Quasilinear Partial Differential Equations Research Articles

A numerical method of bicharacteristics For quasi-linear partial functional Differential equations

Abstract Classical solutions of mixed problems for first order partial functional differential equations in several independent variables are approximated by solutions of an Euler-type difference problem. The mesh for the approximate solutions is obtained by the numerical solution of equations of bicharacteristics. The convergence of explicit difference schemes is proved by means of consistency and stability arguments. It is assumed that the given functions satisfy the nonlinear estimates of the Perron type. Differential systems with deviated variables and differential integral systems can be obtained from the general model by specializing the given operators.

Portfolios and Risk Premia for the Long Run

This paper develops a method to derive optimal portfolios and risk premia explicitly in a general diffusion model, for an investor with power utility and a long horizon. The market has several risky assets and is potentially incomplete. Investment opportunities are driven by, and partially correlated with, state variables which follow an autonomous diffusion. The framework nests models of stochastic interest rates, return predictability, stochastic volatility and correlation risk.In models with several assets and a single state variable, long-run portfolios and risk premia admit explicit formulas up the solution of an ordinary differential equation, which characterizes the principal eigenvalue of an elliptic operator. Multiple state variables lead to a quasilinear partial differential equation, which is solvable for many models of interest.The paper derives the long-run optimal portfolio and the long-run optimal pricing measures depending on relative risk aversion, as well as their finite-horizon performance.

Intrinsic Harnack inequalities for quasi-linear singular parabolic partial differential equations

Intrinsic Harnack estimates for non-negative solutions of singular, quasi-linear, parabolic equations, are established, including the prototype p -Laplacean equation (1.4) below. For p in the super-critical range \frac{2N}{N+1}<p<2 , the Harnack inequality is shown to hold in a parabolic form, both forward and backward in time, and in a elliptic form at fixed time. These estimates fail for the heat equation ( p\to2 ). It is shown by counterexamples, that they fail for p in the sub-critical range 1<p\le \frac{2N}{N+1} . Thus the indicated super-critical range is optimal for a Harnack estimate to hold. The novel proofs are based on measure theoretical arguments, as opposed to comparison principles and are sufficiently flexible to hold for a large class of singular parabolic equation including the porous medium equation and its quasi-linear versions.

Identification of the hydraulic conductivities in a saltwater intrusion problem

In this paper, the identification of the hydraulic conductivities in a seawater intrusion problem is investigated. The system of two coupled quasilinear partial differential equations describing the direct problem are solved on the basis of an established maximum principle using a fixed point theorem. The existence of an optimal control is proved, even if the conductivities are discontinuous functions, and the necessary conditions for optimality are obtained. Finally, some numerical results are presented for the sake of illustration.

Analysis of the validity of the asymptotic techniques in the lower hybrid wave equation solution for reactor applications

Knowing that the lower hybrid (LH) wave propagation in tokamak plasmas can be correctly described with a full wave approach only, based on fully numerical techniques or on semianalytical approaches, in this paper, the LH wave equation is asymptotically solved via the Wentzel-Kramers-Brillouin (WKB) method for the first two orders of the expansion parameter, obtaining governing equations for the phase at the lowest and for the amplitude at the next order. The nonlinear partial differential equation (PDE) for the phase is solved in a pseudotoroidal geometry (circular and concentric magnetic surfaces) by the method of characteristics. The associated system of ordinary differential equations for the position and the wavenumber is obtained and analytically solved by choosing an appropriate expansion parameter. The quasilinear PDE for the WKB amplitude is also solved analytically, allowing us to reconstruct the wave electric field inside the plasma. The solution is also obtained numerically and compared with the analytical solution. A discussion of the validity limits of the WKB method is also given on the basis of the obtained results.

A posteriori error analysis of hp-version discontinuous Galerkin finite-element methods for second-order quasi-linear elliptic PDEs

We develop the a-posteriori error analysis of hp-version interior-penalty discontinuous Galerkin finite element methods for a class of second-order quasilinear elliptic partial differential equations. Computable upper and lower bounds on the error are derived in terms of a natural (mesh-dependent) energy norm. The bounds are explicit in the local mesh size and the local degree of the approximating polynomial. The performance of the proposed estimators within an automatic hp-adaptive refinement procedure is studied through numerical experiments.

Ground states of a prescribed mean curvature equation

We study the existence of radial ground state solutions for the problem − div ( ∇ u 1 + | ∇ u | 2 ) = u q , u > 0 in R N , u ( x ) → 0 as | x | → ∞ , N ⩾ 3 , q > 1 . It is known that this problem has infinitely many ground states when q ⩾ N + 2 N − 2 , while no solutions exist if q ⩽ N N − 2 . A question raised by Ni and Serrin in [W.-M. Ni, J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations, Atti Convegni Lincei 77 (1985) 231–257] is whether or not ground state solutions exist for N N − 2 < q < N + 2 N − 2 . In this paper we prove the existence of a large, finite number of ground states with fast decay O ( | x | 2 − N ) as | x | → + ∞ provided that q lies below but close enough to the critical exponent N + 2 N − 2 . These solutions develop a bubble-tower profile as q approaches the critical exponent.

State estimation of a stratified storage tank

Two approaches for the state estimation of the spatial temperature profile in a stratified storage tank, i.e. the state of the art domestic hot water storage system, are presented. There are a distributed parameter observer as a late lumping and an Unscented Kalman filter (UKF) as an early lumping approach which are investigated with respect to applicability and reconstruction convergence. Both, the distributed parameter observer and the UKF are designed based on a hybrid distributed parameter model of the stratified storage tank. The hybrid model is composed of a finite state automaton interacting with an underlying thermal heat conduction–convection system described by a quasi-linear partial differential equation. The performance of the estimation algorithms is illustrated by simulation studies and measurement data, showing excellent convergence results.

Rational expectations models: An approach using forward–backward stochastic differential equations

We show that a general class of continuous time rational expectations models can be reformulated as forward–backward stochastic differential equations (FBSDEs). Using this connection we obtain results on the conditions under which paths leading to, or keeping close to equilibrium exist, as well as their qualitative properties. We also provide a method for the construction of such paths through the connection of FBSDEs with quasilinear partial differential equations (PDEs). The theory is applied to specific macroeconomic models.

Dressing method based on the homogeneous Fredholm equation: quasilinear PDEs in multidimensions

In this paper we develop a dressing method for constructing and solving some classes of matrix quasilinear partial differential equations (PDEs) in arbitrary dimensions. This method is based on a homogeneous integral equation with a nontrivial kernel, which allows one to reduce the nonlinear PDEs to systems of non-differential (algebraic or transcendental) equations for the unknown fields. In the simplest examples, the above dressing scheme captures matrix equations integrated by the characteristics method and nonlinear PDEs associated with matrix Hopf–Cole transformations.

Numerical pricing of options using high-order compact finite difference schemes

We consider high-order compact (HOC) schemes for quasilinear parabolic partial differential equations to discretise the Black–Scholes PDE for the numerical pricing of European and American options. We show that for the heat equation with smooth initial conditions, the HOC schemes attain clear fourth-order convergence but fail if non-smooth payoff conditions are used. To restore the fourth-order convergence, we use a grid stretching that concentrates grid nodes at the strike price for European options. For an American option, an efficient procedure is also described to compute the option price, Greeks and the optimal exercise curve. Comparisons with a fourth-order non-compact scheme are also done. However, fourth-order convergence is not experienced with this strategy. To improve the convergence rate for American options, we discuss the use of a front-fixing transformation with the HOC scheme. We also show that the HOC scheme with grid stretching along the asset price dimension gives accurate numerical solutions for European options under stochastic volatility.

ROBUST FAULT DETECTION AND HANDLING IN CONTROL OF UNCERTAIN TRANSPORT-REACTION PROCESSES

This paper presents an integrated robust fault detection and fault-tolerant control architecture for transport-reaction processes modeled by quasi-linear parabolic partial differential equations with uncertain variables, control constraints and control actuator faults. Using an appropriate reduced-order model that captures the dominant process dynamics, the proposed architecture comprises a family of robustly stabilizing bounded feedback controllers with explicitly characterized stability and uncertainty attenuation properties, a performance- based fault detection scheme and a high-level supervisor that reconfigures the control actuators upon fault detection in a way that maintains robust closed-loop stability. The key idea is to shape the closed-loop performance via robust control in a way that facilitates the design of robust fault detection rules that are less sensitive to the adverse effects of uncertainty. The results are demonstrated using a non-isothermal tubular reactor example with recycle.

Discretization of forward–backward stochastic differential equations and related quasi-linear parabolic equations

Efficient numerical algorithms for a class of forward-backward stochastic differential equations (FBSDEs) and related quasi-linear parabolic partial differential equations are proposed. The quasi-linear parabolic equation is solved by new layer methods which are constructed by means of a probabilistic approach. The proposed algorithms for solving FBSDEs are based on the four-step scheme of Ma, Protter and Yong. Convergence theorems are proved. Results of some numerical experiments are presented.

Spreading of a thin film with suction or blowing including surface tension effects

In this paper we derive a fourth-order nonlinear partial differential equation modelling the effects of suction/blowing on the evolution of the free surface of a thin viscous film on a porous base. Gravity and surface tension effects are included. The suction/blowing is modelled as an arbitrary function v n ( t , r ) . The Lie group classification method is used to determine a first-order quasi-linear partial differential equation satisfied by v n ( t , r ) . This equation is coupled with the model equation and solved numerically. New results are obtained which show how the free surface and the apparent contact angle vary with suction/blowing.

Formal Gevrey Theory for Singular First Order Quasi-Linear Partial Differential Equations

This paper is concerned with the existence, the uniqueness, convergence and divergence of formal power series solutions of singular first order quasi-linear partial differential equations. Firstly we give the condition under which the formal solution exists uniquely. However, this formal power series solution does not necessarily converge. So we characterize the rate of divergence of the formal solution via the Gevrey order of formal power series. The Gevrey orders of formal solutions are determined by the Newton polyhedrons of nonlinear partial differential operators.

-solutions with singularities as almost solutions

Almost solutions of quasilinear partial differential equations of elliptic type are introduced. The class of equations under consideration contains, in particular, the equations of -harmonic functions, the gas dynamic equation, and some other equations. Conditions ensuring that singular solutions of equations are almost solutions are indicated. Applications to the problem of removable singularities of solutions of elliptic equations and quasiregular maps are presented. The proofs are based on connections with differential forms.

Nonlinear evolution PDEs in [formula omitted]: existence and uniqueness of solutions, asymptotic and Borel summability properties

We consider a system of n-th order nonlinear quasilinear partial differential equations of the formut+P(∂xj)u+g(x,t,{∂xju})=0;u(x,0)=uI(x) with u∊Cr, for t∊(0,T) and large |x| in a poly-sector S in Cd (∂xj≡∂x1j1∂x2j2⋯∂xdjd and j1+⋯+jd⩽n). The principal part of the constant coefficient n-th order differential operator P is subject to a cone condition. The nonlinearity g and the functions uI and u satisfy analyticity and decay assumptions in S.The paper shows existence and uniqueness of the solution of this problem and finds its asymptotic behavior for large |x|.Under further regularity conditions on g and uI which ensure the existence of a formal asymptotic series solution for large |x| to the problem, we prove its Borel summability to the actual solution u.The structure of the nonlinearity and the complex plane setting preclude standard methods. We use a new approach, based on Borel–Laplace regularization and Écalle acceleration techniques to control the equation.These results are instrumental in constructive analysis of singularity formation in nonlinear PDEs with prescribed initial data, an application referred to in the paper.In special cases motivated by applications we show how the method can be adapted to obtain short-time existence, uniqueness and asymptotic behavior for small t, of sectorially analytic solutions, without size restriction on the space variable.

Quasilinear hyperbolic equations with hysteresis

This paper is devoted to the summary of results about a quasilinear hyperbolic partial Differential equation of first order with a hysteresis operator v = ℱ[u]: ∂(u + v)/∂t + ∂u/∂x = 0. Hysteresis is represented by functional describing adsorption and desorption on the particles of the substance and is a possibly discontinuous generalized play operator. The results can be extended to possibly discontinuous generalized Prandtl-Ishlinskii operators of play type.

Determination of a time-dependent parameter in a one-dimensional quasi-linear parabolic equation with temperature overspecification

In this work, the Adomian decomposition method is used to find the solution to a one-dimensional quasi-linear parabolic partial differential equation with a time-dependent unknown function. A wide class of physical phenomena is modelled by non-classical parabolic initial-boundary value problems. Thus the theoretical behaviour and numerical approximation of these problems have been active areas of research. The decomposition procedure first proposed by the American mathematician G. Adomian (1923–1996) is useful for obtaining both exact solutions to, and numerical approximations of, various kinds of linear and nonlinear problem. The Adomian decomposition method, which accurately computes the series solution, is of great interest in science and engineering. It provides a solution as a convergent series with components that can be elegantly computed. Sufficient conditions for convergence and stability of the approximate solution are given and the results of numerical experiments are presented.

Homeomorphism of solutions to backward SDEs and applications

In this paper we study the homeomorphic properties of the solutions to one dimensional backward stochastic differential equations under suitable assumptions, where the terminal values depend on a real parameter. Then, we apply them to the solutions for a class of second order quasilinear parabolic partial differential equations.