Original Partial Differential Equation Research Articles

A hierarchically normalized physics-informed neural network for solving differential equations: Application for solid mechanics problems

Physics-informed neural networks (PINNs) usually confront significant difficulties to accurately solve partial differential equations (PDEs) due to many pathologies caused by gradient failures during training process. In this paper, a gradient disease regarding the ill-conditioned loss function of the PINN methodology is intensively investigated in both theoretical and experimental aspects. Particularly, the regular PINN approaches can obtain good predictions for trivial problems but may fail to learn solutions when their PDE coefficients and/or physical domain sizes vary. Besides, the influences of neural network structure to the PINN training quality are inferred from the theoretical analyses and also confirmed by relating experiments, in which using large-width multi-layer perceptrons (MLPs) is proved to be beneficial in stabilizing PINN training process. To overcome the abovementioned restrictions, a novel PINN methodology using hierarchically normalized technique (hnPINN) motivated from the developed theory is devised. The key idea of the proposed hnPINN is, on the one hand, to transform the original PDE system into one of two proposed dimensionless forms to alleviate the negative effects of PDE coefficients and domain size; on the other hand, to use secondary output scalers to flexibly calibrate the gradient flow for training effectively and improving the solution preciseness. The determination of the secondary output scaler is formulated by a heuristic framework inspired from theoretical analyses on the hnPINN gradient flow. The obtained results from some typical PDEs and common problems in solid mechanics strongly confirm the high effectiveness of the hnPINN in terms of solution accuracy, superior convergence and performance stability in comparison to those of the vanilla PINN and the non-dimensionalized PINN (ndPINN). As a preprocessing procedure, the hnPINN method is independent on network architectures and also greatly potential to combine with other state-of-the-art PINN models for effectively solving challenged issues in practice.

Analytical solutions to the compressible Euler equations with cylindrical symmetry and free boundary

In this paper, we study the analytical solutions to the compressible Euler equations with cylindrical symmetry and free boundary. We assume that the free boundary is moving in the radial direction with the radial velocity, which will affect the angular velocity but does not affect the axial velocity. By using some ansatzs, we reduce the original partial differential equations into an ordinary differential equation about the free boundary. We prove that the free boundary grows linearly in time by constructing some new physical functionals. Furthermore, the analytical solutions to the compressible Euler equations with time-dependent damping are also considered and the spreading rate of the free boundary is investigated according to the various sizes of the damping coefficients.

Engine oil blended Casson hybrid nanofluid flow along a uniformly heated curved surface with Arrhenius activation energy and suction: A computational study

The research studies the numerical analysis of an incompressible Casson hybrid nanofluid, composed of Fe3O4–Cu nanoparticles blended with engine oil, flowing over an exponentially stretched curved surface. It explores the effects of an aligned magnetic field, a uniform heat source, activation energy, and suction while maintaining a low Prandtl number. By employing appropriate similarity transformations, the original partial differential equations are converted into ordinary ones and solved using bvp4c in MATLAB. Graphs and tables depict the influence of dimensionless factors on skin friction coefficient, velocity, thermal, and concentration profiles, as well as Nusselt and Sherwood numbers. The computational findings reveal that magnetic field alignment significantly modifies velocity, decreasing it while uniformly enhancing temperature and concentration profiles. Casson fluid behavior in engine oil amplifies temperature and concentration profiles, thereby enhancing heat and mass transfer rates. Moreover, an increased heat source elevates temperature but reduces overall heat transfer. The chemical reaction parameter diminishes concentration, whereas activation energy enhances it, albeit impacting the mass transfer rate conversely. Additionally, suction increases flow resistance, while the stretching parameter exhibits an inverse effect, both positively influencing heat and mass transfer rates at higher values. Practically, these findings offer insights crucial for optimizing heat exchangers, biomedical devices, and cooling systems, enhancing their efficiency and performance. Additionally, they inform the design of industrial processes involving fluid flow, aiding in the development of more effective and sustainable solutions.

Entropy optimization on magnetized radiative bioconvective flow of Carreau-Yasuda hybrid CNTs through a spherical surface in a non-Darcian porous medium with activation energy

This work inspects entropy generation and heat transfer induced by a bioconvection slip flow of nonlinear radiative Carreau-Yasuda hybrid nanofluid (NF) over a convectively heated sphere. Activation energy for microbes is contemplated in order to comprehend its contribution toward flow features. The original partial differential equations (PDEs) are rendered non-dimensional through appropriate conversions, and the resulting PDEs are solved using the overlapping grid spectral collocation algorithm. The impact of diverse flow factors on flow profiles, entropy generation, skin friction, heat, mass, and motile microbes transport rates is analyzed. Key outcomes reveal that hybrid NF flow demonstrates a supplementary indispensable feature in the operation of heat transport compared to mono NF flow. More entropy is generated in the system by using the hybrid NF model along with magnetic field, heat source, convective heating, nonlinear radiation, and viscous dissipation. Thermal fields and rate of heat transport are improved by including nonlinear radiative heat flux in the system. Mass and motile microbes transport rates are respectively enhanced by chemical and microbial reactions, but both quantities are reduced by activation energy. The effects of microbial reactions on flow quantities substantiate the significance of these features for the dynamics of microorganisms. The findings of this study can be useful in the upsurge of thermal performance of the working fluid and contribute toward the improvement of microbial fuel cell performance.

Дополнительные граничные условия в задачах теплопроводности с переменным по координате начальным условием

It is exceedingly difficult to obtain mathematically accurate analytical solutions of heat conduction problems with a variable initial condition. Known solutions of these problems are expressed by cumbersome functional series that converge poorly in the range of small values of time and space variables. Thus, to obtain simpler and more effective solutions of these problems is an urgent issue. The authors have used an additional required function and additional boundary conditions to obtain solutions of the problem. Application of the additional required function allows us to reduce the original partial differential equation to the integration of an ordinary differential equation. Additional boundary conditions are in such a form that their fulfillment using the resulting solution is equivalent to the fulfillment of the equation at the boundary points. The authors have developed a technique to obtain an analytical solution of the heat conduction problem under a linear change of the initial condition, based on an additional required function and additional boundary conditions. Solution of an ordinary differential equation with respect to the additional required function determines the eigenvalues. In classical methods these eigenvalues are found in the solution of the Sturm–Liouville boundary value problem. The authors have proposed another, simpler solution to determine eigenvalues. An accurate analytical solution of the heat conduction problem for an unbounded plate with a coordinate-variable initial condition is obtained. The scientific and practical value of the proposed analytical solution is the development of an innovative approach to determine eigenvalues, as well as elimination of complex integrals when we solve the equation and initial conditions of the boundary value problem. It makes possible to simplify the use of the solution obtained in engineering applications.

Invariance Analysis and Closed-form Solutions for The Beam Equation in Timoshenko Model

Our research focuses on a fourth-order partial differential equation (PDE) that arises from the Timoshenko model for beams. This PDE pertains to situations where the elastic moduli remain constant and an external load, represented as F, is applied. We thoroughly analyze Lie symmetries and categorize the various types of applied forces. Initially, the principal Lie algebra is two-dimensional, but in certain noteworthy cases, it extends to three dimensions or even more. For each specific case, we derive the optimal system, which serves as a foundation for symmetry reductions, transforming the original PDE into ordinary differential equations. In certain instances, we successfully identify exact solutions using this reduction process. Additionally, we delve into the conservation laws using a direct method proposed by Anco, with a particular focus on specific classes within the equation. The findings we have presented in our study are indeed original and innovative. This study serves as compelling evidence for the robustness and efficacy of the Lie symmetry method, showcasing its ability to provide valuable insights and solutions in the realm of mathematical analysis.

Magneto-convective hybrid nanofluid slip flow over a moving inclined thin needle in a Darcy-Forchheimer porous medium with viscous dissipation

PurposeThe purpose of this study is to provide that porous media and viscous dissipation are crucial considerations when working with hybrid nanofluids in various applications.Recent years have witnessed significant progress in optimizing these fluids for enhanced heat transfer within porous (Darcy–Forchheimer) structures, offering promising solutions for various industries seeking improved thermalmanagement and energy efficiency.Design/methodology/approachThe first step is to transform the original partial differential equations into a system of first-order ordinary differential equations (ODEs). The fourth-order Runge–Kutta method is chosen for its accuracy in solving ODEs. The present study investigates the free convective boundary layer flow of hybrid nanofluids over a moving thin inclined needle with the slip flow brought about by inclined Lorentz force and Darcy–Forchheimer porous matrix, viscous dissipation.FindingsIt is found that slip conditions (velocity and Thermal) exist for a range of the natural convection boundary layer flow. In the hybrid nanofluid flow, which consists of Al2O3 and Fe3O4 are nanoparticles, H2O − C2H6O2 (50:50) are considered as the base fluid. The consequence of the governing parameter on the momentum and temperature profile distribution is graphically depicted. The range of the variables is 1 ≤ M ≤ 4, 1 ≤ d ≤ 2.5, 1 ≤ δ ≤ 4, 1 ≤ Fr ≤ 7, 1 ≤ Kr ≤ 7 and 0.5≤λ ≤ 3.5. The Nusselt number and skin friction factors are used to calculate the numerical values of various parameters, which are displayed in Table 4. These analyses elucidate that upsurges in the value of the Fr noticeably diminish the momentum and temperature. It is investigated to see if the contemporary results are in outstanding promise with the outcomes reported in earlier works.Practical implicationsThe results can be very helpful to improve the energy efficiency of thermal systems.Social implicationsThe hybrid nanofluids in heat transfer have the potential to improve the energy efficiency and performance of a wide range of systems.Originality/valueThis study proposes that in the combined effects of hybrid nanofluid properties, the inclined Lorentz force, the Darcy–Forchheimer model for porous media and viscous dissipation on the boundary layer flow of a conducting fluid over a moving thin inclined needle. Assessing the potential practical applications of the hybrid nanofluids in inclined needles, this could involve areas such as biomedical engineering, drug delivery systems or microfluidic devices. In future should explore the benefits and limitations of using hybrid nanofluids in these applications.

Дополнительные граничные условия в задачах теплопроводности с неоднородными граничными условиями

Using additional boundary conditions (ABC) and an additional sought function (ASF), a solution to the heat conduction problem with non-homogeneous boundary conditions has been obtained. ABC allows satisfying the differential equation at the boundaries, leading to its fulfillment both inside the domain and without resorting to integration over Cartesian coordinates. ASF reduces the original partial differential equation to a temporal ordinary equation, from which the eigenvalues of the boundary value problem are determined, as formulated in the classical methods of the Sturm-Liouville problem, stated in spatial variables. Thus, this work considers an alternative way of determining eigenvalues. The integration constants are found from the initial condition using the least squares method, which allows their values to be determined with a given accuracy. The solution obtained based on ABC and ASF approximates n → ∞ the exact analytical solution in the form of an infinite series, including trigonometric coordinate functions with coefficients that stabilize exponentially in time. In this case, the eigenvalues determined from the solution of the temporal ordinary differential equation regarding the additional sought function coincide with their exact values at any approximation. The accuracy of the integration constants, determined by the method of least squares, is controlled by the number of approximation points in the range of the auxiliary variable's variation. It should be noted that the additional boundary conditions considered in this work hold true for any other method of obtaining a solution to the problem under consideration, including the exact method, as can be verified by direct substitution. Therefore, their introduction does not distort the original mathematical formulation of the problem but only significantly simplifies the process of obtaining its analytical solution.

SONets: Sub-operator learning enhanced neural networks for solving parametric partial differential equations

Complex nonlinear partial differential equations (PDEs) can be decomposed into subsystems comprising certain linear and nonlinear sub-operators. We usually have fundamental solutions to the linear subsystems and have a good knowledge of their uniqueness and regularity. The nonlinear subsystems have more straightforward formulas than the original complex nonlinear PDEs, even with reduced dimensions. Solutions of the complex nonlinear PDEs based on simpler subsystems are investigated in this study. We derive a proper subsystem decomposition of the nonlinear PDEs into linear and nonlinear subsystems to get unique solutions. We propose the sub-operator learning enhanced neural networks (SONets) based on this proper subsystem decomposition to solve nonlinear PDEs. The solutions of nonlinear subsystems with generated force terms are modeled via operator learning. In contrast, only the cumbersome integration and series parts of fundamental solutions of linear subsystems are modeled. The complex nonlinear PDEs are then solved by finding the force terms to minimize the physics-informed loss functions. It is also convenient to generalize SONets for a new PDE parameter, e.g., the diffusion coefficient. Next, we extend the SONets to a meta version (meta-SONets) for parametric PDEs by transfer learning. In meta-SONets, the initial conditions are explicitly encoded via operator learning neural networks (ENNs) to output force terms, followed by the whole SONets to solve PDEs. In the pre-training stage of meta-SONets, parametric PDEs given randomly generated initial conditions are solved by training ENNs to minimize the physics-informed loss functions. At the fine-tuning stage, the trainable parameters of ENNs are fine-tuned to solve the PDE given a new initial condition. Numerical experiments are conducted to solve the Burgers equation, the nonlinear diffusion-reaction system, and the Allen-Cahn equation. The numerical results indicate that SONets with neural network-parameterized force terms obtain solutions with good accuracy. With transfer learning, the computational cost of meta-SONets for solving the nonlinear PDEs with a new initial condition is dramatically reduced.

Nonparametric probabilistic forecasting based stochastic optimal scheduling of integrated electricity and gas systems

The volatility and sporadic availability of renewable energy create significant challenges to the optimal scheduling of integrated electricity and gas systems (IEGS). This paper develops a nonparametric probabilistic forecasting based stochastic scheduling approach of IEGS. The quantile at a series of quantile levels can be generated by direct quantile regression method. Given the set of predicted quantiles, a set of representative scenarios for wind power uncertainty can be obtained by using Monte Carlo simulation method and scenario reduction approach. Based on the implicit finite difference scheme, the original partial differential equations of the gas network are discretized to establish an algebraic model, which provide possibility for efficient solution. Then, the nonconvexity caused by the momentum equation is eliminated by the second-order core relaxation. Finally, the stochastic optimal scheduling model is reformulated as a second-order core programming problem. Numerical simulations are performed to showcase the superiority of the established stochastic optimal scheduling model.

Entropy optimization in mixed convective boundary layer flow of CNTs Casson nanoliquid through a vertical cylinder moving with nonlinear velocity

This article inspects entropy generation and mixed convection boundary layer flow of carbon nanotubes (CNTs) Casson nanoliquid via semi-infinite vertical cylinder which moves with nonlinear velocity in Darcy-Forchheimer porous medium. Appropriate similarity variables have been employed to convert the original partial differential equations into ordinary differential equations that have been solved using overlapping grid spectral quasilinearisation method (SQLM). Comparison of accuracy, convergence and stability between the two methods is made. Rate of entropy generated, flow fields and engineering quantities are discussed for different embedding parameters. It is revealed that single-wall CNTs are more efficacious to improve heat transport features than multi-wall CNTs. The fluid flow and thermal dispersion processes improve with consideration of curved surface, CNTs, non-Newtonian fluid and injection of the fluid. The curvature surface and suction of the fluid contribute towards growth of skin friction factor and rate of thermal transference. Entropy generated expands by accounting for viscous nature of the fluid, strong nonlinearity, suction, non-Newtonian fluid, convective boundary condition and CNTs. The study finds applications in various processes, which are massively impacted by heat transport enhancement and high porosity. Boundary layer flow and heat transfer analysis through cylinders are relevant to various metallurgical and engineering solicitations.

Lie analysis and nonlinear propagating waves of the (3 + 1)-dimensional generalized Boiti–Leon–Manna–Pempinelli equation

The (3+1)-dimensional generalized Boiti-Leon-Manna-Pempinelli equation, which describes the evolution of Riemann wave propagation in an in-compressible fluid along three mutually perpendicular axes, is investigated in this research article. The study encompasses the analysis of Lie symmetries, corresponding symmetry reductions, and conservation laws associated with the equation. The Lie symmetry method is employed to determine the infinitesimal generators of the considered equation. Furthermore, commutator table is generated, and symmetry groups corresponding to each infinitesimal generator are derived. By utilizing the similarity reduction technique, the original nonlinear partial differential equation is transformed into nonlinear ordinary differential equations. Then, generalized exp(−ϕ(ζ)) expansion technique is utilized to solve the reduced equations and estimate specific traveling wave solutions for the equation. Moreover, graphical representations are employed to illustrate the traveling wave solutions, employing suitable parameter values. Additionally, the multiplier approach is utilized to calculate conserved vectors for the equation under consideration.

Bilateral boundary control of an input delayed 2-D reaction–diffusion equation

In this paper, a delay compensation design method based on PDE backstepping is developed for a two-dimensional reaction–diffusion partial differential equation (PDE) with bilateral input delays. The PDE is defined in a rectangular domain, and the bilateral control is imposed on a pair of opposite sides of the rectangle. To represent the delayed bilateral inputs, we introduce two 2-D transport PDEs that form a cascade system with the original PDE. A novel set of backstepping transformations is proposed for delay compensator design, including one Volterra integral transformation and two affine Volterra integral transformations. Unlike the kernel equation for 1-D PDE systems with delayed boundary input, the resulting kernel equations for the 2-D system have singular initial conditions governed by the Dirac Delta function. Consequently, the kernel solutions are written as a double trigonometric series with singularities. To address the challenge of stability analysis posed by the singularities, we prove a set of inequalities by using the Cauchy–Schwarz inequality, the 2-D Fourier series, and the Parseval’s theorem. A numerical simulation illustrates the effectiveness of the proposed delay-compensation method.

The damped vibrating string equation on the positive half-line

In this paper, the existence of a solution to the problem describing the small vertical vibration of an elastic string on the positive half-line is investigated in the case when both viscous and material damping coefficients are present. The result is obtained by transforming the original partial differential equation into an appropriate abstract second-order ordinary differential equation in a suitable infinite dimensional space. The abstract problem is then studied using the combination of the Kakutani fixed point theorem together with the approximation solvability method and the weak topology. The applied procedure enables obtaining the existence result also for problems depending on the first derivative, without any strict compactness assumptions put on the right-hand side and on the fundamental system generated by the linear term. The paper ends by applying the obtained result to the studied mathematical model describing the small vertical vibration of an elastic string with a nonlinear Balakrishnan–Taylor-type damping term.

Unified solution of some problems of rectangular plates with four free edges based on symplectic superposition method

PurposeA unified framework for solving the bending, buckling and vibration problems of rectangular thin plates (RTPs) with four free edges (FFFF), including isotropic RTPs, orthotropic rectangular thin plates (ORTPs) and nano-rectangular plates, is established by using the symplectic superposition method (SSM).Design/methodology/approachThe original fourth-order partial differential equation is first rewritten into Hamiltonian system. The class of boundary value problems of the original equation is decomposed into three subproblems, and each subproblem is given the corresponding symplectic eigenvalues and symplectic eigenvectors by using the separation variable method in Hamiltonian system. The symplectic orthogonality and completeness of symplectic eigen-vectors are proved. Then, the symplectic eigenvector expansion method is applied to solve the each subproblem. Then, the symplectic superposition solution of the boundary value problem of the original fourth-order partial differential equation is given through superposing analytical solutions of three foundation plates.FindingsThe bending, vibration and buckling problems of the rectangular nano-plate/isotropic rectangular thin plate/orthotropic rectangular thin plate with FFFF can be solved by the unified symplectic superposition solution respectively.Originality/valueThe symplectic superposition solution obtained is a reference solution to verify the feasibility of other methods. At the same time, it can be used for parameter analysis to deeply understand the mechanical behavior of related RTPs. The advantages of this method are as follows: (1) It provides a systematic framework for solving the boundary value problem of a class of fourth-order partial differential equations. It is expected to solve more complicated boundary value problems of partial differential equations. (2) SSM uses series expansion of symplectic eigenvectors to accurately describe the solution. Moreover, symplectic eigenvectors are orthogonal and directly reflect the orthogonal relationship of vibration modes. (3) The SSM can be carried to bending, buckling and free vibration problems of the same plate with other boundary conditions.

An Alternative Method for Analytical Solutions of Two-Dimensional Black-Scholes-Merton Equation

We present a method of deriving analytical solutions for a two-dimensional Black-Scholes-Merton equation. The method consists of three changes of variables in order to reduce the original partial differential equation (PDE) to a normal form and then solve it. Analytical solutions for two cases of option pricing on the minimum and maximum of two assets are derived using our method and are shown to agree with previously published results. The advantage of our solution procedure is the ability of splitting the original problem into several components in order to demonstrate some solution properties. The solutions of the two cases have a total of five components; each is a particular solution of the PDE itself. Due to the linearity of the two-dimensional Black-Scholes-Merton equation, any linear combination of these components constitutes another solution. Some other possible solutions as well as the solution properties are discussed.

Backstepping Control for Vibration Suppression of 2-D Euler–Bernoulli Beam Based on Nonlinear Saturation Compensator

In this article, a boundary control scheme is proposed to suppress 2-D vibration of Euler–Bernoulli beam with output constraints and input saturation. The original partial differential equations (PDEs) model is transformed to a new form containing virtual control. Then a boundary controller is designed via the backstepping method to suppress the coupled vibration. The hyperbolic tangent function and Nussbaum function are employed to deal with the input saturation. A barrier Lyapunov function with time adjusting function is introduced to suppress the structural vibration of Euler–Bernoulli beam with arbitrary initial conditions. The disturbance observer is designed to deal with the unknown boundary disturbance. Finally, the simulation results show the effectiveness of the proposed vibration controller.

BI-GreenNet: Learning Green’s Functions by Boundary Integral Network

Green’s function plays a significant role in both theoretical analysis and numerical computing of partial differential equations (PDEs). However, in most cases, Green’s function is difficult to compute. The troubles arise in the following threefold. Firstly, compared with the original PDE, the dimension of Green’s function is doubled, making it impossible to be handled by traditional mesh-based methods. Secondly, Green’s function usually contains singularities which increase in the difficulty to get a good approximation. Lastly, the computational domain may be very complex or even unbounded. To override these problems, we develop a new framework for computing Green’s function leveraging the fundamental solution, boundary integral method and neural networks with reasonably high accuracy in this paper. We focus on Green’s function of Poisson and Helmholtz equations in bounded domains, unbounded domains. We also consider Poisson equation and Helmholtz equations in domains with interfaces. Extensive numerical experiments illustrate the efficiency and accuracy of our method for solving Green’s function. In addition, Green’s function provides the operator from the source term and boundary condition to the PDE solution. We apply Green’s function to solve PDEs with different sources, and obtain reasonably high-precision solutions, which shows the good generalization ability of our method. However, the requirements for explicit fundamental solutions to remove the singularity of Green’s function hinder the application of our method in more complex PDEs, such as variable coefficient equations, which will be investigated in our future work.

Optimal Gas-Power Flow Calculation With Lightweight Gas Flow Dynamics

Optimal gas-power flow (OGPF) is one of the most fundamental problems for the operation of integrated electric-gas systems (IEGSs), where the gas flow dynamics partial differential equations (PDEs) are computational challenging. This paper aims to propose a lightweight and accurate model for gas flow dynamics for the OGPF problem. Firstly, the second-order hyperbolic PDEs are equivalently derived based on the original PDEs. Then, the second-order hyperbolic PDEs are discretized using the tri-layer explicit difference scheme. Given some mild boundary and initial conditions, the time-domain two-port model for gas flow dynamics is established, where the state variables of the in-between segments of the pipeline are eliminated. Finally, the iterative parameter updating procedure is devised to improve the modeling accuracy. The lightweight feature of the proposed gas flow dynamics model is verified by the simulation results of several test systems.

Near-resonance approximation of rotating Navier–Stokes equations

We formalise the concept of near resonance for the rotating Navier–Stokes equations, based on which we propose a novel way to approximate the original partial differential equation (PDE). The spatial domain is a three-dimensional flat torus of arbitrary aspect ratios. We prove that the family of proposed PDEs are globally well-posed for any rotation rate and initial datum of any size in any H s space with . Such approximations retain many more 3-mode interactions, and are thus more accurate, than the conventional exact-resonance approach. Our approach is free from any limiting argument that requires physical parameters to tend to zero or infinity, and is free from any use of small divisors (so that all estimates depend smoothly on the torus’s aspect ratios). The key estimate hinges on the counting of integer solutions of Diophantine inequalities rather than Diophantine equations. Using a range of novel ideas, we handle rigorously and optimally challenges arising from the non-trivial irrational functions in these inequalities. The main results and ingredients of the proofs can form part of the mathematical foundation of a non-asymptotic approach to nonlinear, oscillatory dynamics in real-world applications.