Higher-order Partial Differential Equations Research Articles

SHoP: A Deep Learning Framework for Solving High-Order Partial Differential Equations

Solving partial differential equations (PDEs) has been a fundamental problem in computational science and of wide applications for both scientific and engineering research. Due to its universal approximation property, neural network is widely used to approximate the solutions of PDEs. However, existing works are incapable of solving high-order PDEs due to insufficient calculation accuracy of higher-order derivatives, and the final network is a black box without explicit explanation. To address these issues, we propose a deep learning framework to solve high-order PDEs, named SHoP. Specifically, we derive the high-order derivative rule for neural network, to get the derivatives quickly and accurately; moreover, we expand the network into a Taylor series, providing an explicit solution for the PDEs. We conduct experimental validations four high-order PDEs with different dimensions, showing that we can solve high-order PDEs efficiently and accurately. The source code can be found at https://github.com/HarryPotterXTX/SHoP.git.

Computational Approach on Convective-Magneto Trihybrid Nanoflow with Space Dependent Energy Source/Sink: Drug Delivery

Iron oxide, Silver, Aluminium oxide, these nanoparticles individually or combined help in drug delivery especially aluminium oxide nanofluid used in an anti-blood pressure drug called 'Telmisartan'. Alumina and silver particles are used in manufacturing nanocomposites which have more antimicrobial properties. This is the reason for the current study of ternary nanofluids natural convective flow and efficacy of energy transfer in a bi-directionally sheet. Currently, ternary nanofluids (Fe3O4, Ag, Al2O3) are being taken for analysis. Usual water (H2O) is the conventional base fluid. Two different combinations of ternary nanofluids are used to get the average heat transfer rate, mixture ratios (Fe3O4 + Ag) and (Fe3O4 + Ag + Al2O3) when subjected to a variety of physical influences, including thermal radiation, magnetic fields, heat production and absorption, and nanoparticle volume amount. It is possible to solve the developed set of equations numerical results using the Keller box (finite differences) method with the help of MATLAB programming. This method helps in solving higher order partial differential equation (PDEs) to ordinary differential equation (ODEs). The investigations findings demonstrated that the ternary hybrid nanofluids specific heat capacity is directly impacted by temperature. Numerical solutions for Nusselt number, velocity profile, skin friction coefficient, temperature profiles have represented with the help of graphs and tables. The ternary hybrid nanoflow (Fe3O4 + Ag +  Al2O3/H2O) transmits more energy for increasing volume fractions, comparing to the hybrid nanofluid (Fe3O4 + Ag/H2O). The study reveals the fact that metal oxides transfer more heat from the system than that of metals. The estimated error of heat transfer rate is higher in alumina nanoflow followed by silver and iron oxide nanofluid flows. The streamlines are equally spaced, but the energy flow amount is higher for the case M, S = 1 than M, S = 2. But in the case of stretching ratio parameter, the amount energy flow in α = 0.5 > α = 1.0. An equal amount of energy flow is observed for varied Biot number.

Hutchinson Trace Estimation for high-dimensional and high-order Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) have proven effective in solving partial differential equations (PDEs), especially when some data are available by seamlessly blending data and physics. However, extending PINNs to high-dimensional and even high-order PDEs encounters significant challenges due to the computational cost associated with automatic differentiation in the residual loss function calculation. Herein, we address the limitations of PINNs in handling high-dimensional and high-order PDEs by introducing the Hutchinson Trace Estimation (HTE) method. Starting with the second-order high-dimensional PDEs, which are ubiquitous in scientific computing, HTE is applied to transform the calculation of the entire Hessian matrix into a Hessian vector product (HVP). This approach not only alleviates the computational bottleneck via Taylor-mode automatic differentiation but also significantly reduces memory consumption from the Hessian matrix to an HVP’s scalar output. We further showcase HTE’s convergence to the original PINN loss and its unbiased behavior under specific conditions. Comparisons with the Stochastic Dimension Gradient Descent (SDGD) highlight the distinct advantages of HTE, particularly in scenarios with significant variability and variance among dimensions. We further extend the application of HTE to higher-order and higher-dimensional PDEs, specifically addressing the biharmonic equation. By employing tensor-vector products (TVP), HTE efficiently computes the colossal tensor associated with the fourth-order high-dimensional biharmonic equation, saving memory and enabling rapid computation. The effectiveness of HTE is illustrated through experimental setups, demonstrating comparable convergence rates with SDGD under memory and speed constraints. Additionally, HTE proves valuable in accelerating the Gradient-Enhanced PINN (gPINN) version as well as the Biharmonic equation. Overall, HTE opens up a new capability in scientific machine learning for tackling high-order and high-dimensional PDEs.

A gradient reproducing kernel based stabilized collocation method for the 5th order Korteweg–de Vries equations

This paper presents a gradient reproducing kernel based stabilized collocation method (GRK-SCM) to solve the generalized nonlinear 5th order Korteweg–de Vries (KdV) equations. By introducing gradient reproducing kernel (GRK) approximations, one can circumvent the intricacy of high-order derivatives in RK approximations, while still satisfying high-order consistency requirements. This leads to the high accuracy and efficiency. GRKs are very good approximation candidates for addressing high order partial differential equations that require higher order derivatives formulated as strong form. The stabilized collocation method (SCM) effectively meets high-order integration constraints which can achieve accurate integration in the subdomains. This ensures both high precision and optimal convergence. Von Neumann analysis is utilized to establish the stability criteria for GRK-SCM when combined with forward difference temporal discretization. Numerical solutions for Sawada–Kotera (SK) equation and Kaup-Kupershmidt (KK) equation are studied where the solitary wave migration and collision, and periodic waves are represented. A fifth order forced KdV equation is also considered. The presented method's high accuracy and convergence are demonstrated through comparative studies with analytical solutions, and investigations of invariants and corresponding errors determine the good conservation properties of this algorithm.

New Analytic Free Vibration Solutions of L-Shaped Moderately Thick Plates by Symplectic Superposition

L-shaped moderately thick plates have widespread applications in diverse engineering structures. Exploring the benchmark analytic solutions for the free vibration of L-shaped moderately thick plates is important in order to accurately analyze and efficiently design structures. Nevertheless, analytical solutions, which serve as the benchmarks, have been rarely documented in previous literature due to the challenge of finding suitable solutions that satisfy both the governing higher-order partial differential equations (PDEs) and the boundary conditions of the plates. The symplectic superposition method was employed in our recently published study to present the benchmark vibration solutions of clamped rectangular moderately thick plates. In this study, we expand upon this method to solve the free vibration problem of L-shaped moderately thick plates. By employing domain decomposition, we construct an irregular domain by combining multiple rectangular domains. First, the construction of the superposition system is carried out, followed by the import of the sub-problems into the Hamiltonian system, utilizing the fundamental governing equations of the plate. Then, the sub-problems are resolved through the application of the symplectic geometry methodology in an analytical manner. Ultimately, the analytical solutions for frequencies and mode shapes are derived through ensuring the equivalence between the initial problem and the combination of sub-problems. The finite element method is used to validate and present a comprehensive analysis of the natural frequencies and mode shapes obtained from this method. This method possesses the benefits of rapid convergence and accurate precision, rendering it well suited for the analytic modeling of a broader range of plate-related problems.

Applications of regime-switching in the nonlinear double-diffusivity (D-D) model

The linear double-diffusivity (D-D) model of Aifantis, comprising two coupled Fick-type partial differential equations and a mass exchange term connecting the diffusivities, is a paradigm in modeling mass transport in inhomogeneous media, e.g., fissures or fractures. Uncoupling of these equations led to a higher order partial differential equation that reproduced the non-classical transport terms, analyzed independently through Barenblatt’s pseudoparabolic equation and the Cahn–Hilliard spinodal decomposition equation. In the present article, we study transport in a nonlinearly coupled D-D model and determine the regime-switching of the associated diffusive processes using a revised formulation of the celebrated Lux method that combines forward Fourier transform with a Laplace transform followed by an Inverse Fourier transform of the governing reaction–diffusion (R–D) equations. This new formulation has key application possibilities in a wide range of non-equilibrium biological and financial systems by approximating closed-form analytical solutions of nonlinear models.

Hamiltonian system-based analytic thermal buckling solutions of orthotropic rectangular plates

New analytic solutions for the thermal buckling of orthotropic rectangular plates are presented using the symplectic superposition method (SSM). The SSM deals with higher-order partial differential equations within the Hamiltonian framework, distinguishing it from conventional methods in the Lagrangian framework. The solution procedure involves dividing the original problem into two subproblems, utilizing the techniques in the symplectic space, like separating variables and symplectic eigen expansion, to achieve analytic solutions, and using superposition to obtain the final solutions. One notable advantage of the SSM is the absence of predetermined assumptions, which is a departure from conventional semi-inverse methods. The new analytic solutions under challenging non-Lévy-type boundary conditions (BCs) were rarely addressed in prior research due to their complex mathematical characteristics, which reveals the main novelty. Comprehensive thermal buckling results are provided for isotropic/orthotropic plates under various BCs, validated by those from the finite element method with a maximum difference being 2.7374% and literature with a maximum difference being 5.9201%. Parameter analyses quantify the effects of BCs, aspect ratio, and thickness-to-length ratio on critical buckling temperatures and mode shapes.

MHD flow of a rotating micropolar fluid between two plates by using DTM

The primary purpose of this work is to investigate the influence of a magnetic field on a micropolar fluid, which is rotating between two horizontally arranged parallel plates. A uniform magnetic field acts perpendicularly to the porous plate, which consumes fluid with a steady suction velocity. The obtained higher-order nonlinear partial differential equations (PDEs) are converted into nonlinear ordinary differential equations (ODEs) by using similarity transformation. A semi-analytical methodology has been applied to obtain the result to the problem. The influence of various flow parameters of the flow field like coupling parameter, viscosity parameter, Reynolds number, porosity parameter, rotation parameter and magnetic field parameter has been examined and explained by graphs. A resemblance between the results obtained by solving by the differential transform method (DTM) and the outcomes of the former research works has also been displayed which shows the correctness and efficiency of the used analytical method. The numerical values of the coefficient of skin friction at the lower plate are also displayed in tabular form and compared with the literature.

Computational analysis of melting radiative heat transfer for solar Riga trough collectors of Jeffrey hybrid-nanofluid flow: A new stochastic approach

Current investigation deals with the melting heat transfer for the Jeffrey hybrid-nanofluid flow in parabolic trough solar collectors through Darcy Forchheimer porous media over a variable thick vertical elongation Riga surface under the effect of solar radiation. The impacts of viscous dissipation and Joule heating are also investigated. Equations governing the Jeffrey’s hybrid nanofluid flow are higher-order non-linear partial differential equations (PDEs). These governing PDEs are transformed into the non-linear ordinary differential equations (ODEs) by introducing appropriate similarity transformations and dimensionless parameters. Runga Kutta’s fourth-order numerical scheme is implemented with the shooting technique to solve coupled higher-order ODEs. Results for velocity profile, temperature profile, drag coefficient and Nusselt number are discussed for various influential parameters. Artificial neural networking is also performed to predict Nusselt numbers in different cases and scenarios. The Artificial Neural Network gives the desired outputs with the highest possible accuracy. It is observed that the temperature profile rises with increase in Hartmann number, porosity parameter, and Forchheimer number. However thermal profile worsens with escalates in the exponential index. Radiant energy of the Sun is a renewable energy source available in considerable amounts in our living environment. A parabolic trough solar collector is an efficient solar collector that stores the concentrated incoming radiant energy from the Sun to fulfill the requirement of high temperatures in thermal energy storage systems. It has various applications in solar-powered appliances for cooking and air conditioning, solar-powered systems for treating wastewater, photovoltaic lighting, solar-powered cars, aircraft, etc.

New Analytical Solutions for Elastoplastic Buckling of Non-Lévy‐Type Rectangular Plates

Analytical solutions for elastoplastic buckling of plates play a crucial role in providing benchmark results and facilitating fast structural analyses for preliminary designs. However, the analytical solutions for elastoplastic buckling of plates remain incomplete due to the inherent mathematical difficulties associated with higher-order partial differential equations and material nonlinearity. Consequently, the existing analytical solutions for rectangular plates are only applicable to those with Lévy‐type boundary conditions. To address the limitation, this study extends a novel symplectic superposition method to obtain new analytical elastoplastic buckling solutions of non-Lévy‐type rectangular plates, where both the incremental theory (IT) and deformation theory (DT) are adopted. Comprehensive benchmark elastoplastic buckling loads are presented and validated by the modified differential quadrature method. The plastic buckling paradox is explicitly observed, which highlights a significant disparity between the IT and DT in predicting buckling loads for relatively thick plates. Furthermore, our analysis reveals that the DT provides an optimal load ratio for buckling resistance, while no such finding is observed with the IT. The stability criterion curves are plotted using the elastic theory and plastic theories (IT and DT) to further reveal the importance of incorporating the effect of plasticity as well as to provide a useful guideline for the relevant analyses and designs.

Straightforward Free Vibration Solutions of Open Cylindrical Shells by the Finite Integral Transform Method

Straightforward analytical free vibration solutions of non-Lévy-type open cylindrical shells are obtained in this study by the finite integral transform method, which was not achieved by conventional semi-inverse methods. Three double finite integral transforms are imposed on the high-order partial differential equations (PDEs) for the problems, with some boundary conditions input, by which the relations between the transformed quantities and specific unknowns are given. Substituting the corresponding inverse transforms into the other boundary conditions provides systems of linear algebraic equations for determining the final analytical solutions. Comprehensive benchmark results for representative open cylindrical shells under different boundary conditions are shown. The results are well verified by other solution methods. With the present solutions, the effects of geometric parameters and boundary conditions on the vibration behaviors are quantitatively revealed by a parametric study. Due to its rigorous and straightforward solution procedure, this study presents a solid easy-to-implement way to explore new analytical solutions of similar problems.

Two-dimensional (p,q)-heat polynomials of Gould–Hopper type

We introduce a new class of holomorphic polynomials extending the classical Gould–Hopper to two complex variables. The considered polynomials include the 1-D and 2-D holomorphic and polyanalytic Itô–Hermite polynomials as particular cases. We emphasize studying their operational representation, various generating functions, and recurrence relations. We also establish some special identities including multiplication, Runge addition and Nielson types formulas. Higher-order partial differential equations are analyzed and the connection to Gould-Hopper polynomials and hypergeometric functions are investigated.

Intersection numbers from higher-order partial differential equations

We propose a new method for the evaluation of intersection numbers for twisted meromorphic n-forms, through Stokes’ theorem in n dimensions. It is based on the solution of an n-th order partial differential equation and on the evaluation of multivariate residues. We also present an algebraic expression for the contribution from each multivariate residue. We illustrate our approach with a number of simple examples from mathematics and physics.

Stability and nonlinear vibration of carbon nanotubes-reinforced composite pipes conveying fluid

This paper focuses on the stability and nonlinear vibration characteristics of carbon nanotubes-reinforced composite pipes conveying fluid, and the mechanism of the combined effects of multiple factors. Based on the assumption of Euler beam and Von-karman nonlinear stress-strain relationship, the dynamic governing equations and generalized boundary conditions of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) pipes are established by Hamiltonian variational principle. Firstly, the modal function suitable for generalized boundary conditions is derived by Galerkin method. The stability of FG-CNTRC pipes system and the relationship between natural frequency and flow velocity are analyzed in this paper. The effect of flow velocity on the natural frequency of pipelines is examined under various carbon nanomaterial distribution patterns and wall thicknesses in this research. Then, the nonlinear vibration characteristics of FG-CNTRC pipes are analyzed by multi-scale method. The method of deducing nonlinear frequency through high-order partial differential equations is also extended in nonlinear analysis. The results show that, in the case of uniform distribution of carbon nanotubes, increasing the wall thickness can improve the stability of the pipe system. The influence of amplitude on the nonlinear frequency is more obvious at the second-order frequency.

Numerical investigation of the magnetohydrodynamic hybrid nanofluid flow over a stretching surface with mixed convection: A case of strong suction

The present work explores the physical aspects of the alumina and silver nanoparticles on the magnetohydrodynamic (MHD) flow of mixed convection micropolar hybrid nanofluid with ethylene glycol + water [Formula: see text] base fluid via stretching surface embedded in a porous medium. A strong magnetic field is employed normally in the flow direction. The behavior of the suction on the presented flow analysis is discussed strongly. Heat transport phenomena are analyzed. The current model’s mathematical modeling is based on higher-order nonlinear partial differential equations, which are then translated into higher-order nonlinear ordinary differential equations using appropriate similarity transformations. The modeled higher-order nonlinear ordinary differential equations are solved using NDSolve technique. The physical significance of the different flow parameters on the velocity, microrotation, and temperature profiles of the hybrid nanofluid are described in a graphical form. In a tabular form, the skin friction coefficients for nanofluid and hybrid nanofluid against various flow parameters are calculated. Some important results from this investigation are demonstrated that the velocity of the hybrid nanofluid is higher for the stretching ratio parameter and it is detected that the suction parameter enhanced the microrotation profile of the hybrid nanofluid. From the comparison, it is noted that the velocity, microrotation, and temperature of the hybrid nanofluid are higher as compared to the velocity, microrotation, and temperature of the alumina nanofluid and silver-nanofluid.

Investigating and Mitigating Failure Modes in Physics-Informed Neural Networks (PINNs)

This paper explores the difficulties in solving partial differential equations (PDEs) using physics-informed neural networks (PINNs). PINNs use physics as a regularization term in the objective function. However, a drawback of this approach is the requirement for manual hyperparameter tuning, making it impractical in the absence of validation data or prior knowledge of the solution. Our investigations of the loss landscapes and backpropagated gradients in the presence of physics reveal that existing methods produce non-convex loss landscapes that are hard to navigate. Our findings demonstrate that high-order PDEs contaminate backpropagated gradients and hinder convergence. To address these challenges, we introduce a novel method that bypasses the calculation of high-order derivative operators and mitigates the contamination of backpropagated gradients. Consequently, we reduce the dimension of the search space and make learning PDEs with non-smooth solutions feasible. Our method also provides a mechanism to focus on complex regions of the domain. Besides, we present a dual unconstrained formulation based on Lagrange multiplier method to enforce equality constraints on the model's prediction, with adaptive and independent learning rates inspired by adaptive subgradient methods. We apply our approach to solve various linear and non-linear PDEs.

Adaptive isogeometric methods with C1 (truncated) hierarchical splines on planar multi-patch domains

Isogeometric analysis is a powerful paradigm which exploits the high smoothness of splines for the numerical solution of high order partial differential equations. However, the tensor-product structure of standard multivariate B-spline models is not well suited for the representation of complex geometries, and to maintain high continuity on general domains special constructions on multi-patch geometries must be used. In this paper, we focus on adaptive isogeometric methods with hierarchical splines, and extend the construction of [Formula: see text] isogeometric spline spaces on multi-patch planar domains to the hierarchical setting. We replace the hypothesis of local linear independence for the basis of each level by a weaker assumption, which still ensures the linear independence of hierarchical splines. We also develop a refinement algorithm that guarantees that the assumption is fulfilled by [Formula: see text] splines on certain suitably graded hierarchical multi-patch mesh configurations, and prove that it has linear complexity. The performance of the adaptive method is tested by solving the Poisson and the biharmonic problems.

Vibration Analysis of a Unimorph Nanobeam with a Dielectric Layer of Both Flexoelectricity and Piezoelectricity.

In this study, for the first time, free and forced vibrational responses of a unimorph nanobeam consisting of a functionally graded base, along with a dielectric layer of both piezoelectricity and flexoelectricity, is investigated based on paradox-free local/nonlocal elasticity. The formulation and boundary conditions are attained by utilizing the energy method Hamilton's principle. In order to set a comparison, the formulation of a model in the framework of differential nonlocal is first presented. An effective implementation of the generalized differential quadrature method (GDQM) is then utilized to solve higher-order partial differential equations. This method can be utilized to solve the complex equations whose analytic results are quite difficult to obtain. Lastly, the impact of various parameters is studied to characterize the vibrational behavior of the system. Additionally, the major impact of flexoelectricity compared to piezoelectricity on a small scale is exhibited. The results show that small-scale flexoelectricity, rather than piezoelectricity, is dominant in electromechanical coupling. One of the results that can be mentioned is that the beams with higher nonlocality have the higher voltage and displacement under the same excitation amplitude. The findings can be helpful for further theoretical as well as experimental studies in which dielectric material is used in smart structures.

On the Non-Fourier’s Heat and Non-Fick’s Mass Flux for the Quadratic Convection Flow of a Couple Stress Nanofluid with Wu’s Slip

The overall goal of this paper is to look at how non-Fick’s and non-Fourier’s concepts affect the quadratic convection flow of a couple stress nanofluid across an extending surface under Wu’s slip, convective heating, and convective mass transfer conditions. The governing equations in the investigation are also initially stated as higher-order partial differential equations and thereafter rehabilitated into ordinary differential equations via similarity conversions. Finally, the numerical approach of the bvp5c solver in MATLAB R2019a is engaged to solve the resultant equations. It has been researched and shown using graphs and tables how physical characteristics affect the velocity, temperature, and concentration fields. The results show that the velocity profile along the x-direction grows with intensifying values of the couple stress, quadratic convection, buoyancy, and third-order slip constraints, whereas it declines with intensifying values of the fourth-order slip and magnetic field constraints. Also, the results show that as the velocity ratio constraint is increased, the velocity field along the y-axis increases. Furthermore, an augment in the values of the quadratic convection constraint and Schmidt number, respectively, causes a diminution in temperature and concentration distributions. The results of the current study can be used to increase lubricant production and reduce machinery deterioration. Furthermore, this theoretical and numerical deliberation has the potential to be beneficial in the field of bio-fluid dynamics and biomedicine.

Fictitious finite integration method for solving high order partial differential equations

In this paper, we combine the finite integration method (FIM) with the technique of fictitious points to solve high order partial differential equations (HOPDEs). Two type of fictitious finite integration method (FFIM) are proposed and analyzed. The first type FFIM (FFIM-I) constructs the first order integration matrices by shifted Chebyshev polynomials so that the analytical high order integration matrices can be recursively obtained. Compared with the classical FIM, whose high order integration matrices are approximated directly through low order integration matrices, this FFIM-I leads to higher accuracy. Based on FFIM-I, the second type FFIM (FFIM-II) is proposed in which the fictitious points are located freely either inside or outside the problem domain. The FFIM-II can further improve the performance of FFIM-I in solving HOPDEs. The advantage of this fictitious finite integration method will be demonstrated by making comparisons among FIM, FFIM-I and FFIM-II through several 2D and 3D numerical experiments.