Determination of a control parameter in a one-dimensional parabolic equation using the moving least-square approximation

In this paper the approximation of moving least-square (MLS) is used for finding the solution of a one-dimensional parabolic inverse problem with source control parameter. Comparing with other numerical methods based on meshes such as finite difference method, finite element method and boundary element method, etc. the MLS approximation has merits of simpler numerical procedures, lower computation cost and arbitrary nodes. The result of a numerical example is presented.     

New generalized method to construct new non-travelling wave solutions and travelling wave solutions of K–D equations

With the aid of computerized symbolic computation, we obtain new types of general solution of a first-order nonlinear ordinary differential equation with six degrees of freedom and devise a new generalized method and its algorithm, which can be used to construct more new exact solutions of general nonlinear differential equations. The (2+1)-dimensional K–D equation is chosen to illustrate our algorithm such that more families of new exact solutions are obtained, which contain non-travelling wave solutions and travelling wave solutions.     

Image processing applications via time-multiplexing cellular neural network simulator with numerical integration algorithms

A novel approach to simulate cellular neural networks (CNN) is presented in this paper. The approach, time-multiplexing simulation, is prompted by the need to simulate hardware models and test hardware implementations of CNN. For practical applications, due to hardware limitations, it is impossible to have a one-to-one mapping between the CNN hardware processors and all the pixels of the image. This simulator provides a solution by processing the input image block by block, with the number of pixels in a block being the same as the number of CNN processors in the hardware. The algorithm for implementing this simulator is presented along with popular numerical integration algorithms. Some simulation results and comparisons are also presented.     

Multi-symplectic integration of coupled non-linear Schrödinger system with soliton solutions

Systems of coupled non-linear Schrödinger equations with soliton solutions are integrated using the six-point scheme which is equivalent to the multi-symplectic Preissman scheme. The numerical dispersion relations are studied for the linearized equation. Numerical results for elastic and inelastic soliton collisions are presented. Numerical experiments confirm the excellent conservation of energy, momentum and norm in long-term computations and their relations to the qualitative behaviour of the soliton solutions.     

Chebyshev polynomial solution of the system of Cauchy-type singular integral equations of the first kind

This paper presents a Chebyshev series method for the numerical solutions of system of the first kind Cauchy type singular integral equation (SIE). The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density functions. It is shown that the numerical solution of system of characteristic SIEs is identical to the exact solution when the force functions are cubic functions.     

A convergence analysis of the Adomian decomposition method for an abstract Cauchy problem of a system of first-order nonlinear differential equations

In this article, we study some fundamental results concerning the convergence of the Adomian decomposition method (ADM) for an abstract Cauchy problem of a system of first-order nonlinear differential equations. Under certain conditions, we obtain upper estimates for the norm of solutions of this system. We also obtain results about the error estimates for the approximate solutions by the ADM and discuss their applications.     

A linearly semi-implicit compact scheme for the Burgers–Huxley equation

In this paper, a linearly semi-implicit compact scheme is developed for the Burgers–Huxley equation. The equation is decomposed into two subproblems, i.e. a Burgers equation and a nonlinear ODE, by the operator splitting technique. The Burgers equation is solved by a linearly self-starting compact scheme which is fourth-order accurate in space and second-order accurate in time. The nonlinear ODE is discretized by a third-order semi-implicit Runge–Kutta method, which possesses good numerical stability with low computational cost. The numerical experiments show that the scheme provides the expected convergence order. Finally, several experiments are conducted to simulate the solutions of the Burgers–Huxley equation to validate our numerical method.     

Necessary and sufficient conditions for the existence of positive definite solutions of the matrix equation X+A T X −2 A=I

Necessary and sufficient conditions for the matrix equation X+A ^T X ^?2 A=I to have a real symmetric positive definite solution X are derived. Based on these conditions, some properties of the matrix A as well as relations between the solution X and A are derived.     

Convergence and stability of numerical methods with variable step size for stochastic pantograph differential equations

The stochastic pantograph equations (SPEs) are very special stochastic delay differential equations (SDDEs) with unbounded memory. When the numerical methods with a constant step size are applied to the pantograph equations, the most difficult problem is the limited computer memory. In this paper, we construct methods with variable step size to solve SPEs. The analysis is motivated by the example of a mean-square stable linear SPE for which the Euler–Maruyama (EM) method with variable step size fails to reproduce this behaviour for any nonzero timestep. Then we consider the Backward Euler (BE) method with variable step size and develop the fundamental numerical analysis concerning its strong convergence and mean-square linear stability. It is proved that the numerical solutions produced by the BE method with variable step size converge to the exact solution under the local Lipschitz condition and the Bounded condition. Furthermore, the order of convergence p=½ is given under the Lipschitz condition. The result of the mean-square linear stability is given. Some illustrative numerical examples are presented to demonstrate the order of strong convergence and the mean-square linear stability of the BE method.