System Of Linear Partial Differential Equations Research Articles

Variational solution to the lattice Boltzmann method for Couette flow.

Literature studies of the lattice Boltzmann method (LBM) demonstrate hydrodynamics beyond the continuum limit. This includes exact analytical solutions to the LBM, for the bulk velocity and shear stress of Couette flow under diffuse reflection at the walls through the solution of equivalent moment equations. We prove that the bulk velocity and shear stress of Couette flow with Maxwell-type boundary conditions at the walls, as specified by two-dimensional isothermal lattice Boltzmann models, are inherently linear in Mach number. Our finding enables a systematic variational approach to be formulated that exhibits superior computational efficiency than the previously reported moment method. Specifically, the number of partial differential equations(PDEs) in the variational method grows linearly with quadrature order while the number of moment method PDEs grows quadratically. The variational method directly yields a system of linear PDEs that provide exact analytical solutions to the LBM bulk velocity field and shear stress for Couette flow with Maxwell-type boundary conditions. It is anticipated that this variational approach will find utility in calculating analytical solutions for novel lattice Boltzmann quadrature schemes and other flows.

Fast particle-mesh code for Milgromian dynamics

Context. Modified Newtonian dynamics (MOND) is a promising alternative to dark matter. To further test the theory, there is a need for fluid- and particle-dynamics simulations. The force in MOND is not a direct particle-particle interaction, but derives from a potential for which a nonlinear partial differential equation (PDE) needs to be solved. Normally, this makes the problem of simulating dynamical evolution computationally expensive. Aims. We intend to develop a fast particle-mesh (PM) code for MOND (the AQUAL formalism). Methods. We transformed the nonlinear equation for MOND into a system of linear PDEs plus one algebraic equation. An iterative scheme with the fast Fourier transform (FFT) produces successively better numerical approximations. Results. The algorithm was tested for dynamical systems in MOND where analytical solutions are known: the two-body problem, a body with a circular ring, and a spherical distribution of particles in thermal equilibrium in the self-consistent potential. Conclusions. The PM code can accurately calculate the forces at subpixel scale and reproduces the analytical solutions. Four iterations are required for the potential, but when the spatial steps are small compared to the kernel width, one iteration is suffices. The use of a smoothing kernel for the accelerations is inevitable in order to eliminate the self-gravity of the point particles. Our PDE solver is 15 to 42 times as slow as a standard Poisson solver. However, the smoothing and particle propagation takes up most of the time above one particle per 103 pixels. The FFTs, the smoothing, and the propagation part in the code can all be parallelized.

Integral Quadratic Constraints with Infinite-Dimensional Channels.

Modern control theory provides us with a spectrum of methods for studying the interconnection of dynamic systems using input-output properties of the interconnected subsystems. Perhaps the most advanced framework for such input-output analysis is the use of Integral Quadratic Constraints (IQCs), which considers the interconnection of a nominal linear system with an unmodelled nonlinear or uncertain subsystem with known input-output properties. Although these methods are widely used for Ordinary Differential Equations (ODEs), there have been fewer attempts to extend IQCs to infinite-dimensional systems. In this paper, we present an IQC-based framework for Partial Differential Equations (PDEs) and Delay Differential Equations (DDEs). First, we introduce infinite-dimensional signal spaces, operators, and feedback interconnections. Next, in the main result, we propose a formulation of hard IQC-based input-output stability conditions, allowing for infinite-dimensional multipliers. We then show how to test hard IQC conditions with infinite-dimensional multipliers on a nominal linear PDE or DDE system via the Partial Integral Equation (PIE) state-space representation using a sufficient version of the Kalman-Yakubovich-Popov lemma (KYP). The results are then illustrated using four example problems with uncertainty and nonlinearity.

A new solution for the deformations of an initially elliptical elastic-walled tube

SummaryWe investigate the small-amplitude deformations of a long thin-walled elastic tube having an initially axially uniform elliptical cross-section. The tube is deformed by a (possibly non-uniform) transmural pressure. At leading order, its deformations are shown to be governed by a single partial differential equation (PDE) for the azimuthal displacement as a function of the axial and azimuthal co-ordinates and time. Previous authors have obtained solutions to this PDE by making ad hoc approximations based on truncating an approximate Fourier representation. In this article, we instead write the azimuthal displacement as a sum over the azimuthal eigenfunctions of a generalised eigenvalue problem and show that we are able to derive an uncoupled system of linear PDEs with constant coefficients for the amplitude of the azimuthal modes as a function of the axial co-ordinate and time. This results in a formal solution of the whole system being found as a sum over the azimuthal modes. We show that the $n$th mode’s contribution to the tube’s relative area change is governed by a simplified second-order PDE and examine the case in which the tube’s deformations are driven by a uniform transmural pressure. The relative errors induced by truncating the series solution after the first and second terms are then evaluated as a function of both the ellipticity and pre-stress of the tube. After comparing our results with Whittaker et al. (A rational derivation of a tube law from shell theory, Q. J. Mech. Appl. Math. 63 (2010) 465–496), we find that this new method leads to a significant simplification when calculating contributions from the higher-order azimuthal modes, which in turn makes a more accurate solution easier to obtain.

NON-POLYNOMIAL CUBIC SPLINE APPROACH FOR NUMERICAL APPROXIMATION OF SECOND ORDER LINEAR KLEIN-GORDON EQUATION

Most of the fundamental theories and mathematical models of engineering and physical sciences are expressed in terms of partial differential equations (PDEs). Several studies were carried out for the numerical approximation of the second order linear Klein-Gordon equation. This study constructed a new numerical technique for the numerical approximation of second order linear Klein-Gordon equation. The new constructed scheme was based on employing non-polynomial cubic spline method (NPCSM). The second order time derivatives involved in the linear Klein-Gordon equation were decomposed into the first order derivatives. The decomposition generated a linear system of PDEs, where the first order time derivatives were approximated by the central finite differences of . Three test problems were considered for the numerical illustration of the developed scheme. For different values of spatial displacement , step size , and time step , the developed numerical technique produced encouraging results which were very much close to the analytical solution. For , , and , the best observed accuracy was close to the machine precision.

Control of Partial Differential Equations via Physics-Informed Neural Networks

This paper addresses the numerical resolution of controllability problems for partial differential equations (PDEs) by using physics-informed neural networks. Error estimates for the generalization error for both state and control are derived from classical observability inequalities and energy estimates for the considered PDE. These error bounds, that apply to any exact controllable linear system of PDEs and in any dimension, provide a rigorous justification for the use of neural networks in this field. Preliminary numerical simulation results for three different types of PDEs are carried out to illustrate the performance of the proposed methodology.

A New Explicit Numerical Schemes for Time-Dependent PDEs with Application to Pressure Driven Fluid Flow in a Rectangular Duct

A modified class of temporal discretization schemes for partial differential equations (PDEs) is proposed, explicit and second to fifth-order accurate in time. In time, the stability region of the proposed modified second-order scheme is larger than the standard second-order Adams–Bashforth method constructed on two time levels. A modification made for the Du Fort–Frankel method was also implemented in the proposed second-order scheme, which permits the little larger stability region, but the scheme becomes first-order accurate. Since the Du Fort–Frankel method cannot be employed without a modification of averaging in time levels, the proposed second-order scheme can be used without any modification. The proposed modified scheme with different orders in space and second orders in time was implemented for heat and mass transfer of chemically reactive fluid flow in a rectangular duct. The flow is generated due to applying different pressure gradients. The contour plots of velocity, temperature, and concentration profiles are portrayed at different pressure gradients; Péclet number in heat transfer, Péclet number in mass transfer, reaction parameter, and at different times. In addition, stability and convergence conditions for the considered system of linear and non-linear PDEs consisting of non-dimensional momentum, energy, and concentration equations were found for two cases. The displayed graphs depict the transfer of heat in the fluid, which rises due to heated boundaries, and the transfer of mass in the fluid at various moments. Classical models can be solved using the proposed method, which has a faster convergence rate than the standard or classical approach. This approach is illustrated through computer simulations that demonstrate its key computational features. It is believed that the data presented in this study will serve as a useful source for future fluid flow investigations to be conducted in an industrial setting within an enclosed area.

Numerical Solution Methods for a Nonlinear Operator Equation Arising in an Inverse Coefficient Problem

We consider the inverse problem of determining two unknown coefficients in a linearsystem of partial differential equations using additional information about one of the solutioncomponents. The problem is reduced to a nonlinear operator equation for one of the unknowncoefficients. The successive approximation method and the Newton method are used to solve thisoperator equation numerically. Results of calculations illustrating the convergence of numericalmethods for solving the inverse problem are presented.

Symmetry structure of multi-dimensional time-fractional partial differential equations

In this paper, we concentrate on the Lie symmetry structure of a system of multi-dimensional time-fractional partial differential equations (PDEs). Specifically, we first give an explicit prolongation formula involving Riemann–Liouville time-fractional derivative for the Lie infinitesimal generator in multi-dimensional case, and then show that the infinitesimal generator has an elegant structure. Furthermore, we present two simple conditions to determine the infinitesimal generators where one is a system of linear time-fractional PDEs, the other is a system of integer-order PDEs and plays the dominant role in finding the infinitesimal generators. We study three time-fractional PDEs to illustrate the efficiencies of the results.

Differential invariant method for seeking nonlocally related systems and nonlocal symmetries. II: Connections with the conservation law method

In this paper, we show direct connections between the conservation law (CL)-based method and the differential invariant (DI)-based method for obtaining nonlocally related systems and nonlocal symmetries for a given partial differential equation (PDE) system. For a PDE system with two independent variables, we show that the CL method is a special case for the DI method. For a PDE system with at least three independent variables, we show that the CL method, for a curl-type CL, is a special case for the DI method. We also consider the situation for a self-adjoint, i.e. variational, linear PDE system. Here, a solution of the linear PDE system yields a nonlocally related system for both approaches. In particular, the resulting nonlocally related systems need not be invertibly equivalent. Through an example, we show that three distinct nonlocally related systems can be obtained from an admitted point symmetry.

Numerical Approximation of Port-Hamiltonian Systems for Hyperbolic or Parabolic PDEs with Boundary Control

We consider the design of structure-preserving discretization methods for the solution of systems of boundary controlled Partial Differential Equations (PDEs) thanks to the port-Hamiltonian formalism. We first provide a novel general structure of infinite-dimensional port-Hamiltonian systems (pHs) for which the Partitioned Finite Element Method (PFEM) straightforwardly applies. The proposed strategy is applied to abstract multidimensional linear hyperbolic and parabolic systems of PDEs. Then we show that instructional model problems based on the wave equation, Mindlin equation and heat equation fit within this unified framework. Secondly, we introduce the ongoing project SCRIMP (Simulation and Control of Interactions in Multi-Physics) developed for the numerical simulation of infinite-dimensional pHs. SCRIMP notably relies on the FEniCS open-source computing platform for the finite element spatial discretization. Finally, we illustrate how to solve the considered model problems within this framework by carefully explaining the methodology. As additional support, companion interactive Jupyter notebooks are available.

Maurice Janet’s algorithms on systems of linear partial differential equations

This article describes the emergence of formal methods in theory of partial differential equations (PDE) in the French school of mathematics through Janet’s work in the period 1913–1930. In his thesis and in a series of articles published during this period, Janet introduced an original formal approach to deal with the solvability of the problem of initial conditions for finite linear PDE systems. His constructions implicitly used an interpretation of a monomial PDE system as a generating family of a multiplicative set of monomials. He introduced an algorithmic method on multiplicative sets to compute compatibility conditions, and to study the problem of the existence and the uniqueness of a solution to a linear PDE system with given initial conditions. The compatibility conditions are formulated using a refinement of the division operation on monomials defined with respect to a partition of the set of variables into multiplicative and non-multiplicative variables. Janet was a pioneer in the development of these algorithmic methods, and the completion procedure that he introduced on polynomials was the first one in a long and rich series of works on completion methods which appeared independently throughout the twentieth-century in various algebraic contexts.

How symmetries yield non-invertible mappings of linear partial differential equations

Nonlocally related partial differential equation (PDE) systems are important in the analysis of a given PDE system. Useful nonlocally related systems can be constructed through conservation law and symmetry based methods. In this paper, we focus on an application of the symmetry-based method to linear PDE systems. In particular, we show how to obtain systematically non-invertible mappings of linear PDEs to linear PDEs. As examples, we obtain non-invertible mappings of the Kolmogorov equation with variable coefficients to the backward heat equation (a PDE with constant coefficients) as well as non-invertible mappings of linear hyperbolic PDEs with variable coefficients to linear hyperbolic PDEs with constant coefficients.

ExaHyPE: An engine for parallel dynamically adaptive simulations of wave problems

ExaHyPE (“An Exascale Hyperbolic PDE Engine”) is a software engine for solving systems of first-order hyperbolic partial differential equations (PDEs). Hyperbolic PDEs are typically derived from the conservation laws of physics and are useful in a wide range of application areas. Applications powered by ExaHyPE can be run on a student’s laptop, but are also able to exploit thousands of processor cores on state-of-the-art supercomputers. The engine is able to dynamically increase the accuracy of the simulation using adaptive mesh refinement where required. Due to the robustness and shock capturing abilities of ExaHyPE’s numerical methods, users of the engine can simulate linear and non-linear hyperbolic PDEs with very high accuracy. Users can tailor the engine to their particular PDE by specifying evolved quantities, fluxes, and source terms. A complete simulation code for a new hyperbolic PDE can often be realised within a few hours — a task that, traditionally, can take weeks, months, often years for researchers starting from scratch. In this paper, we showcase ExaHyPE’s workflow and capabilities through real-world scenarios from our two main application areas: seismology and astrophysics. Program summaryProgram title: ExaHyPE-EngineProgram Files doi:http://dx.doi.org/10.17632/6sz8h6hnpz.1Licensing provisions: BSD 3-clauseProgramming languages: C++, Python, FortranNature of Problem: The ExaHyPE PDE engine offers robust algorithms to solve linear and non-linear hyperbolic systems of PDEs written in first order form. The systems may contain both conservative and non-conservative terms.Solution method: ExaHyPE employs the discontinuous Galerkin (DG) method combined with explicit one-step ADER (arbitrary high-order derivative) time-stepping. An a-posteriori limiting approach is applied to the ADER-DG solution, whereby spurious solutions are discarded and recomputed with a robust, patch-based finite volume scheme. ExaHyPE uses dynamical adaptive mesh refinement to enhance the accuracy of the solution around shock waves, complex geometries, and interesting features.

On constructing boundaries for boundary value problems defined by continuous 2-D autonomous systems

Abstract In this paper, we provide a constructive way of specifying initial/boundary data for a given continuous 2-D autonomous system described by a set of linear partial differential equations (PDEs) with real constant coefficients. One of the ways of specifying initial/boundary data is by specifying the values of various derivatives of the solution trajectories at the origin; the derivatives correspond to a standard monomial set obtained using Grobner basis. However, such an initial/boundary data often lacks physical interpretation. In this paper, we consider subsets of the domain having some algebraic structure (in the form of subspaces and strips of finite width around such subspaces) such that trajectories restricted to these subsets, often called characteristic sets, serve as initial/boundary conditions for the given autonomous system of linear PDEs. We provide a systematic way to construct such characteristic sets with the help of Grobner bases and Oberst-Riquier algorithm. Thus we bridge the gap between initial/boundary conditions involving standard monomials and more conventional initial/boundary conditions in the form of restrictions on characteristic sets. We also show that every scalar system of PDEs admits such a characteristic set given by a rectangular strip of finite width around a subspace whose dimension equals the Krull dimension of the system’s quotient ring.

Spectral synthesis and partial differential equations

We offer a new approach to spectral synthesis on affine groups. The idea is to consider solution spaces of systems of linear partial differential equations corresponding to invariant differential operators with respect to some closed subgroup of the general linear group on $$\mathbb{R}^n$$. We show that a certain type of spherical spectral analysis and spectral synthesis holds for such solution spaces.

Hodograph solutions of the wave equation of nonlinear electrodynamics in the quantum vacuum

The process of photon-photon scattering in vacuum is investigated analytically in the long-wavelength limit within the framework of the Euler-Heisenberg Lagrangian. In order to solve the nonlinear partial differential equations (PDEs) obtained from this Lagrangian use is made of the hodograph transformation. This transformation makes it possible to turn a system of quasilinear PDEs into a system of linear PDEs. Exact solutions of the equations describing the nonlinear interaction of electromagnetic waves in vacuum in a one-dimensional configuration are obtained and analyzed.

Three-Phases Dynamic Modelling of Column Flotation Process

In this work, a three-phases discrete dynamic model of column flotation which accounts for the interface and froth regions is developed. The system is described by transport hyperbolic nonlinear partial differential equations (PDEs). The steady state profiles are utilized to linearize the original nonlinear system. The Cayley-Tustin time discretization transformation is applied to the linear hyperbolic PDEs system and maps the continuous infinite dimensional system to a discrete infinite dimensional system without spatial discretization. The final discretized model is structure preserving and does not imply any model reduction. Discrete model dynamics are compared with high fidelity numerical simulations of continuous linearized model in order to demonstrate applicability of the proposed discrete model development.

The Maxwell equations as a Bäcklund transformation

Bäcklund transformations (BTs) are a useful tool for integrating nonlinear partial differential equations (PDEs). However, the significance of BTs in linear problems should not be ignored. In fact, an important linear system of PDEs in Physics, namely, the Maxwell equations of Electromagnetism, may be viewed as a BT relating the wave equations for the electric and the magnetic field, these equations representing integrability conditions for solution of the Maxwell system. We examine the BT property of this system in detail, both for the vacuum case and for the case of a linear conducting medium.

How to calculate all point symmetries of linear and linearizable differential equations

The systematic method for calculating all point symmetries of partial differential equations (PDEs) with non-trivial Lie point symmetry algebras does not currently apply to linear PDEs. Consider any given locally solvable system of linear homogeneous PDEs, each of order 2 or higher, of general Kovalevskaya form. Suppose that its Lie point symmetry generators' characteristics are linear in the dependent variables, while those that are homogeneous of degree 1 in the dependent variables contain a finite number of arbitrary parameters (this is common for systems that arise from applications). Herein, every point symmetry is shown to be projectable/fibre-preserving: the transformed independent variables depend only on the original independent variables. Moreover, two (conditional) Lie-algebraic characterizations of the generators of scalings and superpositions of solutions are derived. One of them holds if the system cannot be decoupled (with a smooth, locally invertible, linear change of dependent variables). If any such characterization holds, every point symmetry maps the dependent variables to functions that are linear in the original dependent variables. The aforementioned systematic method is adapted to calculate these symmetries. This version of the method also applies to linearizable systems.