Discontinuous Oblique Derivative Problem for Second Order Equations of Mixed Type in General Domains

In [L. Bers (1958). Mathematical Aspects of Subsonic and Transonic Gas Dynamics . Wiley, New York; A.V. Bitsadze (1988). Some Classes of Partial Differential Equations . Gordon and Breach, New York; J.M. Rassias (1990). Lecture Notes on Mixed Type Partial Differential Equations. World Scientific, Singapore; H.S. Sun (1992). Tricomi problem for nonlinear equation of mixed type. Sci. in China ( Series A ), 35 , 14-20], the authors proposed and discussed the Tricomi problem of second order equations of mixed type in a special domain, and in [G.C. Wen (1998). Oblique derivative problems for linear mixed equations of second order. Sci. in China ( Series A ), 41 , 346-356], the author discussed the oblique derivative problem of second order equations of mixed type in a special domain. The present article deals with the discontinuous oblique derivative problem for quasilinear second order equations of mixed (elliptic-hyperbolic) type in general domains. Firstly, we give the formulation of the above boundary value problem, and then prove the existence of solutions for the above problem in general domains, in which the complex analytic method is used.     

General Tricomi-Rassias problem and oblique derivative problem for generalized Chaplygin equations

Many authors have discussed the Tricomi problem for some second order equations of mixed type, which has important applications in gas dynamics. In particular, Bers proposed the Tricomi problem for Chaplygin equations in multiply connected domains [L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Wiley, New York, 1958]. And Rassias proposed the exterior Tricomi problem for mixed equations in a doubly connected domain and proved the uniqueness of solutions for the problem [J.M. Rassias, Lecture Notes on Mixed Type Partial Differential Equations, World Scientific, Singapore, 1990]. In the present paper, we discuss the general Tricomi-Rassias problem for generalized Chaplygin equations. This is one general oblique derivative problem that includes the exterior Tricomi problem as a special case. We first give the representation of solutions of the general Tricomi-Rassias problem, and then prove the uniqueness and existence of solutions for the problem by a new method. In this paper, we shall also discuss another general oblique derivative problem for generalized Chaplygin equations.     

Point‐wise decay estimate for the global classical solutions to quasilinear hyperbolic systems

In this paper, we first consider the Cauchy problem for quasilinear strictly hyperbolic systems with weak linear degeneracy. The existence of global classical solutions for small and decay initial data was established in (Commun. Partial Differential Equations 1994; 19 :1263–1317; Nonlinear Anal. 1997; 28 :1299–1322; Chin. Ann. Math. 2004; 25B :37–56). We give a new, very simple proof of this result and also give a sharp point‐wise decay estimate of the solution. Then, we consider the mixed initial‐boundary‐value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant. Under the assumption that the positive eigenvalues are weakly linearly degenerate, the global existence of classical solution with small and decay initial and boundary data was established in (Discrete Continuous Dynamical Systems 2005; 12 (1):59–78; Zhou and Yang, in press). We also give a simple proof of this result as well as a sharp point‐wise decay estimate of the solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Notice of retraction and apology

It has come to the attention of the editors and publisher that an article published in Numerical Methods and Partial Differential Equations, Second‐order Galerkin‐Lagrange method for the Navier‐Stokes equations, by Mohamed Bensaada, Driss Esselaoui, and Pierre Saramito, Numer Methods Partial Differential Eq 21(6) (2005), 1099–1121, Numerical Methods for Partial Differential Equations Numer Methods Partial Differential Eq(2005)21(6)1099 included large portions that were copied from the following paper without proper citation: Convergence and nonlinear stability of the Lagrange‐Galerkin method for the Navier‐Stokes equations, Endre Suli, Numerische Mathematik, Vol. 53, No. 4, pp. 459–486 (July, 1988). We have retracted the paper and apologize to Dr. Suli.     

OBLIQUE DERIVATIVE PROBLEMS FOR SECOND ORDER NONLINEAR MIXED EQUATIONS WITH DEGENERATE LINE

The present article deals with oblique derivative problems for some nonlinear mixed equations with parabolic degeneracy, which include the Tricomi problem as a special case. First, the formulation of the problems for the equations is given; next, the representation and estimates of solutions for the above problems are obtained; finally, the existence of solutions for the problems is proved by the successive iteration and the compactness principle of solutions of the problems. In this article, the author uses the complex method, namely, the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain are used.     

The Exterior Tricomi Problem for Generalized Mixed Equations with Parabolic Degeneracy

This paper deals with the exterior Tricomi problem for generalized mixed equations with parabolic degeneracy. Firstly the representation of solutions of the problem for the equations is given, and then the uniqueness and existence of solutions are proved by a new method.     

Boundary Value Problems for Degenerate Hyperbolic Systems of First Order Equations

In this paper we discuss some boundary value problems for degenerate hyperbolic complex equations of first order in a simply connected domain, in which the boundary value problems include the Riemann-Hilbert problem and the Cauchy problem. We first give the representation of solutions of the boundary value problems for the equations, and then prove the uniqueness and existence of solutions for the problems. In [A.V. Bitsadze (1988). Some Classes of Partial Differential Equations . Gordon and Breach, New York; A.V. Bitsadze and A.N. Nakhushev (1972). Theory of degenerating hyperbolic equations. Dokl. Akad. Nauk, SSSR , 204 , 1289-1291 (Russian); M.H. Protter (1954). The Cauchy problem for a hyperbolic second order equation. Can. J. Math ., 6 , 542-553], the authors discussed some boundary value problems for hyperbolic equations of second order.     

A Joint DAE/PDE Model for Interconnected Electrical Networks

To model transmission lines effects in integrated circuits, we couple the network equations for the circuits with the telegrapher's equations for the transmission lines. This results in an initial/boundary value problem for a mixed system of Differential-Algebraic Equations (DAEs) and hyperbolic Partial Differential Equations (PDEs). By semidiscretization the system is transformed into differential-algebraic equations in time only. We apply this modeling approach to a CMOS ring oscillator, an oscillatory circuit with transmission lines as coupling units, and discuss the simulation results.     

On stable implicit difference scheme for hyperbolic–parabolic equations in a Hilbert space

The first‐order of accuracy difference scheme for approximately solving the multipoint nonlocal boundary value problem for the differential equation in a Hilbert space H, with self‐adjoint positive definite operator A is presented. The stability estimates for the solution of this difference scheme are established. In applications, the stability estimates for the solution of difference schemes of the mixed type boundary value problems for hyperbolic–parabolic equations are obtained. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Solving nonlocal initial‐boundary value problems for parabolic and hyperbolic integro‐differential equations in reproducing kernel hilbert space

This article is concerned with a method for solving nonlocal initial‐boundary value problems for parabolic and hyperbolic integro‐differential equations in reproducing kernel Hilbert space. Convergence of the proposed method is studied under some hypotheses which provide the theoretical basis of the proposed method and some error estimates for the numerical approximation in reproducing kernel Hilbert space are presented. Finally, two numerical examples are considered to illustrate the computation efficiency and accuracy of the proposed method. © 2016 The Authors Numerical Methods for Partial Differential Equations Published by Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 174–198, 2017     

Positive numerical splitting method for the Hull and White 2D Black–Scholes equation

We consider the locally one‐dimensional backward Euler splitting method to solve numerically the Hull and White problem for pricing European options with stochastic volatility in the presence of a mixed derivative term. We prove the first‐order convergence of the time‐splitting. The parabolic equation degenerates on the boundary x = 0 and we apply a fitted finite volume scheme to the equation to resolve the degeneracy and derive the fully discrete problem as we also investigate the discrete maximum principle. Numerical experiments illustrate the efficiency of our difference scheme. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 822–846, 2015     

A Note on Semiconcave Function

This article gives a simple proof of an equivalent proposition on semiconcave function (see [L.C. Evans (1998). Partial Differential Equations. American Mathematical Society; p. 130]). The proof of sufficiency of the proposition can be easily obtained. We prove its necessity by three steps: First, we prove that the equivalent proposition holds for discrete points <artwork name="GAPA31045ei1">; Secondly, we obtain continuity of semiconcave function; Finally, by using the fact that the sequences λ_m ^k are dense in the interval (0, 1), we prove that the equivalent proposition holds for each λ ∈ (0, 1).     

Optimal convergence analysis of mixed finite element methods for fourth‐order elliptic and parabolic problems

By using a special interpolation operator developed by Girault and Raviart (finite element methods for Navier‐Stokes Equations, Springer‐Verlag, Berlin, 1986), we prove that optimal error bounds can be obtained for a fourth‐order elliptic problem and a fourth‐order parabolic problem solved by mixed finite element methods on quasi‐uniform rectangular meshes. Optimal convergence is proved for all continuous tensor product elements of order k ≥ 1. A numerical example is provided for solving the fourth‐order elliptic problem using the bilinear element. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Jiang Zhu,Abimael F. D. Loula, “Mixed finite element analysis of a thermally nonlinear coupled problem,” Numerical Methods for Partial Differential Equations (2006) 22(1)180–196

The original article to which this erratum refers was published in Numerical Methods for Partial Differential Equations Numer Methods Partial Differential Eq(2006)22(1)180     

Optimal control of n x n parabolic system involving time lag

In this paper, we study the optimal control problem for an nx n coupled system of second-order parabolic partial differentialequations with infinitely many variables and constant time lag.Making use of the Lions scheme (Lions, 1971, Optimal Controlof Systems Governed by Partial Differential Equations), necessaryand sufficient condition of optimality for the Neumann problemwith quadratic performance functional and constraint controlis derived. Finally, several mathematical examples for derivedoptimality conditions are presented.     

Uniqueness of Oblique Derivative Problems for Second Order Equations of Mixed Type with Nonsmooth Degenerate Line

In [], the authors discussed the Tricomi problem of second order mixed equations with nonsmooth degenerate line, but they only consider some special mixed equations. In [], the author discussed the uniqueness of solutions of the Tricomi problem for some second order mixed equations with nonsmooth degenerate line. The present article deals with oblique derivative problems for general second order mixed equations with nonsmooth parabolic degenerate line, and prove the uniqueness of solutions of the problems. We first give the formulation of the problems for the equations, and then prove the uniqueness of solutions for the above problems.     

About the Tricomi problem for the mixed parabolic–hyperbolic type equation with complex spectral parameter

The unique solvability of the Tricomi problem for the parabolic–hyperbolic equation with complex spectral parameter is proved. Uniqueness of the solution is shown by the method of energy integral and existence by the method of integral equations.     

Continuous Dependence Estimates for Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Equations

Using the maximum principle for semicontinuous functions (Differential Integral Equations3 (1990), 1001-1014; Bull. Amer. Math. Soc. (N.S)27 (1992), 1-67), we establish a general “continuous dependence on the non- linearities” estimate for viscosity solutions of fully nonlinear degenerate parabolic equations with time- and space-dependent nonlinearities. Our result generalizes a result by Souganidis (J. Differential Equations56 (1985), 345-390) for first- order Hamilton-Jacobi equations and a recent result by Cockburn et al. (J. Differential Equations170 (2001), 180-187) for a class of degenerate parabolic second-order equations. We apply this result to a rather general class of equations and obtain: (i) Explicit continuous dependence estimates. (ii) L^∞ and Hölder regularity estimates. (iii) A rate of convergence for the vanishing viscosity method. Finally, we illustrate results (i)-(iii) on the Hamilton-Jacobi- Bellman partial differential equation associated with optimal control of a degenerate diffusion process over a finite horizon. For this equation such results are usually derived via probabilistic arguments, which we avoid entirely here.     

Crank-Nicolson method for the numerical solution of models of excitability

We analyze a Crank-Nicolson scheme for a family of nonlinear parabolic partial differential equations. These equations cover a wide class of models of excitability, in particular the Hodgkin Huxley equations. To do the analysis, we have in mind the general discretization framework introduced by López-Marcos and Sanz-Serna [in Numerical Treatment of Differential Equations, K. Strehemel, Ed., Teubner-Texte zur Mathematik, Leipzig, 1988, p. 216]. We study consistency, stability and convergence properties of the scheme. We use a technique of modified functions, introduced by Strang [Numer. Math. 6 , 37 (1964)], in the study of consistency. Stability is derived by means of the energy method. Finally we obtain existence and convergence of numerical approximations by means of a result due to Stetter (Analysis of Discretization Methods for Ordinary Differential Equations. Springer-Verlag, Berlin, 1973). We show that the method has optimal order of accuracy in the discrete H¹ norm. © 1994 John Wiley & Sons, Inc.     

Mixed Problem for Systems of Hyperbolic Equations with Nonlinear Boundary Dissipation and a Nonlinear Source of Variable Growth Order

Differential Equations - We study a mixed problem for systems of one-dimensional semilinear hyperbolic equations with variable nonlinearity growth rate and nonlinear boundary conditions. Theorems...