Variable-coefficient Partial Differential Equations Research Articles

VC-PINN: Variable coefficient physics-informed neural network for forward and inverse problems of PDEs with variable coefficient

The paper proposes a deep learning method specifically dealing with the forward and inverse problem of variable coefficient partial differential equations – variable coefficient physics-informed neural network (VC-PINN). The shortcut connections (ResNet structure) introduced into the network alleviate the “vanishing gradient” problem and unify linear and nonlinear coefficients. The developed method was applied to four equations including the variable coefficient Sine–Gordon (vSG), the generalized variable coefficient Kadomtsev–Petviashvili equation (gvKP), the variable coefficient Korteweg–de Vries equation (vKdV), the variable coefficient Sawada–Kotera equation (vSK). Numerical results show that VC-PINN is successful in the case of high dimensionality, various variable coefficients (polynomials, trigonometric functions, fractions, oscillation attenuation coefficients), and the coexistence of multiple variable coefficients. We also conducted an in-depth analysis of VC-PINN in a combination of theory and numerical experiments, including four aspects: the necessity of ResNet; the relationship between the convexity of variable coefficients and learning; anti-noise analysis; the unity of forward and inverse problems/relationship with standard PINN.

Dynamic Modeling and Simulation of Integrated Electricity and Gas Systems

The dynamic simulation of the integrated electricity and gas system (IEGS) is effective for accurate operational guidance. However, the IEGS model governed by partial and ordinary differential-algebraic equations (PDAE) is challenging for analysis, because it poses heterogeneous properties, high dimensionality, and strong nonlinearity. To eliminate the nonlinear terms, this paper firstly derives the expression of average flow velocity (AFV) and reformulates the gas flow dynamics into the variable-coefficient partial differential equations (VC-PDE), whose consistency, stability, and convergence can be theoretically guaranteed. Then, a novel adaptive step size control (ASC) strategy is developed for VC-PDE solvers to improve computational efficiency and accuracy. Finally, this paper builds a framework for the simulation of IEGS considering the dynamics in the natural gas network (NGN) and coupling units, where the bidirectional interaction between the electric power system (EPS) and NGN is investigated. Case studies verify the superiority of the proposed VC-PDE solver with ASC regarding accuracy, stability, convergence, and efficiency in different systems.

Mathematical and Physical Analyses of Middle/Neutral Surfaces Formulations for Static Response of Bi-Directional FG Plates with Movable/Immovable Boundary Conditions

This article is prompted by the existing confusion about correctness of responses of beams and plates produced by middle surface (MS) and neutral surface (NS) formulations. This study mathematically analyzes both formulations in the context of the bending of bi-directional functionally graded (BDFG) plates and discusses where the misconceptions are. The relation between in-plane displacement field variables on NS and on MS are derived. These relations are utilized to define a modified set of boundary conditions (BCs) for immovable simply supported plates that enables either formulation to apply fixation conditions on the refence plane of the other formulation. A four-variable higher order shear deformation theory is adopted to present the displacement fields of BDFG plates. A 2D plane stress constitution is used to govern stress–strain relations. Based on MS and NS, Hamilton’s principles are exploited to derive the equilibrium equations which are described by variable coefficient partial differential equations. The governing equations in terms of stress resultants are discretized by the differential quadrature method (DQM). In addition, analytical expressions that relate rigidity terms and stress resultants associated with the two formulations are proved. Both the theoretical analysis and the numerical results demonstrate that the responses of BDFG plates based on MS and NS formulations are identical in the cases of clamped BCs and movable simply supported BCs. However, the difference in responses of immovable simply supported BCs is expected since each formulation assumes plate fixation at different planes. Further, numerical results show that the responses of immovable simply supported BDFG plates obtained using the NS formulation are identical to those obtained by the MS formulation if the transferred boundary condition (from NS- to MS-planes) are applied. Theoretical and numerical results demonstrate also that both MS and NS formulations are correct even for immovable simply supported BCs if fixation constraints at different planes are treated properly.

A Unified Lattice Boltzmann Model for Fourth Order Partial Differential Equations with Variable Coefficients.

In this work, a unified lattice Boltzmann model is proposed for the fourth order partial differential equation with time-dependent variable coefficients, which has the form . A compensation function is added to the evolution equation to recover the macroscopic equation. Applying Chapman-Enskog expansion and the Taylor expansion method, we recover the macroscopic equation correctly. Through analyzing the error, our model reaches second-order accuracy in time. A series of constant-coefficient and variable-coefficient partial differential equations are successfully simulated, which tests the effectiveness and stability of the present model.

A unified method for in-plane vibration analysis of double-beam systems with translational springs

Double-beam structures with many translational springs are a class of significant engineering structures, which have been widely used in civil, aerospace, and mechanical engineering. The dynamic characteristics of such structures is significant to their structural design, vibration abatement and condition assessment. Since the motion differential equation of a double-beam system is a system of complex fourth-order variable coefficient partial differential equations. The existing analytical methods either need to simplify the structural mechanics model or make some prior assumptions about the form of the solution of the equation, which is convenient for the simplification and solution of the equation, and finally reduces the universality of the whole analysis process, and has many limitations in the practical application. An alternative solution is the finite element method, but it can only give an approximate solution to distributed parameter systems. In view of this, this article developed a unified approach for the free vibration of the double-beam structures by employing the dynamic stiffness method (DSM), which has no restrictions on the geometric characteristics and boundary conditions of the structure, and can obtain the exact dynamic characteristics of the system without any approximation. The correctness of the proposed method is validated by comparison with literatures and the finite element solutions. Finally, a comprehensive parametric study is performed to investigate the influence of key parameters on modal frequencies and mode shapes.

Free vibration analysis of rotating functionally graded conical shells reinforced by anisogrid lattice structure

This paper investigates the free vibration behavior of a rotating functionally graded conical shell, reinforced by an anisogrid lattice structure. The material properties of the shell are assumed to be graded in the thickness direction. The governing equations have been derived based on classical shell theory and considering the effects of centrifugal and Coriolis accelerations as well as initial hoop tension due to shell rotation. The smeared method is also employed to superimpose the stiffness contribution of the stiffeners with those of the shell to obtain the whole structure's equivalent stiffness parameters. The resulting equations, which are the coupled set of three variable coefficient partial differential equations in terms of displacement components, are solved by the Galerkin method for different boundary conditions. The obtained frequencies are compared with the available literature and the finite element software results. Finally, new results are discussed to show the effect of various parameters such as shell geometrical and material properties, stiffeners, rotating speed, and boundary conditions on natural frequencies.

Equivalent transformations, bifurcations and exact solutions to a class of variable-coefficient PDE

In this paper, the equivalent transformations (ETs) of the generalized variable-coefficient KdV (vc-KdV) types of equations are constructed, so these variable-coefficient partial differential equations (vc-PDEs) are transformed into constant-coefficient partial differential equations (cc-PDEs) under some conditions, the vc-KdV and vc-mKdV equations are transformed into classical KdV and mKdV equations accordingly. Furthermore, the dynamical behavior is provided by the bifurcation analysis method, the exact parametric representations are investigated. Then the explicit solutions to the variable-coefficient equations are presented in terms of the ETs and parametric representations.

Equivalent transformations and exact solutions to the generalized cylindrical KdV type of equation

In this paper, by constructing equivalent transformations (ETs) of the generalized cylindrical KdV (cKdV) types of equations, we transform the variable-coefficient partial differential equations (vc-PDEs) into constant-coefficient PDEs (cc-PDEs) under some conditions. Particularly, the classical cKdV equation is transformed into the classical KdV equation accordingly, then the exact solutions to the vc-PDEs are provided in terms of the ETs. Thus, an effective approach to getting exact solutions to vc-PDEs is presented based on the solutions to cc-PDEs.

A Novel Lie Group Classification Method for Generalized Cylindrical KdV Type of Equation: Exact Solutions and Conservation Laws

This paper is concerned with the generalized cylindrical KdV (cKdV) equation, which includes the cKdV types of equations. By a novel Lie group classification method, all of the point symmetries of the variable-coefficient partial differential equations are obtained under the compatibility condition, which is necessary and the basis for the complete group classification. Then, the symmetries of the corresponding nonlinear constant-coefficient PDE and the classical cKdV equation are presented accordingly. Furthermore, the exact solutions to the generalized cKdV type of equation are provided in general case, and the explicit conservation laws of the equation are given through the nonlinearly self-adjoint procedure.

Extending the unified transform: curvilinear polygons and variable coefficient PDEs

Abstract We provide the first significant extension of the unified transform (also known as the Fokas method) applied to elliptic boundary value problems, namely, we extend the method to curvilinear polygons and partial differential equations (PDEs) with variable coefficients. This is used to solve the generalized Dirichlet-to-Neumann map. The central component of the unified transform is the coupling of certain integral transforms of the given boundary data and of the unknown boundary values. This has become known as the global relation and, in the case of constant coefficient PDEs, simply links the Fourier transforms of the Dirichlet and Neumann boundary values. We extend the global relation to PDEs with variable coefficients and to domains with curved boundaries. Furthermore, we provide a natural choice of global relations for separable PDEs. These generalizations are numerically implemented using a method based on Chebyshev interpolation for efficient and accurate computation of the integral transforms that appear in the global relation. Extensive numerical examples are provided, demonstrating that the method presented in this paper is both accurate and fast, yielding exponential convergence for sufficiently smooth solutions. Furthermore, the method is straightforward to use, involving just the construction of a simple linear system from the integral transforms, and does not require knowledge of Green’s functions of the PDE. Details on the implementation are discussed at length.

Stress singularity of a notch in a higher-order elastic solid under anti-plane deformation

The problem of the stress singularity of a notch in a nonlinear solid under finite anti-plane deformation is investigated using higher-order elasticity. The equilibrium equations are written in terms of the first Piola–Kirchhoff stresses, which are replaced by displacements up to the third-order. The resulting variable-coefficient partial differential equations are solved numerically, subject to vanishing out-of-plane shear tractions on the notch faces. The key results are: (i) the stress exponent characterizing the variation of stress with distance from the notch tip may be positive or negative, implying that the stresses can be non-singular or singular, (ii) the stress exponent becomes more positive with a decrease in the notch angle, (iii) a single dimensionless reduced elastic parameter determines the stress exponent, and (iv) the stress exponent varies with the elastic constants, unlike the case in linear elasticity. Specifically, for a given notch angle the stress exponent becomes more negative with decrease in the first Lamé constant λ and the third-order elastic constant n, and with the increase in the magnitude of the negative third-order constant m, while it varies non-monotonically with the second Lamé constant (shear modulus) µ.These results have significant implications for notch-like defects in soft solids, e.g., replacement tissues, industrial robots and devices in biomedical applications.

Modeling of first-order photobleaching kinetics using Krylov subspace spectral methods

We solve the first order 2-D reaction–diffusion equations which describe binding-diffusion kinetics using the photobleaching scanning profile of a confocal laser scanning microscope, approximated by a Gaussian laser profile. We show how to solve the first-order photobleaching kinetics partial differential equations (PDEs) using a time-stepping method known as a Krylov subspace spectral (KSS) method. KSS methods are explicit methods for solving time-dependent variable-coefficient partial differential equations. They approximate Fourier coefficients of the solution using Gaussian quadrature rules in the spectral domain. In this paper, we show how a KSS method can be used to obtain not only an approximate numerical solution, but also an approximate analytical solution when using initial conditions that come from pre-bleach steady states and also general initial conditions, to facilitate asymptotic analysis. Analytical and numerical results are presented. It is observed that although KSS methods are explicit, it is possible to use a time step that is far greater than what the CFL condition would indicate.

Solution of PDEs for First-Order Photobleaching Kinetics using Krylov Subspace Spectral Methods

We solve the first order reaction-diffusion equations which describes binding-diffusion kinetics using photobleaching scanning profile of confocal laser scanning Microscopes approximated by a Gaussian laser profile. We show how to solve the first order photobleaching kinetics partial differential equations with prebleach steady state initial conditions using a time-domain method known as a Krylov Subspace Spectral method (KSS method). KSS are explicit methods for solving time-dependent variable-coefficient partial differential equations (PDEs). KSS methods are advantageous compared to other methods because of its high resolution and its superior scalability. We will apply Gaussian Quadrature rules in the spectral domain developed by Golub and Meurant to solve PDEs. We present a simple rough analytical solution, as well as a computational solution that is first-order accurate. We then use this solution to examine short and long time behaviors.

Analytical investigation on the free vibration behavior of rotating FGM truncated conical shells reinforced by orthogonal eccentric stiffeners

ABSTRACTThis article presents an analytical investigation on the free vibration behavior of rotating functionally graded truncated conical shells reinforced by stringers and rings with the change of spacing between stringers. Using the Donnell shell theory, smeared stiffeners technique, and taking into account the influences of centrifugal force and Coriolis acceleration, the governing equations are derived. These variable coefficient partial differential equations are studied by the Galerkin method. The sixth-order polynomial equation of natural frequency is obtained. Numerical results show effects of stiffener and input parameters on the frequency of shell.

A preliminary study on the meshless local exponential basis functions method for nonlinear and variable coefficient PDEs

Purpose The purpose of this paper is to extend the meshless local exponential basis functions (MLEBF) method to the case of nonlinear and linear, variable coefficient partial differential equations (PDEs).

Analytical investigation on mechanical buckling of FGM truncated conical shells reinforced by orthogonal stiffeners based on FSDT

This work presents an analytical investigation for analyzing the mechanical buckling of truncated conical shells made of functionally graded materials, subjected to axial compressive load and external uniform pressure. Shells are reinforced by closely spaced stringers and rings. The change of spacing between stringers in the meridional direction also is taken into account. Using the adjacent equilibrium criterion, the first order shear deformation theory (FSDT) and Lekhnitskii smeared stiffener technique, the linearization stability equations have been established. The resulting equations which they are the system of five variable coefficient partial differential equations in terms of displacement components are investigated by Galerkin method. The closed-form expression for determining the critical buckling load is obtained. The effects of material properties, dimensional parameters, stiffeners and semi-vertex angle on buckling behaviors of shell are considered. Shown that for thick conical shells, the use of FSDT for determining their critical buckling load is necessary and more suitable.

On the stability of functionally graded truncated conical shells reinforced by functionally graded stiffeners and surrounded by an elastic medium

This paper presents an analytical approach to investigate the mechanical buckling load of eccentrically stiffened functionally graded truncated conical shells surrounded by elastic medium and subjected to axial compressive load and external uniform pressure. Shells are reinforced by stringers and rings in which material properties of shell and stiffeners are graded in the thickness direction according to a volume fraction power-law distribution. The elastic medium is assumed as two-parameter elastic foundation model proposed by Pasternak. The change of spacing between stringers in the meridional direction is taken into account. The equilibrium and linearized stability equations for stiffened shells are derived based on the classical shell theory and smeared stiffeners technique. The resulting equations which they are the couple set of three variable coefficient partial differential equations in terms of displacement components are investigated by Galerkin method and the closed-form expression for determining the buckling load is obtained. Four cases of stiffener arrangement are analyzed. Carrying out some computations, effects of foundation, stiffener and input factors on stability of shell have been studied. The effectiveness of FGM stiffeners in enhancing the stability of cylindrical shells comparing with homogenuos stiffener is shown.

Instability of eccentrically stiffened functionally graded truncated conical shells under mechanical loads

This paper is concerned with the mechanical buckling load of an eccentrically stiffened truncated conical shells made of functionally graded materials and subjected to axial compressive load and external uniform pressure by analytical method. Shells are reinforced by stringers and rings. The change of spacing between stringers in the meridional direction is taken into account. Material properties of shell are graded in the thickness direction according to a volume fraction power-law distribution. The equilibrium and linearized stability equations for stiffened shells are derived based on the classical shell theory and smeared stiffeners technique. The resulting equations which they are the couple set of three variable coefficient partial differential equations in terms of displacement components are investigated by Galerkin method and the closed-form expression for determining the buckling load is obtained. The effects of stiffeners, material and dimensional parameters are analyzed in detail.

Discussion on integrable properties for higher-dimensional variable-coefficient nonlinear partial differential equations

In this paper we introduce two new higher-dimensional variable-coefficient partial differential equations. One is a (2+1)-dimensional equation which can be reduced to the well-known KP equation which first occurs to the paper B. B. Kadomtsev and V. I. Petviashvili, “On the stability of solitary waves in weakly dispersive media,” Sov. Phys. Dokl. 15, 539 (1970), whose bilinear representation, Lax pairs, Bäcklund transformations, and infinite conservation laws are obtained respectively by using the Bell polynomials. Another one is a (3+1)-dimensional equation whose integrability is also investigated by us and whose Lax pairs, Bäcklund transformations, and infinite conservation laws are obtained, respectively.

Derivation of Strictly Stable High Order Difference Approximations for Variable-Coefficient PDE

A new way of deriving strictly stable high order difference operators for partial differential equations (PDE) is demonstrated for the 1D convection diffusion equation with variable coefficients. The derivation is based on a diffusion term in conservative, i.e. self-adjoint, form. Fourth order accurate difference operators are constructed by mass lumping Galerkin finite element methods so that an explicit method is achieved. The analysis of the operators is confirmed by numerical tests. The operators can be extended to multi dimensions, as we demonstrate for a 2D example. The discretizations are also relevant for the Navier---Stokes equations and other initial boundary value problems that involve up to second derivatives with variable coefficients.