as the non-dimensional quantities. In addition, the impact of 7 problem parameters, including the mass transfer factor, characteristic velocity and temperature nonlinearity of sheet, as well as the nanoparticles concentration, the magnetic field strength, the medium permeability and the base fluid Prandtl number are addressed. The results indicate that the sheet mass transfer parameter has the highest effect on the thermo-hydraulic performance of flow, competing with the influence of Prandtl number on the system heat transfer. Indeed, the wall suction significantly increases both the heat transfer rate and pressure loss, and the Prandtl number is an upward function of the former parameter. The boundary layer thickness varies from 15 \\% to 100 \\% if f 0 changes from - 2 to C . Additionally, when f 0 = - 2 , heat transfer performance is threefold greater than its value under f 0 = 0 . This performance also increases from 2 to 13 by the increase in the Prandtl number from 1 to 6.5 at f 0 = - 2 . Eventually, the range of wall mass transfer factor for which there are exact solutions relies on all the parameters existing in the momentum equation, which is also completely discussed.

In this study, a bilinear neural network method is used to solve the exact solutions of the (2 + 1)-dimensional Kadomtsev–Petviashvili equation, which is a new geophysical fluid mechanics model. This equation is derived from Charney equation of geomagnetic field by integrating the multiscale expansion and the perturbation method. To obtain the exact solutions of the model, we built test functions by using the bilinear neural network method. Compared to conventional methods, this method is shown to have faster results and better accuracy. Based on the construction of single-layer models, the rational solution, rogue wave solution, breather type solution and mixed solutions are obtained, and exact solution is also produced based on double-layer models. By increasing the number of activation functions or the number of layers in a neural network, the method's effectiveness is tested from different angles, and more importantly, the method of solving nonlinear equations is expanded. Then, various three-dimensional graphs, contour plots, and density plots with time and selection of activation function are used to depict the typical evolution of these waves.

In the present paper, an MHD three-dimensional non-Newtonian fluid flow over a porous stretching/shrinking sheet in the presence of mass transpiration and thermal radiation is examined. This problem mainly focusses on an analytical solution; graphene water is immersed in the flow of a fluid to enhance the thermal efficiency. The given non-linear PDEs are mapped into ODEs via suitable transformations, then the solution is obtained in terms of incomplete gamma function. The momentum equation is analyzed, and to derive the mass transpiration analytically, this mass transpiration is used in the heat transfer analysis and to find the analytical results with a Biot number. Physical significance parameters, including volume fraction, skin friction, mass transpiration, and thermal radiation, can be analyzed with the help of graphical representations. We indicate the unique solution at stretching sheet and multiple solution at shrinking sheet. The physical scenario can be understood with the help of different physical parameters, namely a Biot number, magnetic parameter, inverse Darcy number, Prandtl number, and thermal radiation; these physical parameters control the analytical results. Graphene nanoparticles are used to analyze the present study, and the value of the Prandtl number is fixed to 6.2. The graphical representations help to discuss the results of the present work. This problem is used in many industrial applications such as Polymer extrusion, paper production, metal cooling, glass blowing, etc. At the end of this work, we found that the velocity and temperature profile increases with the increasing values of the viscoelastic parameter and solid volume fraction; additionally, efficiency is increased for higher values of thermal radiation.