Simple Partial Differential Equation Research Articles

Investigating the ability of PINNs to solve Burgers’ PDE near finite-time blowup

Physics Informed Neural Networks (PINNs) have been achieving ever newer feats of solving complicated Partial Differential Equations (PDEs) numerically while offering an attractive trade-off between accuracy and speed of inference. A particularly challenging aspect of PDEs is that there exist simple PDEs which can evolve into singular solutions in finite time starting from smooth initial conditions. In recent times some striking experiments have suggested that PINNs might be good at even detecting such finite-time blow-ups. In this work, we embark on a program to investigate this stability of PINNs from a rigorous theoretical viewpoint. Firstly, we derive error bounds for PINNs for Burgers’ PDE, in arbitrary dimensions, under conditions that allow for a finite-time blow-up. Our bounds give a theoretical justification for the functional regularization terms that have been reported to be useful for training PINNs near finite-time blow-up. Then we demonstrate via experiments that our bounds are significantly correlated to the ℓ2 -distance of the neurally found surrogate from the true blow-up solution, when computed on sequences of PDEs that are getting increasingly close to a blow-up.

Continuum Modeling and Boundary Control of a Satellite with a Large Space Truss Structure

Due to its advantages of easy deployment and high stiffness-to-mass ratio, the utilization of truss structures for constructing large satellites presents an appealing solution for modern space missions, including Earth observation and astronomy. However, the dimensions of the traditional finite element model for a satellite with a large space truss structure become exceedingly large as the structure’s size increases. The control system design process based on the finite element model is complex and time-consuming. This paper employs the continuum modeling method to represent the truss structure as a continuous entity. The bending vibrations of the truss structure are encapsulated by a simplified partial differential equation (PDE), as opposed to the more intricate traditional finite element model. Simultaneously, the satellite’s attitude motion is characterized by an ordinary differential equation (ODE). Building upon this coupled PDE-ODE model, a boundary control law that only requires sensors/actuators at the boundary is formulated to effectively mitigate structural vibrations and regulate the satellite’s attitude. The exponential stability of this closed-loop system is scrutinized using Lyapunov’s direct method. The simulation results affirm that the continuum modeling method is indeed well-suited for satellites endowed with substantial truss structures, and the proposed boundary law proves to be highly effective in both attitude tracking and vibration suppression.

Physics-informed kernel function neural networks for solving partial differential equations

This paper proposes an improved version of physics-informed neural networks (PINNs), the physics-informed kernel function neural networks (PIKFNNs), to solve various linear and some specific nonlinear partial differential equations (PDEs). It can also be considered as a novel radial basis function neural network (RBFNN). In the proposed PIKFNNs, it employs one-hidden-layer shallow neural network with the physics-informed kernel functions (PIKFs) as the customized activation functions. The PIKFs fully or partially contain PDE information, which can be chosen as fundamental solutions, green's functions, T-complete functions, harmonic functions, radial Trefftz functions, probability density functions and even the solutions of some linear simplified PDEs and so on. The main difference between the PINNs and the proposed PIKFNNs is that the PINNs add PDE constraints to the loss function, and the proposed PIKFNNs embed PDE information into the activation functions of the neural network. The feasibility and accuracy of the proposed PIKFNNs are validated by some benchmark examples.

Nonlinear Kelvin wakes and exponential asymptotics

A notable feature of Kelvin surface-wave patterns due to submerged sources at small Froude numbers (F≪1) is that the wave amplitudes are exponentially small with respect to F. Moreover, as F→0 the assumption of linearization is not appropriate. Thus, the role of nonlinear effects in Kelvin wake patterns at small F hinges on exponential (‘beyond-all-orders’) asymptotics. This outstanding theoretical issue is addressed here in the context of a simple partial differential equation with a quadratic nonlinear term, controlled by a parameter ɛ, and a locally confined forcing term in two spatial dimensions. The linear far-field wave response of this model equation is akin to a Kelvin wake whose wave amplitudes are exponentially small with respect to a parameter μ, analogous to F in water waves. To understand how nonlinearity affects this Kelvin-like pattern for ɛ, μ≪1, we develop an exponential asymptotics technique motivated by our earlier analysis in the wavenumber domain of the fKdV equation. Results are presented for three forcing terms that mimic a submerged source, a submerged doublet and a surface pressure. While the linear wave-crest geometry remains intact, the wave amplitudes in the Kelvin wake are significantly enhanced/suppressed by nonlinearity, depending on the type, strength and sign (polarity) of the forcing term. The asymptotic predictions generally are supported by direct numerical computations for various ɛ and μ.

Nonisothermal optimal power and gas flow considering energy storage

The isothermal assumption that treats the gas temperature as a constant is widely adopted in the traditional research. However, the gas temperature has a noticeable effect on the optimal power and gas flow (OPGF) calculation. Moreover, the rational allocation of energy storage can enhance the flexibility and economy of the power and gas system. This paper presents a nonisothermal OPGF considering energy storage, which effectively avoids the constant assumptions on the gas temperature. A simplified partial differential equation which only has bilinear constraints is introduced to depict the temperature change of natural gas. Then, the partial differential equation is discretized by the finite difference scheme and an algebraic pipeline thermal model is developed to avoid the mathematical complexity of the energy conservation equation. The energy storage model is embedded into the OPGF model to enhance the system economy and flexibility. Coupling the DC power flow model and the nonisothermal gas flow model, the nonisothermal OPGF model considering energy storage is formulated. Comprehensive simulations illustrates that the proposed nonisothermal OPGF model considering energy storage can better evaluate the operation state of the integrated energy system. In addition, the energy storage can improve the operation flexibility of systems, thereby significantly reducing the operation costs of the integrated energy systems.

A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source

In this discussion, a new numerical algorithm focused on the Haar wavelet is used to solve linear and nonlinear inverse problems with unknown heat source. The heat source is dependent on time and space variables. These types of inverse problems are ill‐posed and are challenging to solve accurately. The linearization technique converted the nonlinear problem into simple nonhomogeneous partial differential equation. In this Haar wavelet collocation method (HWCM), the time part is discretized by using finite difference approximation, and space variables are handled by Haar series approximation. The main contribution of the proposed method is transforming this ill‐posed problem into well‐conditioned algebraic equation with the help of Haar functions, and hence, there is no need to implement any sort of regularization technique. The results of numerical method are efficient and stable for this ill‐posed problems containing different noisy levels. We have utilized the proposed method on several numerical examples and have valuable efficiency and accuracy.

Holographic β Function in de Sitter Space

The scale invariance of the universe is slightly broken by slow roll parameters. It is likely the slow roll is dual to the random walk. We investigate the distribution function of the conformal zeromode. We identify de Sitter entropy $S_{dS}$ with the distribution entropy of the conformal zeromode $\rho(\omega)$. We have collected convincing support on our postulate. The semiclassical evidence is that the both are given by the gravitational coupling $1/g=\log N/2$ where $g=G_NH^2/\pi$ and $N$ is the e-folding number. We show the renormalized distribution function obeys gravitational Fokker-Planck equation (GFP) and Langevin equations . Under the Gaussian approximation, they boil down to a simple first order partial differential equation. The identical equation is derived by the thermodynamic arguments in the inflationary space-time. GFP determines the evolution of de Sitter entropy of the universe. It coincides with $\beta$ function of $g$. We find two types of the solutions of GFP:(1) UV complete spacetime and (2) inflationary spacetime with power potentials. The maximum entropy principle favors the scenario: (a) born small $\epsilon$ and (b) grow large by inflation. We like to convey the emerging notion of de Sitter duality. The inflationary universe: (bulk/geometrical) is dual to the stochastic space-time on the boundary (cosmological horizon ) as the both are the solutions of GFP.

Unsteady inclined stagnation point flow and thermal transmission of Maxwell fluid on a stretched/contracted plate with modified pressure field

Purpose The purpose of this study is to investigate the two-dimensional unsteady inclined stagnation point flow and thermal transmission of Maxwell fluid on oscillating stretched/contracted plates. First, based on the momentum equation at infinity, pressure field is modified by solving first-order differential equation. Meanwhile, thermal relaxation characteristic of fluid is described by Cattaneo–Christov thermal diffusion model. Design/methodology/approach Highly coupled model equations are transformed into simpler partial differential equations (PDE) via appropriate dimensionless variables. The approximate analytical solutions of unsteady inclined stagnation point flow on oscillating stretched and contracted plates are acquired by homotopy analysis method for the first time, to the best of the authors’ knowledge. Findings Results indicate that because of tensile state of plate, streamline near stagnation point disperses to both sides with stagnation point as center, while in the case of shrinking plate, streamline near stagnation point is concentrated near stagnation point. The enhancement of velocity ratio parameter leads to increasing of pressure variation rate, which promotes flow of fluid. In tensile state, surface friction coefficient on both sides of stagnation point has opposite symbols; when the plate is in shrinkage state, there is reflux near the right side of the stagnation point. In addition, although the addition of unsteady parameters and thermal relaxation parameters reduce heat transfer efficiency of fluid, heat transfer of fluid near the plate can also be enhanced by considering thermal relaxation effect when plate shrinks. Originality/value First, approximate analytical solutions of unsteady inclined stagnation point flow on oscillating stretched and contracted plates are researched, respectively. Second, pressure field is further modified. Finally, based on this, thermal relaxation characteristic of fluid is described by Cattaneo–Christov thermal diffusion model.

Quadratic Convective Nanofluid Flow at a Three-Dimensional Stagnation Point with the g-Jitter Effect

The nonlinear density variation with temperature happens in many thermal applications like solar collectors, energy production, heat exchangers, and combustions, and it gives a significant impact on heat transfer and fluid flows. Thus, a nonlinear convective of an unsteady stagnation point flow under the influence of gravity modulation in the presence of water-based nanoparticles alumina (Al2O3) is studied here. Suitable variables are utilized to reduce the highly coupled nonlinear governing equations into a system of dimensionless simple partial differential equations. The Keller-box method is then applied to solve the consequent governing equations. Velocity and temperature profiles for various values of pertinent parameters are displayed graphically and discussed. The results indicate that the quadratic convection has enhanced the fluid flow and heat transport. Furthermore, the nonlinear convection parameter and the nanoparticles volume fractions have delivered a positive effect on the skin friction and the rate of heat transfer.

Exploiting self-organized criticality in strongly stratified turbulence

A multiscale reduced description of turbulent free shear flows in the presence of strong stabilizing density stratification is derived via asymptotic analysis of the Boussinesq equations in the simultaneous limits of small Froude and large Reynolds numbers. The analysis explicitly recognizes the occurrence of dynamics on disparate spatiotemporal scales, yielding simplified partial differential equations governing the coupled evolution of slow large-scale hydrostatic flows and fast small-scale isotropic instabilities and internal waves. The dynamics captured by the coupled reduced equations is illustrated in the context of two-dimensional strongly stratified Kolmogorov flow. A noteworthy feature of the reduced model is that the fluctuations are constrained to satisfy quasilinear (QL) dynamics about the comparably slowly varying large-scale fields. Crucially, this QL reduction is not invoked as an ad hoc closure approximation, but rather is derived in a physically relevant and mathematically consistent distinguished limit. Further analysis of the resulting slow–fast QL system shows how the amplitude of the fast stratified-shear instabilities is slaved to the slowly evolving mean fields to ensure the marginal stability of the latter. Physically, this marginal stability condition appears to be compatible with recent evidence of self-organized criticality in both observations and simulations of stratified turbulence. Algorithmically, the slaving of the fluctuation fields enables numerical simulations to be time-evolved strictly on the slow time scale of the hydrostatic flow. The reduced equations thus provide a solid mathematical foundation for future studies of three-dimensional strongly stratified turbulence in extreme parameter regimes of geophysical relevance and suggest avenues for new sub-grid-scale parametrizations.

A Linear Approach Suitable for a Class of Steady-State Heat Transfer Problems with Temperature-Dependent Thermal Conductivity

This work presents an useful tool for constructing the solution of steady-state heat transfer problems, with temperature-dependent thermal conductivity, by means of the solution of Poisson equations. Specifically, it will be presented a procedure for constructing the solution of a nonlinear second-order partial differential equation, subjected to Robin boundary conditions, by means of a sequence whose elements are obtained from the solution of very simple linear partial differential equations, also subjected to Robin boundary conditions. In addition, an a priori upper bound estimate for the solution is presented too. Some examples, involving temperature-dependent thermal conductivity, are presented, illustrating the use of numerical approximations.

Identifying Berwald Finsler geometries

Berwald geometries are Finsler geometries close to (pseudo)-Riemannian geometries. We establish a simple first order partial differential equation as necessary and sufficient condition, which a given Finsler Lagrangian has to satisfy to be of Berwald type. Applied to (α,β)-Finsler spaces or spacetimes, respectively, this reduces to a necessary and sufficient condition for the Levi-Civita covariant derivative of the geometry defining 1-form. We illustrate our results with novel examples of (α,β)-Berwald geometries which represent Finslerian versions of Kundt (constant scalar invariant) spacetimes. The results generalize earlier findings by Tavakol and van den Bergh, as well as the Berwald conditions for Randers and m-Kropina resp. very special/general relativity geometries.

Spatial localization beyond steady states in the neighbourhood of the Takens–Bogdanov bifurcation

Abstract Double-zero eigenvalues at a Takens–Bogdanov (TB) bifurcation occur in many physical systems such as double-diffusive convection, binary convection and magnetoconvection. Analysis of the associated normal form, in 1D with periodic boundary condition, shows the existence of steady patterns, standing waves, modulated waves (MW) and travelling waves, and describes the transitions and bifurcations between these states. Values of coefficients of the terms in the normal form classify all possible different bifurcation scenarios in the neighbourhood of the TB bifurcation (Dangelmayr, G. & Knobloch, E. (1987) The Takens–Bogdanov bifurcation with O(2)-symmetry. Phil. Trans. R. Soc. Lond. A, 322, 243-279). In this work we develop a new and simple pattern-forming partial differential equation (PDE) model, based on the Swift–Hohenberg equation, adapted to have the TB normal form at onset. This model allows us to explore the dynamics in a wide range of bifurcation scenarios, including in domains much wider than the lengthscale of the pattern. We identify two bifurcation scenarios in which coexistence between different types of solutions is indicated from the analysis of the normal form equation. In these scenarios, we look for spatially localized solutions by examining pattern formation in wide domains. We are able to recover two types of localized states, that of a localized steady state (LSS) in the background of the trivial state (TS) and that of a spatially localized travelling wave (LTW) in the background of the TS, which have previously been observed in other systems. Additionally, we identify two new types of spatially localized states: that of a LSS in a MW background and that of a LTW in a steady state (SS) background. The PDE model is easy to solve numerically in large domains and so will allow further investigation of pattern formation with a TB bifurcation in one or more dimensions and the exploration of a range of background and foreground pattern combinations beyond SSs.

Bioconvection Due to Gyrotactic Microorganisms in Couple Stress Hybrid Nanofluid Laminar Mixed Convection Incompressible Flow with Magnetic Nanoparticles and Chemical Reaction as Carrier for Targeted Drug Delivery through Porous Stretching Sheet.

In this paper, the steady electrically conducting hybrid nanofluid (CuO-Cu/blood) laminar-mixed convection incompressible flow at the stagnation-point with viscous and gyrotactic microorganisms is considered. Additionally, hybrid nanofluid flow over a horizontal porous stretching sheet along with an induced magnetic field and external magnetic field effects that can be used in biomedical fields, such as in drug delivery and the flow dynamics of the microcirculatory system. This investigation can also deliver a perfect view about the mass and heat transfer behavior of blood flow in a circulatory system and various hyperthermia treatments such as the treatment of cancer. The simple partial differential equations (PDEs) are converted into a series of dimensional ordinary differential equations (ODEs), which are determined using appropriate similarities variables (HAM). The influence of the suction or injection parameter, mixed convection, Prandtl number, buoyancy ratio parameter, permeability parameter, magnetic parameter, reciprocal magnetic prandtl number, bioconvection Rayleigh number, coupled stress parameter, thermophoretic parameter, Schmidt number, inertial parameter, heat source parameter, and Brownian motion parameter on the concentration, motile microorganisms, velocity, and temperature is outlined, and we study the physical importance of the present problem graphically.

Elastoplastic topology optimization of cyclically loaded structures via direct methods for shakedown

For the first time, the lower bound shakedown theorem is integrated into a level set–based topology optimization framework to identify lightweight elastoplastic designs. Shakedown is a cyclic elastoplastic behavior in which, upon cycling beyond the elastic limit, the accumulation of plastic strain arrests and purely elastic behavior is recovered. In contrast to most elastoplastic topology optimization, the use of a lower bound shakedown limit allows elastoplastic shakedown limits to be rigorously estimated using only the elastic solution. Under small deformation assumptions, this amounts to solving one simple partial differential equation, avoiding the non-linearity associated with plasticity, and thus simplifying the resolution process. Numerical results are provided for several benchmark examples. The results highlight the design performance enhancements attributed to allowing elastoplastic shakedown to occur instead of designing to first yield. In particular, up to 10% reduction in weight is found for the simple structures considered.

Interaction between vegetation patches and gaps: A self-organized response to water scarcity

The dynamics of ecological systems are often described by integrodifferential equations that incorporate nonlocal interactions associated with facilitative, competitive interactions between plants, and seed dispersion. In the weak-gradient limit, these models can be reduced to a simple partial-differential equation in the form of a nonvariational Swift–Hohenberg equation. In this contribution, we perform this reduction for any type of kernels provided that their Taylor series converge. Some parameters such as linear and nonlinear diffusion coefficients are affected by the spatial form of the kernel. In particular, Gaussian and exponential kernels are used to evaluate all coefficients of the reduced model. This weak gradient approximation is greatly useful for the investigation of periodic and localized vegetation patches, and gaps. Based on this simple model, we investigate the interaction between two-well separated patches and gaps. In the case of patches, the interaction is always repulsive. As a consequence, bounded states of patches are excluded. However, when two gaps are close to one another, they start to interact through their oscillatory tails. The interaction alternates between attractive and repulsive depending on the distance separating them. This allows for the stabilization of bounded gaps and clusters of them. The analytical formula of the interaction potential is derived for both patches and gaps interactions and checked by numerical investigation of the model equation.This volume is dedicated to Professor Ehud Meron on the occasion of his sixtieth birthday. We take this opportunity to express our warmest and most sincere wishes to him.

Nonlinear Oscillations of a Chain of Masses in a Liquid

Nonlinear regular and random vibrations of a metamaterial made as a chain of massive elements immersed in a viscous liquid are studied. The nearest neighbors are connected by nonlinear elastic forces. Stokes friction is taken into account. A system of equations for mass oscillations is obtained. In the continuum long-wave approximation, the system is reduced to a simplified partial differential equation with nonlinear and dissipative terms. We find the exact solutions and show that the competition of nonlinearity and absorption can lead to the formation of a steady wave profile shape. For noise waves, the correlation function and intensity spectrum are given. The process of sawtooth wave excitation by distributed external sources is analyzed. The limiting “peak” amplitude is calculated. Waves in tubes of variable cross section filled with metamaterial are considered. Based on the solution of the equation containing coordinate-dependent tube section, an expression is found for the amplitude of the sawtooth wave.

The spreading of an axisymmetric ion beam by space charge and diffusion

An analysis is presented of the evolution of a narrow ion beam, produced for instance by an electrospray source, in ambient pressure air. To minimize beam broadening, an intense axial electric field E is externally imposed. Still, radial beam broadening due to the ionic diffusivity D and space charge must be accounted for. Ignoring axial contributions of diffusion and space charge, a simple second order partial differential equation is obtained for the space charge induced field. In the planar problem, this field is governed by Burgers's equation, which is exactly solvable via the Cole-Hopf transformation. The axisymmetric problem is not similarly tractable by any familiar transformation, but may be reduced to an ordinary differential equation in terms of a similarity variable, yielding the complete beam structure under all regimes of interest. The beam broadening effect of space charge is governed by the ratio between the beam current I and the group 2πε0DE, where ε0 is the permittivity of the gas.

Macroscale bending large-deformation and microbuckling behavior of a unidirectional fiber-reinforced soft composite

Due to microscale fiber microbuckling, a fiber-reinforced soft composite demonstrates large macroscale bending deformation (e.g. 10% reversible macroscale compressive strain), which is larger than that of a convenient fiber-reinforced plastics (e.g. 1.5–2% elongation/compression at break). To investigate the deformation behavior, a normalized average energy density of a fiber-reinforced soft composite laminate was derived. By using a self-consistent approach according to the minimum energy principle, a series of analytical expressions were derived by a simplified theoretical method through solving simplified partial differential equations of average energy density. Furthermore, an improved numerical calculation method was developed using the full four terms of partial differential equations of average energy density by employing the results of simplified theoretical method as initial calculating values. The dimensionless results demonstrated that the trend correlated well between those two methods, and the improved numerical method obtained more accurate results than those of the simplified theoretical method. Analytical and numerical results in normalized expressions systematically descripted the bending large-deformation behavior including position of neutral surface and critical buckling, wavelength, amplitude, shearing strain, macroscale compressive/tensile strain, buckled fiber strain, and actuation moment. To design a fiber-reinforced soft composite for use in engineering, the simplified theoretical method is used to predict trend and obtain approximate results for preliminary design, and the improved numerical method is further used to check and obtain more accurate results on detailed design stage.

American-type basket option pricing: a simple two-dimensional partial differential equation

We consider the pricing of American-type basket derivatives by numerically solving a partial differential equation (PDE). The curse of dimensionality inherent in basket derivative pricing is circumvented by using the theory of comonotonicity. We start with deriving a PDE for the European-type comonotonic basket derivative price, together with a unique self-financing hedging strategy. We show how to use the results for the comonotonic market to approximate American-type basket derivative prices for a basket with correlated stocks. Our methodology generates American basket option prices which are in line with the prices obtained via the standard Least-Square Monte-Carlo approach. Moreover, the numerical tests illustrate the performance of the proposed method in terms of computation time, and highlight some deficiencies of the standard LSM method.