A logarithmically improved regularity criterion for the surface quasi-geostrophic equation

In this paper, we consider the two-dimensional dissipative surface quasi-geostrophic equation, and establish a logarithmically improved regularity criterion. Consequently, our result extends the regularity criterion result of Dong and Pavlovié (2009). In addition, a logarithmically improved regularity criterion to the three-dimensional Navier–Stokes equations is also derived by the same arguments. Therefore, this result extends and improves many previous works.     

Numerical method using cubic B-spline for the heat and wave equation

In the present paper, the cubic B-splines method is considered for solving one-dimensional heat and wave equations. A typical finite difference approach had been used to discretize the time derivative while the cubic B-spline is applied as an interpolation function in the space dimension. The accuracy of the method for both equations is discussed. The efficiency of the method is illustrated by some test problems. The numerical results are found to be in good agreement with the exact solution.     

The Bang-Bang principle of time optimal controls for the heat equation with internal controls

In this paper, we study a certain time optimal control problem governed by the internal controlled heat equation and with a closed ball centered at 0 as the target set. We derive, by making use of Pontryagin's maximum principle and a special kind of unique continuation property for solutions of the heat equation, the bang-bang property for time optimal controls of the problem.     

Multilevel difference schemes for the heat conduction equation and its application to the dirichlet problem in two and three dimensions

In this paper we have discussed a general method of deriving high order multilevel schemes for the heat conduction equation in higher dimensions. The application of the three level parabolic schemes for the solution of the steady state Dirichlet problem in two and three dimensions is discussed by introducing two parameters. The error bounds given by Hadjidimos are further refined which drastically reduces the theoretical estimate of the number of iteration cycles required for a given accuracy η. We have also defined a new set of iteration parameters. A numerical example is computed and it is seen that the present methods produce accurate results.     

Stability of optimal control of heat equation with singular potential

In this paper, we study stability of an optimal control for a multi-dimensional heat equation with a singular potential term. A family of perturbed optimal control problems with lower power singular potentials are formulated. It is shown that when the lower powers tend to the critical power two, the optimal controls are convergent to the optimal control of the original system.     

Continuous dependence of optimal control to controlled domain of actuator for heat equation

In this paper, we consider continuous dependence of the optimal control with respect to the actuator domain which is varying as open subset in the spatial domain for a multi-dimensional heat equation. Both time optimal control and norm optimal control problems are considered. The reason behind combining these two problems together is that these two problems are actually equivalent: The energy to be used to drive the system to target set in minimal time interval is actually the minimal energy of driving the system to target set in this minimal time interval, and visa versa. It is shown that both optimal control and optimal cost are continuous with respect to open controlled actuator domain under the Lebesgue measure.     

Adaptive error feedback regulation problem for 1D wave equation

By using the adaptive control approach, we solve the error feedback regulator problem for the one‐dimensional wave equation with a general harmonic disturbance anticollocated with control and with two types of disturbed measurements, ie, one collocated with control and the other anti‐collocated with control. Different from the classical error feedback regulator design, which is based on the internal mode principle, we give the adaptive servomechanism design for the system by making use of the measured tracking error (and its time derivative) and the estimation mechanism for the parameters of the disturbance and of the unknown reference. Constructing auxiliary systems and observer and applying the backstepping method for infinite‐dimensional system play important roles in the design. The control objective, which is to regulate the tracking error to zero and to keep the states bounded, is achieved.     

A new unconditionally stable ADI compact scheme for the two-space-dimensional linear hyperbolic equation

We formulate a new alternating direction implicit compact scheme of O(τ²+h ⁴) for the linear hyperbolic equation u _tt +2α u _t +β² u=u _xx +u _yy +f(x, y, t), 0<x, y<1, 0<t≤T, subject to appropriate initial and Dirichlet boundary conditions, where α>0 and β≥0 are real numbers. In this article, we show the method is unconditionally stable by the Von Neumann method. At last, numerical demonstrations are given to illustrate our result.     

A quasi-reversibility regularization method for the Cauchy problem of the Helmholtz equation

In this paper, a Cauchy problem for the Helmholtz equation is considered. It is known that such a problem is severely ill-posed, i.e. the solution does not depend continuously on the given Cauchy data. We propose a quasi-reversibility regularization method to solve it. Convergence estimates are established under two different a priori assumptions for an exact solution. Numerical results obtained by two different schemes show that our proposed methods work well.     

A mollification method for a Cauchy problem for the Helmholtz equation

The Cauchy problem for the Helmholtz equation is considered. This problem is severely ill-posed, that is, the solution does not depend continuously on the data. To solve the problem numerically a mollification method is proposed. Convergence on error estimates between the exact solution and its approximation are obtained. Some numerical examples are given to show that the method works effectively.     

Domain decomposition algorithm based on the group explicit formula for the heat equation

A finite difference domain decomposition algorithm (DDA) for solving the heat equation in parallel is presented. In this procedure, interface values between subdomains are calculated by the group explicit formula, whereas interior values of subdomains are determined by the classical implicit scheme. The stability and convergence for this DDA are proved. The stability bound of the procedure is derived to be eight times that of the classical explicit scheme. Though the truncation error at the interface is O(τ?+?h), L ²-error is proved to be O(τ?+?h ²). Numerical examples confirm the second-order convergence and indicate that the stability condition is sharp. A comparison of the numerical errors of this procedure with other known methods is also included.     

Numerical study of a nonlinear heat equation for plasma physics

This paper is devoted to the numerical approximation of a nonlinear parabolic balance equation, which describes the heat evolution of a magnetically confined plasma in the edge region of a tokamak. The nonlinearity implies some numerical difficulties, in particular for the long-time behaviour approximation, when solved with standard methods. An efficient numerical scheme is presented in this paper, based on a combination of a directional splitting scheme and the implicit–explicit scheme introduced in Filbet and Jin [A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys. 229 (2010), pp. 7625–7648].     

A new filtering method for the Cauchy problem of the Laplace equation

In the present paper, we consider a Cauchy problem for the Laplace equation in a rectangle domain. A new filtering method is presented for approximating the solution of this problem, and the Hölder-type error estimates are obtained by the different parameter choice rules. Numerical illustration shows that the proposed method works effectively.     

Reconstruction of the corrosion boundary for the Laplace equation by using a boundary collocation method

In this paper, we consider the identification of a corrosion boundary for the two-dimensional Laplace equation. A boundary collocation method is proposed for determining the unknown portion of the boundary from the Cauchy data on a part of the boundary. Since the resulting matrix equation is badly ill-conditioned, a regularized solution is obtained by employing the Tikhonov regularization technique, while the regularization parameter is provided by the generalized cross-validation criterion. Numerical examples show that the proposed method is reasonable and feasible.     

Uniqueness and numerical scheme for the Robin coefficient identification of the time-fractional diffusion equation

We study an inverse problem of determining the Robin coefficient of fractional diffusion equation from a nonlocal boundary condition. Based on the property of Caputo fractional derivative, the uniqueness is proved. The numerical schemes for the direct problem and the inverse problem are developed. Three examples are given to show the effectiveness of the presented methods.