Showing papers on "Partial differential equation published in 2022"
TL;DR: A comprehensive review of the literature on physics-informed neural networks can be found in this article , where the primary goal of the study was to characterize these networks and their related advantages and disadvantages, as well as incorporate publications on a broader range of collocation-based physics informed neural networks.
Abstract: Abstract Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, integral-differential equations, and stochastic PDEs. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages. The review also attempts to incorporate publications on a broader range of collocation-based physics informed neural networks, which stars form the vanilla PINN, as well as many other variants, such as physics-constrained neural networks (PCNN), variational hp-VPINN, and conservative PINN (CPINN). The study indicates that most research has focused on customizing the PINN through different activation functions, gradient optimization techniques, neural network structures, and loss function structures. Despite the wide range of applications for which PINNs have been used, by demonstrating their ability to be more feasible in some contexts than classical numerical techniques like Finite Element Method (FEM), advancements are still possible, most notably theoretical issues that remain unresolved.
216 citations
TL;DR: In this paper , a gradient-enhanced physics-informed neural networks (gPINNs) is proposed for improving the accuracy of PINNs, which leverage gradient information of the PDE residual and embed the gradient into the loss function.
57 citations
TL;DR: In this paper , the spectral local linearization (SLLM) algorithm was used to solve the non-linear coupled differential equations of a magneto-hydrodynamics (MHD) Carreau nanofluid bi-convection flow.
55 citations
TL;DR: In this article, the Fokas system that describes the nonlinear pulse propagation in monomode optical fibers was considered and the Exp-function method was used to construct abundant exact solutions for the partial differential equations arising in optics.
42 citations
TL;DR: The backward compatible PINN (bc-PINN) as mentioned in this paper was proposed to solve the Cahn Hilliard and Allen Cahn equations sequentially over successive time segments using a single neural network.
42 citations
TL;DR: In this paper , the authors focused on analyzing the heat and flow movement among two rotating disks inside water-based carbon nanotubes and used the similarity techniques to convert the model to a nonlinear ordinary differential equation.
Abstract: This study is focused towards analyzing the heat and flow movement among two stretching rotating disks inside water-based carbon nanotubes. The idea of thermal boundary conditions and heat convection is used and the system is expressed in partial differential equations. Using the similarity techniques, the model is successfully converted to a nonlinear ordinary differential equation. A familiar collocation method is used to simulate the outcomes of the governed system while the method is validated through a set of tables and assessed with existing literature. The physical aspects of the proposed model have been studied in detail and assisted via graphical diagrams against the variation of different parameters. It is found that the multiple-wall carbon nanotubes intensify the system quickly and improve the rate of heat transmission. It is also noted that the proposed method is in excellent in agreement with already published studies and can be extended for other physical problems. Moreover, when values of Re parameter increase, a drop is noted in the magnitude of radial velocity near the faces of the disks. It is very clear from the tabular comparison that collocation scheme is in good agreement with already published studies and homotopic solutions.
41 citations
TL;DR: In this article , the Fokas system that describes the nonlinear pulse propagation in monomode optical fibers was considered and the Exp-function method was employed to construct six families of exact soliton solutions.
41 citations
TL;DR: In this paper , steady electro-magnetohydrodynamic flow of a micropolar nanofluid in the attendance of reactive Casson fluid passing through parallel plates influenced by the rotating system with the implementation of the Buongiorno model is examined.
Abstract: ABSTRACT Steady electro-magnetohydrodynamic flow of a micropolar nanofluid in the attendance of reactive Casson fluid passing through parallel plates influenced by the rotating system with the implementation of Buongiorno nanofluid model is examined in this study. The momentum transport equation is enhanced by incorporating the electric field. In addition, the influence of reactive species has a vital role that is affecting the flow phenomenon in conjunction with a transverse magnetic field. The physical flow problem is modeled in the form of partial differential equations which are then transformed into nonlinear ordinary differential equations by using appropriate similarity functions and then solved numerically by the usage of the finite element method and procured results are visualized graphically. The outcomes for flow rate, microrotation, temperature, concentration, and engineering quantities distributions are shown in terms of graphical presentation. Momentum and angular momentum transport progressively in nature as the Casson parameter grows. Opposite results of microrotational profiles are found for electric currents in comparison with Hall currents. Both thermophoresis and Brownian motion are found to be significant effects in improving heat transportation phenomena in nanofluids. The existing available literature was utilized to test for validation of the numerical findings.
39 citations
TL;DR: In this paper , two active boundary controllers are proposed to restrain the vibrations both in bending and twisting to ensure the stability of a 3D flexible wing system by using Lyapunov's direct method.
Abstract: This brief mainly considers trajectory tracking and vibration suppression for a 3-D flexible wing. The dynamical model of the flexible wing is regarded as a distributed parameter system, which is described by partial differential equations and ordinary differential equations. A control strategy regulates the flexible wing to track the desired trajectory by controlling two angles. Meanwhile, two active boundary controllers are proposed to restrain the vibrations both in bending and twisting. By using Lyapunov’s direct method, the stability of the flexible wing system can be ensured. Numerical simulations based on the finite-difference method demonstrate the effectiveness of the proposed control schemes.
37 citations
TL;DR: In this article, the exact rogue periodic wave (rogue wave on the periodic background) and periodic wave solutions for the Chen-Lee-Liu equation via the odd-th order Darboux transformation were considered.
36 citations
TL;DR: In this article , a coupled nonlinear modeling for composite cylindrical shells is developed by the improved Donnell nonlinear shell theory and Maxwell static electricity/magnetism equations.
TL;DR: In this paper, the electromagnetic forces on the SWCNT/water flow with microorganisms over a Riga plate subject to slip effects were discussed. And the Runge-Kutta-Fehlberg (RKF-45) method was applied to numerically solve the extremely nonlinear system.
Abstract: Electromagnetohydrodynamic (EMHD) is very important because of its numerous advantages such as flow control in fluidics networks, fluid pumping, thermal reactors, mixing, fluid stirring, liquid chromatography, and micro coolers. Based on the above applications in this article discussed the electromagnetic forces on the SWCNT/water flow with microorganisms over a Riga plate subject to slip effects. In addition, the uniform heat source/sink effect is used in the energy equation, as well as the thermophoretic effect in the concentration equation. The governing nonlinear system of partial differential equations (PDEs) was reduced to ordinary differential equations (ODEs) by applying the appropriate similarity variables. Hence, Runge-Kutta-Fehlberg (RKF-45) method was applied to numerically solve the extremely nonlinear system. Based on the analysis of the results, it is worth concluding that raising the role of slip effects lowers the velocity, temperature, and concentration curves, while increasing the solid volume fraction increases the temperature, concentration, and motile microorganism density.
TL;DR: In this article , the authors investigated heat and mass transport in a pseudo-plastic model past over a stretched porous surface in the presence of the Soret and Dufour effects.
Abstract: The rheology of different materials at the micro and macro levels is an area of great interest to many researchers, due to its important physical significance. Past experimental studies have proved the efficiency of the utilization of nanoparticles in different mechanisms for the purpose of boosting the heat transportation rate. The purpose of this study is to investigate heat and mass transport in a pseudo-plastic model past over a stretched porous surface in the presence of the Soret and Dufour effects. The involvement of tri-hybrid nanoparticles was incorporated into the pseudo-plastic model to enhance the heat transfer rate, and the transport problem of thermal energy and solute mechanisms was modelled considering the heat generation/absorption and the chemical reaction. Furthermore, traditional Fourier and Fick’s laws were engaged in the thermal and solute transportation. The physical model was developed upon Cartesian coordinates, and boundary layer theory was utilized in the simplification of the modelled problem, which appears in the form of coupled partial differential equations systems (PDEs). The modelled PDEs were transformed into corresponding ordinary differential equations systems (ODEs) by engaging the appropriate similarity transformation, and the converted ODEs were solved numerically via a Finite Element Procedure (FEP). The obtained solution was plotted against numerous emerging parameters. In addition, a grid independent survey is presented. We recorded that the temperature of the tri-hybrid nanoparticles was significantly higher than the fluid temperature. Augmenting the values of the Dufour number had a similar comportment on the fluid temperature and concentration. The fluid temperature increased against a higher estimation of the heat generation parameter and the Eckert numbers. The impacts of the buoyancy force parameter and the porosity parameter were quite opposite on the fluid velocity.
TL;DR: In this paper , a non-intrusive surrogate modeling scheme based on deep learning for predictive modeling of complex systems, described by parametrized time-dependent partial differential equations, is presented.
TL;DR: In this article , a modified neural architecture search method (NAS) based physics-informed deep learning model is presented for stochastic analysis in heterogeneous porous material, which can fit different partial differential equations (PDEs) with less calculation.
Abstract: In this work, a modified neural architecture search method (NAS) based physics-informed deep learning model is presented for stochastic analysis in heterogeneous porous material. Monte Carlo method based on a randomized spectral representation is first employed to construct a stochastic model for simulation of flow through porous media. To solve the governing equations for stochastic groundwater flow problem, we build a modified NAS model based on physics-informed neural networks (PINNs) with transfer learning in this paper that will be able to fit different partial differential equations (PDEs) with less calculation. The performance estimation strategies adopted is constructed from an error estimation model using the method of manufactured solutions. A sensitivity analysis is performed to obtain the prior knowledge of the PINNs model and narrow down the range of parameters for search space and use hyper-parameter optimization algorithms to further determine the values of the parameters. Further the NAS based PINNs model also saves the weights and biases of the most favorable architectures, then used in the fine-tuning process. It is found that the log-conductivity field using Gaussian correlation function will perform much better than exponential correlation case, which is more fitted to the PINNs model and the modified neural architecture search based PINNs model shows a great potential in approximating solutions to PDEs. Moreover, a three dimensional stochastic flow model is built to provide a benchmark to the simulation of groundwater flow in highly heterogeneous aquifers. The NAS model based deep collocation method is verified to be effective and accurate through numerical examples in different dimensions using different manufactured solutions.
TL;DR: In this article, the performance of the generalized Fourier's and Fick's laws on the MHD bioconvective aspects of couple-stress nanofluid flows through a convectively heated stretching sheet in the presence of activation energy and multiple stratified boundary conditions is analyzed.
Abstract: The current non-homogeneous nanofluid flow model is carried out to scrutinize the performance of the generalized Fourier's and Fick's laws on the MHD bioconvective aspects of couple-stress nanofluid flows through a convectively heated stretching sheet in the presence of activation energy and multiple stratified boundary conditions. Herein, both the concentrations of solid nanoparticles and motile microorganisms are incorporated explicitly into the nonlinear differential expressions describing the present non-Newtonian nanofluid flow model. Besides, the combined thermal influence of the Cattaneo-Christov heat flux and thermal radiation are also discussed. From a practical point of view, the couple-stress nanofluids are useful for examining different types of thermophysical and rheological features, since this kind of enhanced fluids can clarify realistically the dynamical behavior of various liquids, like the human blood and some polymeric suspensions. For reducing the mathematical complexity of the present physical problem, several effective similarity transformations are introduced formally to simplify the resulting partial differential equations (PDEs) into a nonlinear coupled structure of ordinary differential equations (ODEs). Moreover, the transformed dimensionless self-similarity equations are then numerically solved using the built-in shooting technique with the aid of the bvp4c solver MATLAB package. Furthermore, The obtained results are authenticated with an outstanding agreement. In this respect, the engineering quantities of interest are computed extensively with a higher level of accuracy and then summarized tabularly. To illustrate the impacts of the embedded physical parameters on the profiles of velocity, temperature, nanoparticles concentration, and microorganisms concentration, various illustrations are done successfully along with detailed elucidations. As the main findings, it is found that the temperature distribution and the microorganisms concentration profile can be enhanced with the higher values of the bioconvection Rayleigh number. Similarly, it is revealed that the nanoparticles concentration sketch and the microorganisms concentration profile can be boosted up for the higher magnitudes of the buoyancy ratio parameter.
TL;DR: In this paper , a generalized fifth-order nonlinear partial differential equation for the Sawada-Kotera (SK), Lax, and Caudrey-Dodd-Gibbon (CDG) equations is investigated to study the nonlinear wave phenomena in shallow water, ion-acoustic waves in plasma physics, and other nonlinear sciences.
Abstract: This research aims to investigate a generalized fifth-order nonlinear partial differential equation for the Sawada-Kotera (SK), Lax, and Caudrey-Dodd-Gibbon (CDG) equations to study the nonlinear wave phenomena in shallow water, ion-acoustic waves in plasma physics, and other nonlinear sciences. The Painlevé analysis is used to determine the integrability of the equation, and the simplified Hirota technique is applied to construct multiple soliton solutions with an investigation of the dispersion relation and phase shift of the equation. We utilize a linear combination approach to construct a system of equations to obtain a general logarithmic transformation for the dependent variable. We generate one-soliton, two-soliton, and three-soliton wave solutions using the simplified Hirota method and showcase the dynamics of these solutions graphically through interaction between one, two, and three solitons. We investigate the impact of the system’s parameters on the solitons and periodic waves. The SK, Lax, and CDG equations have a wide range of applications in nonlinear dynamics, plasma physics, oceanography, soliton theory, fluid dynamics, and other sciences.
TL;DR: In this article, the authors investigated heat and mass transport past over a stretched surface having pores in a pseudo-plastic model and derived thermal and mass-transport expressions by engaging the double diffusion theories.
Abstract: Abstract This research is conducted to investigate heat and mass transport past over a stretched surface having pores in a pseudo-plastic model. To study porosity effect, Darcy Forchheimer relation is used. Thermal and mass transport expressions are derived by engaging the double diffusion theories as extensively used by researchers proposed by Cattaneo and Christov. Furthermore, the thermal performance is studied by mixing the tri-hybrid nanoparticles in a pseudo-plastic material. The phenomenon of boundary layer is used to derive the complex model. The correlation for tri-hybrid nanoparticles is used to convert the model partial differential equations into ordinary differential equations (ODE) along with appropriate similarity transformation. The transfigured ODEs are coupled nonlinear in nature, and the exact solution is not possible. To approximate the solution numerically, finite element scheme (FES) is used and code is developed in MAPLE 18.0 for the graphical results, grid independent survey, and tabular results. The obtained results are compared with the published findings that confirm the accuracy and authenticity of the solution and engaged scheme. From the performed analysis, it is concluded that FES can be applied to complex engineering problems. Furthermore, it is monitored that nanoparticles are essential to boost the thermal performance and higher estimation of Schmidt number control the mass diffusion.
TL;DR: A mathematical model for the two-dimensional flow of tangent hyperbolic nanofluid over a Riga plate in the existence of gyrotactic microorganisms is discussed in this paper.
Abstract: A mathematical model for the two-dimensional flow of tangent hyperbolic nanofluid over a Riga plate in the existence of gyrotactic microorganisms is discussed. For velocity, temperature, and concentration, Wu's velocity slip, thermal convection, and zero nanoparticle flux conditions are imposed. Nanofluids gained tremendous notability due to their wide spread industrial, technological, and chemical applications. A set of pertinent transformations has been suggested to transform the governing non-linear partial differential equations into a system of non-linear ordinary differential equations (ODEs). The methodology involves solving the dimensionless equations using the bvp4c technique for computing the numerical solution. Additionally, the numerical results are constructed for these obtained equations by using the bvp4c tool in MATLAB. The effects of numerous prominent parameters on concentration, velocity, temperature, and motile microorganism profiles are executed graphically. From the obtained results it is noted that the velocity of the tangent hyperbolic fluid is boosted up by growing the variations of mixed convection parameters. The temperature of nanofluid behavior is reduced for a larger Prandtl number. Concentration of nano particles is enhanced by growing activation energy parameters. Microorganisms profile is improved with first-order velocity slip parameter while decaying for bio-convection Lewis number.
TL;DR: In this paper , the impacts of ternary nanoparticles on hyperbolic tangent materials to establish their thermal characteristics were investigated. But the authors focused on the thermal properties of hybrid nanoparticles to improve thermal processes.
TL;DR: Nonlocal Kernel Network (NKN) as discussed by the authors is a nonlocal neural operator that is resolution independent, characterized by deep neural networks, and capable of handling a variety of tasks such as learning governing equations and classifying images.
TL;DR: In this paper , the electromagnetic forces on the SWCNT/water flow with microorganisms over a Riga plate subject to slip effects were discussed, and the Runge-Kutta-Fehlberg (RKF-45) method was applied to numerically solve the extremely nonlinear system.
Abstract: Electromagnetohydrodynamic (EMHD) is very important because of its numerous advantages such as flow control in fluidics networks, fluid pumping, thermal reactors, mixing, fluid stirring, liquid chromatography, and micro coolers. Based on the above applications in this article discussed the electromagnetic forces on the SWCNT/water flow with microorganisms over a Riga plate subject to slip effects. In addition, the uniform heat source/sink effect is used in the energy equation, as well as the thermophoretic effect in the concentration equation. The governing nonlinear system of partial differential equations (PDEs) was reduced to ordinary differential equations (ODEs) by applying the appropriate similarity variables. Hence, Runge-Kutta-Fehlberg (RKF-45) method was applied to numerically solve the extremely nonlinear system. Based on the analysis of the results, it is worth concluding that raising the role of slip effects lowers the velocity, temperature, and concentration curves, while increasing the solid volume fraction increases the temperature, concentration, and motile microorganism density.
TL;DR: The intelligence based numerical computation of artificial neural network backpropagated with Levenberg-Marquardt algorithm has been developed to analyze the novel ferrofluid flow model in the presence of magnetic dipole.
Abstract: In the presented research article, the intelligence based numerical computation of artificial neural network backpropagated with Levenberg-Marquardt algorithm has been developed to analyze the novel ferrofluid flow model in the presence of magnetic dipole. Heat transfer effects are also incorporated along the horizontal. The designed fluid flow model initially represented by system of partial differential equations are converted into system of non-linear ordinary differential equations through suitable similarity transformations. The reference dataset of the possible outcomes is obtained from Adam numerical solver for the different scenarios of flow model by variation of co-efficient of the thermal expansion, Eckert number, suction parameter, magnetization and radiation parameter. The approximated solutions are interpreted for designed model by testing, training and validation process of backpropagated neural networks. Furthermore, the comparative studies and performance analysis of used algorithm is validated through regression analysis, histogram studies, correlation index and results of mean square error.
TL;DR: In this paper , the modified Gardner-type equation and its time fractional form were derived from Fermi-Pasta-Ulam (FPU) model and the nonlinear Schrodinger equation (NLS) type equation.
Abstract: • Modified Gardner type equation and its time fractional form are derived. • From these two equations, we derived the nonlinear Schrodinger equation (NLS) type equation. • Symmetry analysis and conservation laws also presented. Differential equations play an important role in many scientific fields. In this work, we study modified Gardner-type equation and its time fractional form. We first derive these two equations from Fermi-Pasta-Ulam (FPU) model, and found that these two equations are related with nonlinear Schr o ¨ dinger equation (NLS) type of equations. Subsequently, symmetries and conservation laws are investigated. Finally, B a ¨ cklund transformation of conservation laws also presented. In this article, we not only derive these two equations, but also use perturbation analysis to find the connection between them and the Schr o ¨ dinger equation. Another key point is that B a ¨ cklund transformation of conservation laws are also obtained. From these results, it is obvious that the Lie group method is a very effective method for dealing with partial differential equations .
TL;DR: In this article, the Burgers fluid with a fractional derivatives model analyzed through a rotating annulus is presented, the governing partial differential equation solved for velocity field and shear stress by using integral transformation method and using Bessel equations.
Abstract: Keeping in view of the complex fluid mechanics in bio-medicine and engineering, the Burgers’ fluid with a fractional derivatives model analyzed through a rotating annulus. The governing partial differential equation solved for velocity field and shear stress by using integral transformation method and using Bessel equations. The transformed equation inverted numerically by using Gaver-Stehfest’s algorithm. The approximate analytical solution for rotational velocity, and shear stress are presented. The influence of various parameters like fractional parameters, relaxation and retardation time parameters material constants, time and viscosity parameters are drawn numerically. It is found that the relaxation time and time helps the flow pattern, on the other hand other material constants resist the fluid rotation. Fractional parameters effect on fluid flow is opposite to each other. Finally, to check the validity of the solution there are comparisons for velocity field and shear stress for obtained results with an other numerical algorithm named Tzou’s algorithm.