Dimensional Partial Differential Equations Research Articles

Lie symmetries, exact solutions and conservation laws of time fractional coupled (2+1)-dimensional nonlinear Schrödinger equations

ABSTRACT In this paper, Lie symmetry analysis method is applied to time fractional coupled (2 + 1)-dimensional nonlinear Schrödinger equations. We obtain all the Lie symmetries and reduce the (2 + 1)-dimensional fractional partial differential equations with Riemann-Liouville fractional derivative to (1 + 1)-dimensional counterparts with Erdélyi-Kober fractional derivative. Then we obtain the power series solutions and prove their convergence. In addition, the conservation laws for the governing equations are constructed by using the generalization of the Noether operators and the new conservation theorem.

Lie symmetries, exact solution and conservation laws of (2 + 1)-dimensional time fractional Kadomtsev–Petviashvili system

Abstract In this paper, Lie symmetry analysis method is applied to the ( 2 + 1 ) (2+1) -dimensional time fractional Kadomtsev–Petviashvili (KP) system, which is an important model in mathematical physics. We obtain all the Lie symmetries admitted by the KP system and use them to reduce the ( 2 + 1 ) (2+1) -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to some ( 1 + 1 ) (1+1) -dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative or Riemann–Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the system studied.

Multiple solitons, multiple lump solutions, and lump wave with solitons for a novel (2+1)-dimensional nonlinear partial differential equation

The primary aim of this paper is to explore exact solutions to a novel (2+1)-dimensional water wave equation that models oceanic wave phenomena. We begin by applying the Hirota bilinear transformation method to derive multi-soliton solutions, including 3-soliton and 4-soliton solutions. Then, utilizing the bilinear form of the equation and the long-wave limit method, we identify multiple lump solutions and interaction solutions between lumps and solitons. These include 1-lump, 2-lump, and 3-lump solutions, as well as interactions between a 1-lump and a 1-soliton, and between a 1-lump and 2-solitons. The physical dynamics of these solutions are visually represented, offering insight into the corresponding oceanic wave phenomena.

Double-loop importance sampling for McKean–Vlasov stochastic differential equation

This paper investigates Monte Carlo (MC) methods to estimate probabilities of rare events associated with the solution to the d-dimensional McKean–Vlasov stochastic differential equation (MV-SDE). MV-SDEs are usually approximated using a stochastic interacting P-particle system, which is a set of P coupled d-dimensional stochastic differential equations (SDEs). Importance sampling (IS) is a common technique for reducing high relative variance of MC estimators of rare-event probabilities. We first derive a zero-variance IS change of measure for the quantity of interest by using stochastic optimal control theory. However, when this change of measure is applied to stochastic particle systems, it yields a P×d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P \ imes d$$\\end{document}-dimensional partial differential control equation (PDE), which is computationally expensive to solve. To address this issue, we use the decoupling approach introduced in (dos Reis et al. 2023), generating a d-dimensional control PDE for a zero-variance estimator of the decoupled SDE. Based on this approach, we develop a computationally efficient double loop MC (DLMC) estimator. We conduct a comprehensive numerical error and work analysis of the DLMC estimator. As a result, we show optimal complexity of OTOLr-4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}\\left( \ extrm{TOL}_{\ extrm{r}}^{-4}\\right) $$\\end{document} with a significantly reduced constant to achieve a prescribed relative error tolerance TOLr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{TOL}_{\ extrm{r}}$$\\end{document}. Subsequently, we propose an adaptive DLMC method combined with IS to numerically estimate rare-event probabilities, substantially reducing relative variance and computational runtimes required to achieve a given TOLr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{TOL}_{\ extrm{r}}$$\\end{document} compared with standard MC estimators in the absence of IS. Numerical experiments are performed on the Kuramoto model from statistical physics.

Lie symmetries, exact solutions and conservation laws of (2+1)-dimensional time fractional cubic Schrödinger equation

Abstract In this paper, Lie symmetry analysis method is applied to ( 2 + 1 ) {(2+1)} -dimensional time fractional cubic Schrödinger equation. We obtain all the Lie symmetries and reduce the ( 2 + 1 ) {(2+1)} -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to ( 1 + 1 ) (1+1) -dimensional counterparts with Erdélyi–Kober fractional derivative. Then we obtain the power series solutions of the reduced equations and prove their convergence. In addition, the conservation laws for the governing model are constructed by the new conservation theorem and the generalization of Noether operators.

Neural network approach to investigate heat transfer in SWCNTs nanofluid within trapezoidal cavity with varied corrugated rod amplitudes

This article presents Artificial Neural Networks (ANN) for convective heat transfer in a trapezoidal cavity subjected to corrugated heated rod inside it. The LevenbergMarquardt algorithm is utilized to optimize the Neural Networks. The trapezoidal cavity has low-temperature inclined walls and adiabatic upper and lower walls compared to the corrugated heated rod. Single-wall carbon (SWCNTs) nanomaterials are submerged in the base liquid water. The flow of SWCNTs-water is generated due to the temperature gradient in the cavity. The system of dimensional partial differential equations has been formulated for the physical setup under investigation. The dimensional system has been converted into a non-dimensional form using dimensionless variables. Finite element is used for the solutions. The dimensionless functions velocity, temperature, and heat transfer rates are studied against the Rayleigh number (Ra). The outcomes are presented in the form of isotherms, contours, tabular values, and graphs. The data for Artificial Neural Networks (ANN) has been generated by FEM against the Nusselt number. The ANN has been trained for a specific amplitude of the rod and predicted heat transfer against a larger amplitude. The results show good agreement for both training and testing data. The outcomes of analysis reveals that convection caused by temperature gradient is dominant for higher values of the Rayleigh number (Ra). Local Nusselt number has been discussed against different amplitudes, and predicted enhancement for the larger amplitude of the rod.

Lie symmetry analysis of (2+1)-dimensional time fractional Kadomtsev-Petviashvili equation

In this paper, Lie symmetry analysis method is applied to the (2+1)-dimensional time fractional Kadomtsev-Petviashvili (KP) equation with the mixed derivative of Riemann-Liouville time-fractional derivative and integer-order $x$-derivative. We obtained all the Lie symmetries admitted by the KP equation and used them to reduce the (2+1)-dimensional fractional partial differential equation with Riemann-Liouville fractional derivative to some (1+1)-dimensional fractional partial differential equations with Erd\'{e}lyi-Kober fractional derivative or Riemann-Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equation studied.

Implementing the predictor-corrector approach to examine the thermo-solutal convection for buongiorno eyring-powell nanofluid model with squeezed microcantilivier surface

The Buongiorno nanofluid model serves as a foundational model for theoretical and experimental research in the field of nanofluids, and this model can be applied to various engineering problems, including heat exchangers, biomedical applications, solar collectors, nuclear reactors, and different cooling systems in the automotive and electronics industries. The use of nanofluids in the Eyring-Powell fluid over a squeezed sensing surfaces is a vital area of research for the design and improvement of microfluidic devices and sensors, which find widespread usage in industrial and medical applications. The main purpose of this work is to note the augmentation in thermal conductivity by using the Buongiorno nanofluid model along with the natural convective flow of non-Newtonian Eyring-Powell fluid that is moving on the squeezed sensory surface along with slip velocity. The viscosity of a fluid is assumed to be dependent on temperature. The Cattaneo-Christov heat and mass flux and chemical reaction are assumed to determine the combined effect of heat and mass transport. The governing model of equations is in the form of dimensional partial differential equations, and we have to convert these equations into ordinary differential equations; therefore, a set of similarity transformations is adopted for making the equations into dimensionless form. The numerical results were obtained by adopting the predictor and corrector multistep method, namely the ‘Adams-Bashforth’ technique, in Matlab software. The use of natural convective flow enhances the velocity region. The velocity slip parameter increases the velocity of the fluid, whereas the viscosity parameter drops the velocity profile. The thermophoresis and Brownian motion parameters are the sources of the increment in the temperature region. The thermal relaxation and solutal relaxation parameters are the sources of decline in the temperature region. The squeezing parameter is the source of reduction in the skin friction, whereas the result indicates that the fluid parameter, Grashof number, and viscosity coefficients increase the skin friction.

Inf-sup neural networks for high-dimensional elliptic PDE problems

Solving high dimensional partial differential equations (PDEs) has historically posed a considerable challenge when utilizing conventional numerical methods, such as those involving domain meshes. Recent advancements in the field have seen the emergence of neural PDE solvers, leveraging deep networks to effectively tackle high dimensional PDE problems. This study introduces Inf-SupNet, a model-based unsupervised learning approach designed to acquire solutions for a specific category of elliptic PDEs. The fundamental concept behind Inf-SupNet involves incorporating the inf-sup formulation of the underlying PDE into the loss function. The analysis reveals that the global solution error can be bounded by the sum of three distinct errors: the numerical integration error, the duality gap of the loss function (training error), and the neural network approximation error for functions within Sobolev spaces. To validate the efficacy of the proposed method, numerical experiments conducted in high dimensions demonstrate its stability and accuracy across various boundary conditions, as well as for both semi-linear and nonlinear PDEs.

An analytical study on the influence of magnetic field-dependent viscosity on viscous and ohmic dissipative second-grade fluid boundary layers

This article aims to investigate the magnetohydrodynamic (MHD) boundary layer flow of a second-grade non-Newtonian fluid over a horizontally stretching sheet, considering magnetic field-dependent (MFD) viscosity, as well as viscous and Ohmic dissipations. Analytical solutions for previously unexplored momentum and heat transfer equations involving MFD viscosity are derived. Two boundary conditions, namely Prescribed Surface Temperature (PST) and Prescribed Heat Flux (PHF), are taken into account. Governing dimensional partial differential equations (PDEs) are transformed into non-dimensional ordinary differential equations (ODEs) using similarity transformations. Closed-form analytical solutions for flow are derived, considering stretching velocity, MFD viscosity, second-grade fluid properties, and suction impacts. Heat transfer equations are transformed into Gauss-hypergeometric form, yielding solutions in terms of confluent hypergeometric functions. Analytical expressions for skin friction, local Nusselt number, and non-dimensional wall temperature are derived. A unique solution is obtained for the flow equation. It is found that both second-grade and magnetic viscosity parameters expand the momentum boundary layer while the magnetic parameter reduces thickness. Thermal boundary layer thickness increases with a higher second-grade parameter, magnetic viscosity, and Eckert number. Moreover, analytical solutions are validated against published results for a special case.

Lie Symmetry Analysis, Power Series Solutions and Conservation Laws of (2+1)-Dimensional Time Fractional Modified Bogoyavlenskii–Schiff Equations

In this paper, Lie symmetry analysis method is applied to the (2+1)-dimensional time fractional modified Bogoyavlenskii–Schiff equations, which is an important model in physics. The one-dimensional optimal system composed by the obtained Lie symmetries is utilized to reduce the system of (2+1)-dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to the system of (1+1)-dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative. Then the power series method is applied to derive explicit power series solutions for the reduced system. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equations studied.

The new kink type and non-traveling wave solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation

In this paper, the new solitary wave solutions of the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation are obtained by Lie group symmetry method and the extended homoclinic test approach. Firstly, the equation can be reduced to (1+1)-dimensional partial differential equation by Lie group symmetry, and corresponding bilinear forms of the equation are given by symmetry functions. Secondly, the extended homoclinic test approach is employed to obtain the new kink type and singular solitary wave solutions. In addition, some new traveling and non-traveling wave solutions with arbitrary functions and oscillating tail are investigated through the special transformations for the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation.

Influence of magnetic field-dependent viscosity on Casson-based nanofluid boundary layers: A comprehensive analysis using Lie group and spectral quasi-linearization method

This study examines the effects of magnetic-field-dependent (MFD) viscosity on the boundary layer flow of a non-Newtonian sodium alginate-based Fe3O4 nanofluid over an impermeable stretching surface. The non-Newtonian Casson and homogeneous nanofluid models are utilized to derive the governing flow and heat transfer equations. Applying Lie group transformations to dimensional partial differential equations yields nondimensional ordinary differential equations, which are then numerically solved using the spectral quasi-linearization technique. The analysis primarily focuses on the impacts of the MFD viscosity parameter, nanoparticle volume fraction of Fe3O4, and magnetic parameters on the flow and heat transfer characteristics. The local skin friction and heat transfer rate behaviors influenced by viscosity changes due to the magnetic field are discussed. It is found that MFD viscosity significantly impacts flow and thermal energies, enhancing skin friction coefficients and reducing Nusselt numbers in the boundary layer region.

Impact of Arrhenius activation energy on magnetic nanofluid flow over a slendering stretchable sheet with nonlinear radiative heat transfer: A machine learning algorithm

Thermal systems in the contemporary world need effective cooling and heating methods. Heat transfer technology plays a vital role in various industries, including aerospace, electronic equipment, automobiles, and transportation. Considering the importance of effective cooling and heating processes, this framework explores the hydromagnetic stagnation point flow with binary chemical reactions, ohmic heating, Arrhenius activation energy, nonlinear thermal radiation, thermophoresis, and Brownian motion. At first, the flow equations are determined in a formal way as dimensional partial differential equations. Then, they are transformed into simpler, dimensionless ordinary differential equations using self-similar variables. The present study describes a novel approach to intelligent numerical computing, utilizing a Levenberg-Marquard algorithm-based MLP (Multi-Layer Perceptron) feed-forward back-propagation ANN (Artificial Neural Networks) implementation. The collection of data was conducted to facilitate the evaluation, validation, and instruction of the artificial neural network model. The utilization of appropriate self-similarity variables has facilitated the conversion of fluid transport equations into ordinary differential equations. The Runge–Kutta–Fehlberg (RKF) technique was used to solve these equations. The numerical results are presented visually for numerous constraints of the governing parameters. The increased values of the velocity ratio enhanced the fluid velocity. The nonlinear thermal radiation and temperature differential characteristics significantly raise the temperature of the fluid. Increasing the nonlinear radiation and magnetic field parameters improves the rate of heat transfer. The temperature of the fluid decreases as the size of stagnation increases. The skin friction coefficients decrease as the wall thickness and magnetic parameters increase.

Entropy analysis in a mixed convective Carreau nanofluid flow around a wedge: impact of activation energy and sinusoidal magnetic field

A wide range of real-world applications have proven the importance of non-Newtonian fluids near a wedge, including the oil and gas industry, the aerospace sector. This study elucidates the dynamics of a Carreau nanofluid around a wedge by combining entropy analysis with periodic magnetohydrodynamics (MHD) and activation energy. The dimensional partial differential equations (PDEs) that describe the fluid flow system undergo non-similar transformations, forming nondimensional PDEs. The numerical solution to these PDEs is obtained by applying quasilinearization followed by the implicit finite difference approach. In the case of n = 0.5 (power-law index), when We (Weissenberg number) improves from 0 to 4, surface friction ( R e 1 / 2 C f ) upsurges by approximately 23% and declines by around 41% in the case of n = 1.5. The mass transport intensity of liquid oxygen is about 18% higher than liquid nitrogen’s. Increasing the wedge angle results in a significant increase in fluid velocity.

Enhancing energy efficiency and heat transfer performance of engine oil flow through hybrid nanoparticles in convergent/divergent channel

Engine oil (EO) is employed as a lubricant in various machinery engines. Efficient heat transfers are a fundamental requirement for all processes. To improve heat transfer rates and reduce the energy loss caused by high temperatures, this study focuses on incorporating Iron Oxide (Fe3O4) and Cupper Oxide (CuO) into the engine oil as suspended particles. However, the focus of this research paper is to analyze entropy generation in a convergent/divergent channel that contains a hybrid nanofluid. Mathematical modeling is utilized to present the continuity, momentum, and energy equations. Using the resemblance transformation variables, the governing dimensional partial differential equations (PDEs) are transformed into a system of non-dimensional ordinary differential equations (ODEs), which are then resolved in ascending order through the BVP4c scheme. The graphical results illustrate the effects of various parameters on fluid velocity, temperature, entropy generation, and Bejan number. The study explores changes in velocity profiles under different Hartmann numbers, revealing opposing variations in divergent and convergent channels. Velocity profiles exhibit similar patterns when varying the porosity parameter. In the convergent passage, a higher porosity parameter results in a higher temperature field. Entropy rises against Eckert and Hartmann numbers in both Convergent and Divergent cases. These insights aid in designing and optimizing machinery engines.

DF-ParPINN: parallel PINN based on velocity potential field division and single time slice focus

Modified Benney-Luke equation (mBL equation) is a three-dimensional temporal-spatial equation with complex structures, that is a high-dimensional partial differential equation (PDE), it is also a new equation of the physical ocean field, and its solution is important for studying the internal wave-wave interaction of inclined seafloor. For conventional PDE solvers such as the pseudo-spectral method, it is difficult to solve mBL equation with both accuracy and speed. Physics-informed neural network (PINN) incorporates physical prior knowledge in deep neural networks, which can solve PDE with relative accuracy and speed. However, PINN is only suitable for solving low-dimensional PDE with simple structures, and not suitable for solving high-dimensional PDE with complex structures. This is mainly because high-dimensional PDEs usually have complex structures and high-order derivatives and are likely to be high-dimensional non-convex functions, and the high-dimensional non-convex optimization problem is an NP-hard problem, resulting in the PINN easily falling into inaccurate local optimal solutions when solving high-dimensional PDEs. Therefore, we improve the PINN for the characteristics of mBL equation and propose “DF-ParPINN: parallel PINN based on velocity potential field division and single time slice focus” to solve mBL equation with large amounts of data. DF-ParPINN consists of three modules: temporal-spatial division module of overall velocity potential field, data rational selection module of multiple time slices, and parallel computation module of high-velocity fields and low-velocity fields. The experimental results show that the solution time of DF-ParPINN is no more than 0.5s, and its accuracy is much higher than that of PINN, PIRNN, cPINN, and DeepONet. Moreover, the relative error of DF-ParPINN after deep training 1000000 epochs can be reduced to less than 0.1. The validity of DF-ParPINN proves that the improved PINN also can solve high dimensional PDE with complex structures and large amounts of data quickly and accurately, which is of great significance to the deep learning of the physical ocean field.

Distinctive approach for MHD natural bioconvection flow of Reiner–Rivlin nano-liquid due to an isothermal sphere

The analysis of heat excess through non-Newtonian fluid is noticed proper and promising arena for biomedical technological applications than Newtonian liquid. Herein, a type of non-Newtonian (Reiner–Rivlin) nanomaterial has been scrutinized with gyrotactic microorganisms around an isothermal sphere with surface of zero mass flux. Being mindful of that the model of Buongiorno’s nanofluid is utilized. Non-dimensional formulas for the basic dimensional partial differential equation system are obtained using appropriate non-similarity transformations. MATLAB function bvp4c code is adopted to get the non-similar solutions. The computations disclosed that the velocity of Reiner–Rivlin nanofluid and its gradient accelerate away from the surface of sphere to infinity and reduces in proximity to the sphere surface for higher values of Reiner–Rivlin factor K. The rate of surface density transport, rate of surface heat transport, and gradient of velocity diminish as the stream wise coordinate boosts. As the problem is subjected to special circumstances, there is a remarkable agreement between the acquired and prior numerical calculations resulting from the comparison.

The exact solutions to the generalized (2+1)-dimensional nonlinear wave equation

Due to the importance of the nonlinear partial differential equations in applied physics and engineering, many mathematicians and physicists are interesting to the nonlinear partial differential equations. One of the main tasks of studying the nonlinear partial differential equations is to obtain the exact solutions for the nonlinear partial differential equations. In this paper, we study the exact solutions of a generalized (2+1)-dimensional nonlinear partial differential equation. According to the modified hyperbolic function method and the traveling transformation method, the generalized (2+1)-dimensional nonlinear partial differential equation is changed into an ordinary differential equation. By taking the homogeneous balance between the highest order nonlinear terms and the highest order derivative terms, we change the differential equation into an algebraic system. By solving the algebraic system, we obtain the solutions for the nonlinear partial differential equation. At last, some solutions and their figures are given by specific parameters. All the solutions conclude the trigonometric function solutions, the positive hyperbolic function solutions, and the hyperbolic trigonometric function solutions. These solutions exhibit the dynamics of nonlinear waves and the solutions are useful for the study on interaction behavior of nonlinear waves in shallow water, plasma, nonlinear optics and Bose–Einstein condensates.

Computational analysis on transient MHD free convection flow in a microchannel in presence of inclined magnetic field with Hall and ion slip current

The present analysis is concerned with the effect of magnetic field inclination on transient MHD flow of Newtonian viscous fluid in a vertical microchannel with the consideration of Hall and ion slip currents as well as induced magnetic field effects. Obtained dimensional partial differential equation are rendered dimensionless by employing suitable parameters and thereafter solved numerically in MATLAB. Relevant actions of parameters on different flow features are depicted explicitly and also using Tables for various applicable parameters. Analysis in this direction is relevant in many MHD controlled applications. Results obtained from the present analysis shows that at the early stages of time and in the simultaneous occurrence of inclined magnetic field as well as Hall and ion slip currents, velocity and induced magnetic field behavior are found to be oscillatory all through the microchannel domain.