Nonlinear First-order Partial Differential Equations Research Articles

Physics-informed neural network for solution of forward and inverse kinematic wave problems

The one-dimensional kinematic wave (KW) model (KWM) is widely used in surface water and water quality hydrology as well as for modeling the movement of traffic on long highways. The KW model involves a first-order nonlinear partial differential equation which is solved numerically for realistic geometric representation, input, and initial and boundary conditions. This paper developed a method to solve the partial differential equation using a neural network structure, called the physics-informed neural network (PINN). The PINN was applied to solve the one-dimensional nonlinear KWM for forward and inverse problems. To enhance the accuracy and convergence and to suppress the spurious oscillation at the steep front, a technique with redistributed collocation points as an improved algorithm of PINN was employed. To deal with the fractional order of the nonlinear advection term, a stop-gradient technique was proposed. For validation, the KWM for steady and unsteady overland flow, and open channel flow without lateral inflow was analyzed for the forward problem. For the inverse problem, the unknown Manning coefficient was correctly computed, demonstrating the capability of the proposed algorithm for the identification of KWM parameters.

The New Adomian Technique for Solving Singular and Non-singular Partial Differential Equations of First Order

In this paper, the proposed modification to Adomian decomposition method (ADM) is introduced. In addition, a new modification to solve singular and nonsingular initial-value problems (IVPs) in first order partial differential equations (PDEs) is investigated.
 Then, a discussion of how ADM can be used to solve singular and nonsingular IVPs is produced. Finally, some example are given to demonstrate the applicability of the results of paper.
 Key words; Adomian decomposition method; Nonlinear first-order PDE; Singular and nonsingular first-order partial differential equations.
 1 Introduction
 Singular nonlinear partial differential equations (PDEs) arise in various physical phe- nomena in applied sciences and engineering from such areas as fluid mechanics and heat-

A harmonic balance solution for the intrinsic 1D nonlinear equations of the beams

In this paper, a harmonic balance method is introduced to solve the intrinsic 1D nonlinear equations presented by Hodges for analyzing initially curved and twisted anisotropic rotating beams. First, the nonlinear first-order partial differential equations are discretized in space domain using the Galerkin approach. Then, the time response of the system of equations is approximated by using first and the second harmonic terms under the harmonic excitations applied to the structure. The harmonic balance algorithm has been developed for the linearized system of equations about the steady-state solution and for fully nonlinear equations. The differences between these two systems have been addressed under consideration of different levels of external loads. Fully analytical formulae have been presented for the HB solution of the linearized equations in this work. In the proposed approach for the nonlinear system, the Jacobian matrix utilized in the numerical solution part is obtained analytically to increase the time efficiency of the solution method. The results of the proposed approach for an initially twisted and curved blade are compared with a different algorithm based on a finite difference approach to discretize the equations and a time marching method to solve the system of equations in time domain. The comparison between the outputs of the harmonic balance method and the time marching method demonstrate high correlations between these distinct methods. It should be noted that the accuracy of the time marching method solution has been evaluated in another publication by the authors using the output of FLIGHTLAB for a blade with different types of boundary conditions. The proposed harmonic balance-based method delivers a very time-efficient approach for time analysis of twisted/curved beams and blades under harmonic loads in comparison with other employed methods. The developed approach can be a powerful computational tool for design optimization of composite blades.

АППРОКСИМАЦИЯ ТЕЧЕНИЯ СЛОЯ ОГРАНИЧЕННОЙ ТОЛЩИНЫ МНОГОСВЯЗНОГО КОНТУРА СОВОКУПНОСТЬЮ ТЕЧЕНИЙ ОДНОСВЯЗНЫХ ОБЛАСТЕЙ

The calculations are based on the one proposed by A.A. Ilyushin. theory of flow in a thin plastic layer, enclosed between two surfaces of tool bodies approaching according to a given law, under the assumption that: the material of the spreading layer is ideally plastic and volumetrically incompressible; elastic deformations of the tool are small and can be neglected; on the contact surfaces of the metal layer with the tool, the classical L. Prandtl flow law is accepted. To study the characteristics of the flow, we use the well-known boundary value problem in the “ideal fluid” formulation, which is described by nonlinear first-order partial differential equations with respect to the contact pressure and components of the flow velocity vector. In the general case, the quantities under study are complex functions that depend on the shape of the deformation zone, the magnitude of the contact pressure at the considered point of the contact surface, the presence, composition and method of applying a lubricant, the roughness of the contacting surfaces and other factors. The paper proposes a method for solving problems of flow of plastic layers in a new formulation. Using the proposed mathematical apparatus, a solution algorithm and a computer model of the flow were constructed.

Solving the eikonal equation for compressional and shear waves in anisotropic media using peridynamic differential operator

SUMMARY Traveltimes of compressional (P) and shear (S) waves have proven essential in many earthquake and exploration seismology applications. An accurate and efficient traveltime computation for P and S waves is crucial for the success of these applications. However, solving the eikonal equation with a complex phase velocity field in anisotropic media is challenging. The eikonal equation is a first-order nonlinear hyperbolic partial differential equation. It represents the high-frequency asymptotic approximation of the wave equation. The fast marching and sweeping methods are commonly used due to their efficiency in numerically solving the eikonal equation. However, these methods suffer from numerical inaccuracy in anisotropic media with sharp heterogeneity, irregular surface topography and complex phase velocity fields. This study presents a new method for the solution of the eikonal equation by employing the peridynamic differential operator (PDDO). The PDDO provides the non-local form of the eikonal equation by introducing an internal length parameter (horizon) and a weight function with directional non-locality. The operator is immune to discontinuities in the form of sharp changes in field or model variables and invokes the direction of traveltime in a consistent manner. The weight function controls the strength of association among points within the horizon. Solutions are constructed in a consistent manner without upwind assumptions through simple discretization. The robustness of this approach is established by considering different types of eikonal equations on complex velocity models in anisotropic media. The examples demonstrate its unconditional numerical stability and results compare well with the reference solutions.

Consumer Optimization and a First-Order PDE with a Non-Smooth System

We study a first-order nonlinear partial differential equation and present a necessary and sufficient condition for the global existence of its solution in a non-smooth environment. Using this result, we prove a local existence theorem for a solution to this differential equation. Moreover, we present two applications of this result. The first concerns an inverse problem called the integrability problem in microeconomic theory and the second concerns an extension of Frobenius’ theorem.

An improved meshless artificial viscosity technology combined with local radial point interpolation method for 2D shallow water equations

The two-dimensional shallow water equations (SWEs) are a hyperbolic system of first-order nonlinear partial differential equations which have a characteristic of strong gradient. In this study, a newly-developed numerical model, based on local radial point interpolation method (LRPIM), is adopted to simulate discontinuity in shallow water flows. In order to accurately capture the information of wave propagation, the LRPIM is combined with the split-coefficient matrix (SCM) method to transform the SWEs into a characteristic form and the selection of the direction of local support domain is introduced into the LRPIM. An improved meshless artificial viscosity (MAV) technique is developed to minimize the non-physical oscillations near the discontinuities. Then, the LRPIM and the second-order Runge–Kutta method are adopted for spatial and temporal discretization of the SWEs, respectively. The feasibility and validity of the proposed numerical model are verified by the classical dam-break problem and the mixed flow pattern problem. The comparison of the obtained results with the analytical solution and other numerical results showed that the MAV method combined with LRPIM can accurately capture the shocks and has high accuracy in dealing with discontinuous flow by adding appropriate viscosity to the equations in the discontinuous region.

Weighted Average Based Differential Quadrature Method for One-Dimensional Homogeneous First Order Nonlinear Parabolic Partial Differential Equation

In this paper, the weighted average-based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation. First, the given solution domain is discretized by using uniform discretization grid point. Next, by using Taylor series expansion we obtain central finite difference discretization of the partial differential equation involving with temporal variable associated with weighted average of partial derivative concerning spatial variable. From this, we obtain the system of nonlinear ordinary differential equations and it is linearized by using the quasilinearization method. Then by using the polynomial-based differential quadrature method for approximating derivative involving with spatial variable at specified grid point, we obtain the system of linear equation. Then they obtained linear system equation is solved by using the LU matrix decomposition method. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm and maximum absolute error norm and the local behavior of the solution is captured exactly. Numerical versus exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root mean square error norm and maximum absolute error norm presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving the non-linear parabolic equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution.

Weighted Average Based Differential Quadrature Method for One-Dimensional Homogeneous First Order Nonlinear Parabolic Partial Differential Equation

In this paper, the weighted average-based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation. First, the given solution domain is discretized by using uniform discretization grid point. Next, by using Taylor series expansion we obtain central finite difference discretization of the partial differential equation involving with temporal variable associated with weighted average of partial derivative concerning spatial variable. From this, we obtain the system of nonlinear ordinary differential equations and it is linearized by using the quasilinearization method. Then by using the polynomial-based differential quadrature method for approximating derivative involving with spatial variable at specified grid point, we obtain the system of linear equation. Then they obtained linear system equation is solved by using the LU matrix decomposition method. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm and maximum absolute error norm and the local behavior of the solution is captured exactly. Numerical versus exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root mean square error norm and maximum absolute error norm presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving the non-linear parabolic equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution.

Anti-eikonal equation of an eigenmirror.

We call a surface that appears undistorted when viewed in a curved mirror an eigensurface and the mirror an eigenmirror. Such pairs are described by a first-order nonlinear partial differential equation of the form a0+a1ux+a2uy+a3uxuy+a4ux2+a5uy2=0, where ai=ai(x,y,u), which we call the anti-eikonal equation. We give examples of symbolic and numerical solutions, including pairs that are geometrically congruent. Ray tracing simulations are included that visually confirm the unusual properties of these surfaces.

A robust numerical scheme for transcritical flow simulation in a Braided river

Flow simulation over a braided channel is a challenging task as in the monsoon season flow moves over an undulating bed while as in non-monsoon season or in lean period it moves through several sub-channel. Therefore a more robust scheme is necessary for simulating such kind flow particularly when transcritical flow occurs in river reach. Shallow water equations can be derived by depth integration of the Navier Stokes equation and assuming hydrostatic pressure distribution. These equations are basically based upon the conservation of mass and momentum approach. Shallow water equations are a set of a first-order nonlinear hyperbolic partial differential equation and do not have closed form of analytical solution without considering a large number of approximation because of which these equation needs to be solved by different numerical techniques like finite difference, finite element, and finite volume. The hyperbolic nature of the equations results in a generation of numerical dispersion in the solution domain because of which proper shock capturing methods need to be implemented along with the scheme. The presence of bed slope term in the governing equations causes some difficulty to apply them in complex bathymetry. Researchers have modified the original form of governing equations so that these equations can be implemented in a complex topography with proper mass conservation. In this paper, the modified form of the 2D governing equation is combined with the proper shock capturing technique to investigate the applicability of the scheme in complex bathymetry. This scheme is used as an integral part of BRAHMA model (Braided River Aid: Hydro-Morphological Analyser) developed by IIT-Guwahati in collaboration with Brahmaputra Board. Governing equations solved by Mac Cormack predictor-corrector scheme and the scheme has been tested with some classical problem including dam break, quiescent water above an irregular bed, steady flow over an irregular bed, transcritical flow over a bump etc. For each test case numerical results are compared with the analytical result and found satisfactory.

Eigensurfaces of eigenmirrors.

Typically, if an observer gazes at a curved reflector, the objects in it will appear to be distorted. We show here that for some mirrors there exist surfaces that do not appear distorted when viewed from a prescribed location. We call such mirrors eigenmirrors and the surfaces eigensurfaces. We first give an analysis of the rotationally symmetric case and verify our work with simulations. In the general three-dimensional (3D) case, if the mirror is given, then one does not expect an eigensurface to exist. On the other hand, if we are given two viewpoints and a correspondence between the ray bundles emanating from each point, and we treat both the eigenmirror and the eigensurface as unknowns, then the problem reduces to solving a first-order nonlinear partial differential equation. We derive this partial differential equation in the 3D case and examine one example in detail.

Mapping of non-autonomous dynamical systems to autonomous ones

Using a proper choice of the dynamical variables, we show that a non-autonomous dynamical system transforming to an autonomous dynamical system with a certain coordinate transformations can be obtained by solving a general nonlinear first-order partial differential equations. We examine some special cases and provide particular physical examples. Our framework constitutes general machineries in investigating other non-autonomous dynamical systems.

Modeling and Analysis of a Nonlinear Age-Structured Model for Tumor Cell Populations with Quiescence

We present a nonlinear first-order hyperbolic partial differential equation model to describe age-structured tumor cell populations with proliferating and quiescent phases at the avascular stage in vitro. The division rate of the proliferating cells is assumed to be nonlinear due to the limitation of the nutrient and space. The model includes a proportion of newborn cells that enter directly the quiescent phase with age zero. This proportion can reflect the effect of treatment by drugs such as erlotinib. The existence and uniqueness of solutions are established. The local and global stabilities of the trivial steady state are investigated. The existence and local stability of the positive steady state are also analyzed. Numerical simulations are performed to verify the results and to examine the impacts of parameters on the nonlinear dynamics of the model.

Numerical solution of special ultra-relativistic Euler equations using central upwind scheme

Abstract This article is concerned with the numerical approximation of one and two-dimensional special ultra-relativistic Euler equations. The governing equations are coupled first-order nonlinear hyperbolic partial differential equations. These equations describe perfect fluid flow in terms of the particle density, the four-velocity and the pressure. A high-resolution shock-capturing central upwind scheme is employed to solve the model equations. To avoid excessive numerical diffusion, the considered scheme avails the specific information of local propagation speeds. By using Runge-Kutta time stepping method and MUSCL-type initial reconstruction, we have obtained 2nd order accuracy of the proposed scheme. After discussing the model equations and the numerical technique, several 1D and 2D test problems are investigated. For all the numerical test cases, our proposed scheme demonstrates very good agreement with the results obtained by well-established algorithms, even in the case of highly relativistic 2D test problems. For validation and comparison, the staggered central scheme and the kinetic flux-vector splitting (KFVS) method are also implemented to the same model. The robustness and efficiency of central upwind scheme is demonstrated by the numerical results.

A comparative study of modified cubic B-spline differential quadrature methods for a class of nonlinear viscous wave equations

PurposeIn this study, a second-order standard wave equation extended to a two-dimensional viscous wave equation with timely differentiated advection-diffusion terms has been solved by differential quadrature methods (DQM) using a modification of cubic B-spline functions. Two numerical schemes are proposed and compared to achieve numerical approximations for the solutions of nonlinear viscous wave equations.Design/methodology/approachTwo schemes are adopted to reduce the given system into two systems of nonlinear first-order partial differential equations (PDE). For each scheme, modified cubic B-spline (MCB)-DQM is used for calculating the spatial variables and their derivatives that reduces the system of PDEs into a system of nonlinear ODEs. The solutions of these systems of ODEs are determined by SSP-RK43 scheme. The CPU time is also calculated and compared. Matrix stability analysis has been performed for each scheme and both are found to be unconditionally stable. The results of both schemes have been extensively discussed and compared. The accuracy and reliability of the methods have been successfully tested on several examples.FindingsA comparative study has been carried out for two different schemes. Results from both schemes are also compared with analytical solutions and the results available in literature. Experiments show that MCB-DQM with Scheme II yield more accurate and reliable results in solving viscous wave equations. But Scheme I is comparatively less expensive in terms of CPU time. For MCB-DQM, less depository requirements lead to less aggregation of approximation errors which in turn enhances the correctness and readiness of the numerical techniques. Approximate solutions to the two-dimensional nonlinear viscous wave equation have been found without linearizing the equation. Ease of implementation and low computation cost are the strengths of the method.Originality/valueFor the first time, a comparative study has been carried out for the solution of nonlinear viscous wave equation. Comparisons are done in terms of accuracy and CPU time. It is concluded that Scheme II is more suitable.

A study of structure-exploiting SQP algorithms for an optimal control problem with coupled hyperbolic and ordinary differential equation constraints

In this article, structure-exploiting optimisation algorithms of the sequential quadratic programming (SQP) type are considered for optimal control problems with control and state constraints. Our approach is demonstrated for a 1D mathematical model of a vehicle transporting a fluid container. The model involves a fully coupled system of ordinary differential equations (ODE) and nonlinear hyperbolic first-order partial differential equations (PDE), although the ideas for exploiting the particular structure may be applied to more general optimal control problems as well. The time-optimal control problem is solved numerically by a full discretisation approach. The corresponding nonlinear optimisation problem is solved by an SQP method that uses exact first and second derivative information. The quadratic subproblems are solved using an active-set strategy. In addition, two approaches are examined that exploit the specific structure of the problem: (A) a direct method for the KKT system, and (B) an iterative method based on combining the limited-memory BFGS method with the preconditioned conjugate gradient method. Method (A) is faster for our model problem, but can be limited by the problem size. Method (B) opens the door for a potential extension of the truck-container model to three space dimensions.

Generalized finite difference method for two-dimensional shallow water equations

A novel meshless numerical scheme, based on the generalized finite difference method (GFDM), is proposed to accurately analyze the two–dimensional shallow water equations (SWEs). The SWEs are a hyperbolic system of first-order nonlinear partial differential equations and can be used to describe various problems in hydraulic and ocean engineering, so it is of great importance to develop an efficient and accurate numerical model to analyze the SWEs. According to split-coefficient matrix methods, the SWEs can be transformed to a characteristic form, which can easily present information of characteristic in the correct directions. The GFDM and the second-order Runge-Kutta method are adopted for spatial and temporal discretization of the characteristic form of the SWEs, respectively. The GFDM is one of the newly-developed domain-type meshless methods, so the time-consuming tasks of mesh generation and numerical quadrature can be truly avoided. To use the moving-least squares method of the GFDM, the spatial derivatives at every node can be expressed as linear combinations of nearby function values with different weighting coefficients. In order to properly cooperate with the split-coefficient matrix methods and the characteristic of the SWEs, a new way to determine the shape of star in the GFDM is proposed in this paper to capture the wave transmission. Numerical results and comparisons from several examples are provided to verify the merits of the proposed meshless scheme. Besides, the numerical results are compared with other solutions to validate the accuracy and the consistency of the proposed meshless numerical scheme.

On stability and convergence of semi-Lagrangian methods for the first-order time-dependent nonlinear partial differential equations in 1D

In this article, one-step semi-Lagrangian method is investigated for computing the numerical solutions of the first-order time-dependent nonlinear partial differential equations in 1D with initial and boundary conditions. This method is based on Lagrangian trajectory or the integration from the departure points to the arrival points (regular nodes) and Runge–Kutta method for ordinary differential equations. The departure points are traced back from the arrival points along the trajectory of the path. The convergence and stability are studied for the implicit and explicit methods. The numerical examples show that those methods work very efficient for the time-dependent nonlinear partial differential equations.

Solutions of the generalized kinetic model of annihilation for a mixture of particles of two types

The evolution of the concentrations of particles of two types that annihilate at collision is considered. The kinetic model describing the dynamics of the mixture is represented by a system of two first-order nonlinear partial differential equations. It is shown that the solutions of this model are related to the solutions of the inhomogeneous transport equations by the Backlund transform. Analytic solutions of the problem about penetration of particles of the first type from the left half-plane into the right half-plane occupied by the particles of the second type (the two-dimensional penetration problem or molecular beam problem) and of the problem of outflow of the particles of the first type from a circular source into a domain occupied by the particles of the second type are obtained. Possible generalizations of the model are discussed.