Numerical solution of the nonlinear Klein-Gordon equation

A numerical method is developed to solve the nonlinear one-dimensional Klein-Gordon equation by using the cubic B-spline collocation method on the uniform mesh points. We solve the problem for both Dirichlet and Neumann boundary conditions. The convergence and stability of the method are proved. The method is applied on some test examples, and the numerical results have been compared with the exact solutions. The L₂, L_∞ and Root-Mean-Square errors (RMS) in the solutions show the efficiency of the method computationally.     

Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations

In this paper, we state and prove a new formula expressing explicitly the derivatives of shifted Chebyshev polynomials of any degree and for any fractional-order in terms of shifted Chebyshev polynomials themselves. We develop also a direct solution technique for solving the linear multi-order fractional differential equations (FDEs) with constant coefficients using a spectral tau method. The spatial approximation with its fractional-order derivatives (described in the Caputo sense) are based on shifted Chebyshev polynomials T_L,n(x) with x ∈ (0, L), L > 0 and n is the polynomial degree. We presented a shifted Chebyshev collocation method with shifted Chebyshev–Gauss points used as collocation nodes for solving nonlinear multi-order fractional initial value problems. Several numerical examples are considered aiming to demonstrate the validity and applicability of the proposed techniques and to compare with the existing results.     

Pressure projection stabilized finite element method for Stokes problem with nonlinear slip boundary conditions

In this paper, we consider the pressure projection stabilized finite element method for the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality problem of the second kind with the Stokes operator. The H¹ and L² error estimates for the velocity and the L² error estimate for the pressure are obtained. Finally, the numerical results are displayed to verify the theoretical analysis.     

Robust L2–L∞ filtering for switched systems under asynchronous switching

This paper investigates the problem of robust L₂–L_∞ filtering for continuous-time switched systems under asynchronous switching. When there exists asynchronous switching between the filter and the system, based on the average dwell time approach, sufficient conditions for the existence of a linear filter that guarantee the filtering error system to be exponentially stable with a prescribed weighted L₂–L_∞ performance for switched systems are derived, and filter parameters can be obtained by solving a set of matrix inequalities. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.     

A combined mixed finite element method and local discontinuous Galerkin method for miscible displacement problem in porous media

A combined method consisting of the mixed finite element method for flow and the local discontinuous Galerkin method for transport is introduced for the one-dimensional coupled system of incompressible miscible displacement problem. Optimal error estimates in L∞(0,T;L2) for concentration c,in L2(0,T;L2)for cxand L∞(0,T;L2) for velocity u are derived. The main technical difficulties in the analysis include the treatment of the inter-element jump terms which arise from the discontinuous nature of the numerical method,the nonlinearity,and the coupling of the models. Numerical experiments are performed to verify the theoretical results. Finally,we apply this method to the one-dimensional compressible miscible displacement problem and give the numerical experiments to confirm the efficiency of the scheme.     

A convergence analysis of nonlinear implicit iterative method for nonlinear ill-posed problems

Implicit iterative method acquires good effect in solving linear ill-posed problems. We have ever applied the idea of implicit iterative method to solve nonlinear ill-posed problems, under the restriction that α is appropriate large, we proved the monotonicity of iterative error and obtained the convergence and stability of iterative sequence, numerical results show that the implicit iterative method for nonlinear ill-posed problems is efficient. In this paper, we analyze the convergence and stability of the corresponding nonlinear implicit iterative method when α_k are determined by Hanke criterion.     

Robust L ∞ reliable control for impulsive switched nonlinear systems with state delay

This paper investigates the problem of robust L _∞ reliable control for a class of uncertain impulsive switched nonlinear systems with time-delay in the presence of actuator failure. Based on the dwell time approach, we firstly obtain a sufficient condition of exponential stability for the impulsive switched nonlinear system with time-delay, and L _∞ performance for the considered system is also analyzed. Then, based on above results, a state feedback controller, which guarantees the exponential stability with L _∞ performance of the corresponding closed-loop system, is constructed. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed design method.     

Taylor–Galerkin and Taylor-collocation methods for the numerical solutions of Burgers’ equation using B-splines

In this paper, for the numerical solution of Burgers’ equation, we give two B-spline finite element algorithms which involve a collocation method with cubic B-splines and a Galerkin method with quadratic B-splines. In time discretization of the equation, Taylor series expansion is used. In order to verify the stabilities of the purposed methods, von-Neumann stability analysis is employed. To see the accuracy of the methods, L₂ and L_∞ error norms are calculated and obtained results are compared with some earlier studies.     

Some fixed point theorems of the Schauder and the Krasnosel'skii type and application to nonlinear transport equations

In [J. Math. Phys. 37 (1996) 1336-1348] the existence of solutions to the boundary value problem (1.1)-(1.2) was analyzed for isotropic scattering kernels on L_p spaces for p∈(1,∞). Due to the lack of compactness in L₁ spaces, the problem remains open for p=1. The purpose of this work is to extend this analysis to the case p=1 for anisotropic scattering kernels. Our strategy consists in establishing new variants of the Schauder and the Krasnosel'skii fixed point theorems in general Banach spaces involving weakly compact operators. In L₁ context these theorems provide an adequate tool to attack the problem. Our analysis uses the specific properties of weakly compacts sets on L₁ spaces and the weak compactness results for one-dimensional transport equations established in [J. Math. Anal. Appl. 252 (2000) 767-789].     

A mesh-free numerical method for solution of the family of Kuramoto-Sivashinsky equations

In this paper, radial basis function (RBFs) based mesh-free method is implemented to find numerical solution of the Kuramoto-Sivashinsky equations. This approach has an edge over traditional methods such as finite-difference and finite element methods because it does not require a mesh to discretize the problem domain, and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method. The accuracy of the method is assessed in terms of the error norms L₂,L_∞, number of nodes in the domain of influence, free parameter, dependent parameter RBFs and time step length. Numerical experiments demonstrate accuracy and robustness of the method for solving a class of nonlinear partial differential equations.     

L 1 Spline Methods for Scattered Data Interpolation and Approximation

We generalize the L ₁ spline methods proposed in [4, 5] for scattered data interpolation and fitting using bivariate spline spaces of any degree d and any smoothness r (of course, r<d) over any triangulation. Some numerical experiments are presented to illustrate the better performance of the L ₁ spline methods as compared to the minimal energy method. We include some extensions for dealing with other surface design problems.     

Convex approximations in stochastic programming by semidefinite programming

The following question arises in stochastic programming: how can one approximate a noisy convex function with a convex quadratic function that is optimal in some sense. Using several approaches for constructing convex approximations we present some optimization models yielding convex quadratic regressions that are optimal approximations in L ₁, L _?? and L ₂ norm. Extensive numerical experiments to investigate the behavior of the proposed methods are also performed.     

Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions

We propose a method to construct numerical solutions of parabolic equations on the unit sphere. The time discretization uses Laplace transforms and quadrature. The spatial approximation of the solution employs radial basis functions restricted to the sphere. The method allows us to construct high accuracy numerical solutions in parallel. We establish L ₂ error estimates for smooth and nonsmooth initial data, and describe some numerical experiments.     

Fredholm-Volterra integral equation with singular kernel

The purpose of this paper is to obtain the solution of Fredholm-Volterra integral equation with singular kernel in the space L₂(?1, 1) × C(0,T), 0 ≤t ≤T < ∞, under certain conditions. The numerical method is used to solve the Fredholm integral equation of the second kind with weak singular kernel using the Toeplitz matrices. Also, the error estimate is computed and some numerical examples are computed using the MathCad package.     

<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> spline fits via sliding window process: continuous and discrete cases

The best L ₁ approximation of the Heaviside function and the best ? ₁ approximation of multiscale univariate datasets by a cubic spline have a Gibbs phenomenon near the discontinuity. We show by numerical experiments that the Gibbs phenomenon can be reduced by using L ₁ spline fits which are the best L ₁ approximations in an appropriate spline space obtained by the union of L ₁ interpolation splines. We prove here the existence of L ₁ spline fits for function approximation which has never previously been done to the best of our knowledge. A major disadvantage of this technique is an increased computation time. Thus, we propose a sliding window algorithm on seven nodes which is as efficient as the global method both for functions and datasets with abrupt changes of magnitude, but within a linear complexity on the number of spline nodes.     

The higher rank numerical range of nonnegative matrices

In this article the rank-k numerical range ∧ _k (A) of an entrywise nonnegative matrix A is investigated. Extending the notion of elements of maximum modulus in ∧ _k (A), we examine their location on the complex plane. Further, an application of this theory to ∧ _k (L(λ)) of a Perron polynomial L(λ) is elaborated via its companion matrix C _L .     

Blow-up solutions for a nonlinear wave equation with boundary damping and interior source

In this paper we consider the wave equation with nonlinear damping and source terms. We are interested in the interaction between the boundary damping −|y_t(L,t)|^m−1y_t(L,t) and the interior source |y(t)|^p−1y(t). We find a sufficient condition for obtaining the blow-up solution of the problem. Furthermore, we also obtain that the solution may blow up even if m≥p.     

L2 − L∞ filtering for Markovian jump systems with time-varying delays and partly unknown transition probabilities

This paper considers the L₂ − L_∞ filtering problem for Markovian jump systems. The systems under consideration involve time-varying delays, disturbance signal and partly unknown transition probabilities. The aim of this paper is to design a filter, which is suitable for exactly known and partly unknown transition probabilities, such that the filtering error system is stochastically stable and a prescribed L₂ − L_∞ disturbance attenuation level is guaranteed. By using the Lyapunov-Krasovskii functional, sufficient conditions are formulated in terms of linear matrix inequalities (LMIs). A numerical example is given to illustrate the effectiveness of the proposed main results. All these results are expected to be of use in the study of filter design for Markovian jump systems with partly unknown transition probabilities.     

On Tsertsvadze's difference scheme for the Kuramoto-Tsuzuki equation

In this paper, we have proved the second-order convergence in L_∞ norm of the Tsertsvadze's difference scheme for the Kuramoto-Tsuzuki equation. The existence, uniqueness and iterative algorithm are also discussed in detail. Furthermore, a L_∞ second order convergent linearized difference scheme is given for inhomogeneous equation. All results are obtained without any restrictions on the meshsizes. At last a numerical example is presented.