First-order Quasilinear Partial Differential Equations Research Articles

The [formula omitted] transformation for solving quasi-linear partial differential equations

The Z1-Z4 transformations proposed in our previous papers have acquired remarkable success in the field of linear partial differential equations, general solutions of a large number of typical equations were obtained for the first time, and exact solutions of many definite solution problems of general significance were also successfully obtained. In this paper we will present the Z5 transformation and use it to get the general solutions of a variety of first-order quasi-linear partial differential equations and, for the first time, to obtain analytical solutions of a class of second-order quasi-linear partial differential equations containing arbitrary functions. Using the general solutions, we obtain exact solutions of two definite solution problems. By comparison we find that the general solutions of some first-order quasi-linear partial differential equations that can be obtained using the characteristic equation method are incomplete.

On constraint preservation and strong hyperbolicity

We use partial differential equations (PDEs) to describe physical systems. In general, these equations include evolution and constraint equations. One method used to find solutions to these equations is the free-evolution approach, which consists in obtaining the solutions of the entire system by solving only the evolution equations. Certainly, this is valid only when the chosen initial data satisfies the constraints and the constraints are preserved in the evolution. In this paper, we establish the sufficient conditions required for the PDEs of the system to guarantee the constraint preservation. This is achieved by considering quasi-linear first-order PDEs, assuming the sufficient condition and deriving strongly hyperbolic first-order partial differential evolution equations for the constraints. We show that, in general, these constraint evolution equations correspond to a family of equations parametrized by a set of free parameters. We also explain how these parameters fix the propagation velocities of the constraints. As application examples of this framework, we study the constraint conservation of the Maxwell electrodynamics and the wave equations in arbitrary space–times. We conclude that the constraint evolution equations are unique in the Maxwell case and a family in the wave equation case.

The nonlocal solvability conditions in original coordinates for a system with constant terms

Conditions of a nonlocal solvability of the Cauchy problem in original coordinates are obtained for a system of two quasilinear first-order partial differential equations with constant terms. The investigation is based on the method of an additional argument. The proof of the nonlocal solvability of the Cauchy problem in original coordinates relies on global estimates.

Numerical Examples of Non-Dissipative Discontinuous Kinematic Waves in Open Channels

The kinematic wave model under the assumption of balanced gravity and friction forces has been applied in open channel hydraulics and surface hydrology. There persists a severe misunderstanding that a discontinuity of a kinematic wave occurs due to a discontinuity of input and then dissipates. This study clarifies that a discontinuity can develop without dissipation under the smoothness of all input. The theory of first-order quasilinear partial differential equations shows that Cauchy problems for the kinematic wave model have unique measurable and bounded solutions, which are possibly discontinuous. Numerical examples are presented to visualize the fundamental properties of discontinuous kinematic waves.

Global Stability and Hopf Bifurcation for a Stage Structured Model with Competition for Food

Considering the mature condition of any individual to have eaten a specific amount of food during the entire period that it can spend at its immature stage, we propose a size-structured model by a first-order quasi-linear partial differential equation. The model can be firstly reduced to a single state-dependent delay differential equation and then to a constant delay differential equation. The state-dependent delay represents intra-specific competition among individuals for limited food resources. A complete analysis of the global dynamics on the positivity and boundedness of solutions, global stability for each equilibrium and Hopf bifurcation is carried out. Our results imply that the delay leads to instability that is shown by a simple example of a certain structured population model.

Classical and weak solutions of the partial differential equations associated with a class of two-point boundary value problems

This paper is concerned with a kind of first-order quasilinear parabolic partial differential equations associated with a class of ordinary differential equations with two-point boundary value problems. We prove that the function given by the solution of an ordinary differential equation is the unique solution of a first-order quasilinear parabolic partial differential equation in both classical and weak senses.

Asymptotic model of size-exclusion grouting

In the construction of foundations of buildings and structures on fragile ground, various technologies of soil grouting are used. When pouring fine-grained concrete into the porous soil, the concrete grains are filtered in the pores of the soil. The filtration process depends on the ratio of the pore sizes of the soil and the solid particles of the injected concrete mortar.Injection of a carrier fluid with small solid particles in a porous medium forms a dynamic concentrations front of suspended and retained particles, separating the suspended particles and the hollow part of the porous frame. The purpose of the study is to construct and calculate an asymptotic model near the concentrations front for the filtration of monodisperse suspension in a porous medium with size-exclusion mechanism of particles retention.The classical mathematical model for one-dimensional filtration of suspensions and colloids in a porous medium is based on the geometric ratio of the particles and pores sizes: the particles freely pass through the large pores and get stuck in the pore throats with sizes smaller than the particles diameter. The model is determined by a system of two quasilinear first-order partial differential equations with the gap between boundary and initial conditions. To construct an asymptotic expansion in the vicinity of the concentrations front, a special small parameter is used that specifies the distance to the front. This parameter provides direct determination of the asymptotic terms from the recurrent system of ordinary linear differential equations of the first order.Near the concentrations front of the suspended and retained particles, a nonlinear high-order asymptotics is constructed for the filtration problem of solid particles transported by a carrier fluid in a porous medium. The obtained solution is zero before the front and nonzero behind the front. Approbation of the asymptotic expansion is carried out. It is shown that, for a linear blocking filtration coefficient, the asymptotics coincides with the exact solution.The asymptotic model of deep bed filtration makes it possible to obtain exact formulas for the high-order asymptotic expansion near the dynamic concentrations front of suspended and retained particles. The asymptotics improves the ability to fine-tune the filtration model depending on the properties of the porous soil and the grout.

Whitham modulation theory for (2 + 1)-dimensional equations of Kadomtsev–Petviashvili type

Whitham modulation theory for certain two-dimensional evolution equations of Kadomtsev–Petviashvili (KP) type is presented. Three specific examples are considered in detail: the KP equation, the two-dimensional Benjamin–Ono (2DBO) equation and a modified KP (m2KP) equation. A unified derivation is also provided. In the case of the m2KP equation, the corresponding Whitham modulation system exhibits features different from the other two. The approach presented here does not require integrability of the original evolution equation. Indeed, while the KP equation is known to be a completely integrable equation, the 2DBO equation and the m2KP equation are not known to be integrable. In each of the cases considered, the Whitham modulation system obtained consists of five first-order quasilinear partial differential equations. The Riemann problem (i.e. the analogue of the Gurevich–Pitaevskii problem) for the one-dimensional reduction of the m2KP equation is studied. For the m2KP equation, the system of modulation equations is used to analyze the linear stability of traveling wave solutions.

Whitham modulation theory for the two-dimensional Benjamin-Ono equation.

Whitham modulation theory for the two-dimensional Benjamin-Ono (2DBO) equation is presented. A system of five quasilinear first-order partial differential equations is derived. The system describes modulations of the traveling wave solutions of the 2DBO equation. These equations are transformed to a singularity-free hydrodynamic-like system referred to here as the 2DBO-Whitham system. Exact reductions of this system are discussed, the formulation of initial value problems is considered, and the system is used to study the transverse stability of traveling wave solutions of the 2DBO equation.

Point Process with Last-arrival-time Dependent Intensity and 1-Dimensional Incompressible Fluid System with Evaporation

We consider an infinite system of quasilinear first-order partial differential equations, generalized to contain spacial integration, which describes an incompressible fluid mixture of infinite components in a line segment whose motion is driven by unbounded and space-time dependent evaporation rates. We prove unique existence of the solution to the initial-boundary value problem, with conservation-of-fluid condition at the boundary. The proof uses a map on the space of collection of characteristics, and a representation based on a non-Markovian point process with last-arrival-time dependent intensity.

Interaction of weak discontinuities and the hodograph method as applied to electric field fractionation of a two-component mixture

The hodograph method is used to construct a solution describing the interaction of weak discontinuities (rarefaction waves) for the problem of mass transfer by an electric field (zonal electrophoresis). Mathematically, the problem is reduced to the study of a system of two first-order quasilinear hyperbolic partial differential equations with data on characteristics (Goursat problem). The solution is constructed analytically in the form of implicit relations. An efficient numerical algorithm is described that reduces the system of quasilinear partial differential equations to ordinary differential equations. For the zonal electrophoresis equations, the Riemann problem with initial discontinuities specified at two different spatial points is completely solved.

Asymptotic model of slow three-dimensional liquid film flow in a space drop catcher

Slow steady-state film flows formed on the inner surface of a drop catcher funnel due to inertial deposition of drops of a dispersed working matter in the spacecraft cooling system are considered. A limiting asymptotic model of slow three-dimensional coolant film flow is constructed assuming that the deposited drops transfer all their mass, momentum, and energy to the film described by the equations of creeping viscous fluid flow in a thin layer of a priori unknown thickness. A first-order quasi-linear partial differential equation for the film thickness is derived. The shape of the film surface is investigated numerically as a function of parameters using the method of characteristics. The range of optimum parameters ensuring the steady-state film flow is found. The limits of existence of the solutions corresponding to the limiting model proposed are investigated.

Conservation Laws, Hodograph Transformation and Boundary Value Problems of Plane Plasticity

For the hyperbolic system of quasilinear first-order partial differential equations, linearizable by hodograph transformation, the conservation laws are used to solve the Cauchy problem. The equivalence of the initial problem for quasilinear system and the problem for conservation laws system permits to construct the characteristic lines in domains, where Jacobian of hodograph transformations is equal to zero. Moreover, the conservation laws give all solutions of the linearized system. Some examples from the gas dynamics and theory of plasticity are considered.

The method of characteristics in ideal plasticity problems

A universal computer algorithm and computer program is developed that implements a direct version of the method of characteristics for solving systems of quasilinear first-order partial differential equations in two independent variables. The algorithm is used to calculate the lattice of slip lines and the velocity vector field in problems of the limiting equilibrium of ideally plastic and friable media. Results of the numerical solution of an ideal plasticity problem for a plane with an aperture loaded with an internal pressure, of the plane problem of the limiting equilibrium of a friable medium and of the axisymmetric problem of a rotating punch on the surface of a plastic half-space are presented to demonstrate the efficiency of the method.

Maximal slicings in spherical symmetry: Local existence and construction

We show that any spherically symmetric spacetime locally admits a maximal space-like slicing and we give a procedure allowing its construction. The designed construction procedure is based on purely geometrical arguments and, in practice, leads to solve a decoupled system of first-order quasi-linear partial differential equations. We have explicitly built up maximal foliations in Minkowski and Friedmann spacetimes. Our approach admits further generalizations and efficient computational implementation. As by-product, we suggest some applications of our work in the task of calibrating numerical relativity complex codes, usually written in Cartesian coordinates.

Identification problems for quasilinear first-order partial differential equations in one space dimension and applications

Abstract We recover unknown nonlinear terms in quasilinear first-order partial differential equations and give some applications to conservation laws.

Equations for the Missing Boundary Values in the Hamiltonian Formulation of Optimal Control Problems

Partial differential equations for the unknown final state and initial costate arising in the Hamiltonian formulation of regular optimal control problems with a quadratic final penalty are found. It is shown that the missing boundary conditions for Hamilton’s canonical ordinary differential equations satisfy a system of first-order quasilinear vector partial differential equations (PDEs), when the functional dependence of the H-optimal control in phase-space variables is explicitly known. Their solutions are computed in the context of nonlinear systems with ℝn-valued states. No special restrictions are imposed on the form of the Lagrangian cost term. Having calculated the initial values of the costates, the optimal control can then be constructed from on-line integration of the corresponding 2n-dimensional Hamilton ordinary differential equations (ODEs). The off-line procedure requires finding two auxiliary n×n matrices that generalize those appearing in the solution of the differential Riccati equation (DRE) associated with the linear-quadratic regulator (LQR) problem. In all equations, the independent variables are the finite time-horizon duration T and the final-penalty matrix coefficient S, so their solutions give information on a whole two-parameter family of control problems, which can be used for design purposes. The mathematical treatment takes advantage from the symplectic structure of the Hamiltonian formalism, which allows one to reformulate Bellman’s conjectures concerning the “invariant-embedding” methodology for two-point boundary-value problems. Results for LQR problems are tested against solutions of the associated differential Riccati equation, and the attributes of the two approaches are illustrated and discussed. Also, nonlinear problems are numerically solved and compared against those obtained by using shooting techniques.

An Approximate Analytical Solution of Imploding Strong Shocks in a Non-Ideal Gas through Lie Group Analysis

The method of Lie group transformation is used to obtain an approximate analytical solution to the system of first-order quasilinear partial differential equations that govern a one-dimensional unsteady planer, cylindrically symmetric and spherically symmetric motion in a non-ideal gas, involving strong shock waves. Invariance groups admitted by the governing system of partial differential equations, which are indeed continuous group of transformations under which the system of partial differential equations remains invariant, are determined, and the complete Lie algebra of infinitesimal symmetries is established. The infinitesimal generators are used to construct the similarity variables. These similarity variables are used to reduce the governing system of partial differential equations into a system of ordinary differential equations.

A new model reduction method for nonlinear dynamical systems

The present research work proposes a new systematic approach to the problem of model-reduction for nonlinear dynamical systems. The formulation of the problem is conveniently realized through a system of singular first-order quasi-linear invariance partial differential equations (PDEs), and a rather general explicit set of conditions for solvability is derived. In particular, within the class of analytic solutions, the aforementioned set of conditions guarantees the existence and uniqueness of a locally analytic solution. The solution to the above system of singular PDEs is then proven to represent the slow invariant manifold of the nonlinear dynamical system under consideration exponentially attracting all dynamic trajectories. As a result, an exact reduced-order model for the nonlinear system dynamics is obtained through the restriction of the original system dynamics on the aforementioned slow manifold. The local analyticity property of the solution’s graph that corresponds to the system’s slow manifold enables the development of a series solution method, which allows the polynomial approximation of the system dynamics on the slow manifold up to the desired degree of accuracy and can be easily implemented with the aid of a symbolic software package such as MAPLE. Finally, the proposed approach and method is evaluated through an illustrative biological reactor example.

Solution of a countable system of quasilinear partial differential equations multiperiodic in a part of variables

We establish sufficient conditions for the existence and uniqueness of solutions of a countable system of first-order quasilinear partial differential equations multiperiodic in a part of variables.