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Abstract: The objective of this paper is to present a numerical study of a class of boundary value problems of singularly perturbed differential difference equations (SPDDE) which arise in computational neuroscience in particular in the modeling of neuronal variability. The mathematical modeling of the determination of the expected time for the generation of action potential in the nerve cells by random synaptic inputs in dendrites includes a general boundary-value problem for singularly perturbed differential difference equation with shifts. The problem considered in this paper exhibit turning point behavior which add to the complexity in the construction of numerical approximation to the solution of the problem as well as in obtaining theoretical estimates on the solution. Exponentially fitted finite difference scheme based on Il’in-Allen-Southwell fitting is used on a specially designed mesh. Some numerical examples are given to validate convergence and computational efficiency of the proposed numerical scheme. Effect of the shifts on the layer structure is illuminated for the considered examples. [Copyright &y& Elsevier]

In this paper, the following inverse heat conduction problem: ut = uxx, x≥0, t≥0, u(x,0) = 0, x≥0 u(1,t) = g(t), t≥0, u&z.sfnc;x→∞ is bounded, is considered again. This problem is severely ill-posed: its solution (if it exists) does not depend continuously on the data; a small perturbation in the data may cause a dramatically large error in the solution for 0 < x < 1. In this paper, a new wavelet regularization method for this problem is given. Moreover, we can easily find the regularization parameter J such that some sharp stable estimates between the exact solution and the approximate one in Hr(R)-norm meaning is given. [Copyright &y& Elsevier]

Solving elliptic equations with interfaces has wide applications in engineering and science. The real world problems are mostly in three dimensions, while an efficient and accurate solver is a challenge. Some existing methods that work well in two dimensions are too complicated to be generalized to three dimensions. Although traditional finite element method using body-fitted grid is well-established, the expensive cost of mesh generation is an issue. In this paper, an efficient non-traditional finite element method with non-body-fitted grids is proposed to solve elliptic interface problems. The special cases when the interface cuts though grid points are handled carefully, rather than perturbing the cutting point away to apply the method for general case. Both Dirichlet and Neumann boundary conditions are considered. Numerical experiments show that this method is approximately second order accurate in the L ∞ norm and L 2 norm for piecewise smooth solutions. The large sparse matrix for our linear system also has nice structure and properties. [ABSTRACT FROM AUTHOR]

Abstract: This paper is an elementary introduction to the concepts of the homotopy perturbation method. Particular attention is paid to giving an intuitive grasp for the solution procedure throughout the paper. [Copyright &y& Elsevier]

Abstract: This paper addresses the motion tracking control for a class of flexible-joint robotic manipulators actuated by brushed direct current motors. This class of electrically driven flexible-joint robots is perturbed by plant uncertainties and external disturbances. Adaptive neural network systems are employed to approximate the behaviors of uncertain mechanical and electrical dynamics. A reduced-order observer is constructed to estimate the velocity signals. Only the measurements of link position and armature current are required for feedback. Consequently, an adaptive neural network-based dynamic feedback tracking controller without velocity measurements is developed such that all the states and signals of the closed-loop system are bounded and the trajectory tracking errors can be made as small as possible. Finally, simulation results are presented to demonstrate the effectiveness of the proposed control algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a regular perturbation method to obtain approximate analytic solutions of exterior and interior Dirichlet problems for Laplace’s equation in planar domains. This method, starting from a geometrical perturbation of these planar domains, reduces our problems to a family of classical Dirichlet problems for Laplace’s equation in a circle. Numerical examples are given and comparisons are made with the solutions obtained by other approximation methods. [Copyright &y& Elsevier]

Abstract: This paper deals with the construction of approximate series solutions of random nonlinear diffusion equations where nonlinearity is considered by means of a frank small parameter and uncertainty is introduced through white noise in the forcing term. For the simpler but important case in which the diffusion coefficient is time independent, we provide a Gaussian approximation of the solution stochastic process by taking advantage of the Wiener–Hermite expansion together with the perturbation method. In addition, approximations of the main statistical functions associated with a solution, such as the mean and variance, are computed. Numerical values of these functions are compared with respect to those obtained by applying the Runge–Kutta second-order stochastic scheme as an illustrative example. [Copyright &y& Elsevier]

Abstract: The present paper is concerned with the approximate analytic series solution of the nonlinear two-point second-order singularly perturbed boundary value problems. In place of the traditional numerical, perturbation or asymptotic methods, a homotopy technique is employed. It is shown that proper choices of an auxiliary linear operator and also an initial approximation during the implementation of the homotopy analysis method (HAM) can yield uniformly valid and accurate solutions. The fast convergence of the method is ensured by the optimal convergence control parameter obtained through the absolute residual error concept. To demonstrate the favor of the HAM over the traditional finite-difference techniques several nonlinear problems have been solved and compared. [Copyright &y& Elsevier]

Abstract: In this paper, axisymmetric flow of two-dimensional incompressible fluids is studied. The Optimal Homotopy Asymptotic Method (OHAM) is applied to derive a solution of the reduced fourth-order nonlinear boundary value problem. For comparison, the same problem is also solved by the Perturbation Method (PM), the Homotopy Perturbation Method (HPM) and the Homotopy Analysis Method (HAM) . OHAM is parameter free and provides better accuracy at lower order of approximation. Moreover we can easily adjust and control the convergence region. As a result it is concluded that the new technique, OHAM, shows fast convergence, simplicity of application and efficiency. [Copyright &y& Elsevier]

Abstract: In this paper, two methods for solving a nonlinear differential equation known as He’s variational iteration and homotopy perturbation methods are applied to derive the approximate kink-type soliton solutions for a new (2+1)-dimensional simplified generalized Broer–Kaup system. Furthermore, the solutions obtained are compared with the corresponding exact solution to show the applicability, accuracy and, finally, efficiency of the present methods in solving a large class of nonlinear physics and engineering problems without substantial noisy sensitivity to the nonlinear terms or any restrictive assumptions or transformations that may change the physical behavior of the problems. [Copyright &y& Elsevier]

Abstract: Continuum hypothesis on nanoscales is invalid, and a differential–difference model is considered as an alternative approach to describing discontinued problems. This paper applies the homotopy perturbation method to a nonlinear differential–difference equation arising in nanotechnology. Comparison of the approximate solution with the exact one reveals that the method is very effective. [Copyright &y& Elsevier]

Abstract: This paper deals with initial value problems for the sine–Gordon equation by using the modified homotopy perturbation method. The advantage of this method that is its ability to provide the analytical or approximate solutions to linear and nonlinear equations makes it reliable for solving the sine–Gordon equation. The numerical results are presented to show the efficiency of this method. [Copyright &y& Elsevier]

Abstract: In this paper, nonlinear oscillators under mixed quadratic and cubic nonlinearities with stochastic inputs are considered. Different methods are used to obtain second order approximations, namely; the Wiener–Hermite and perturbation (WHEP) technique and the homotopy perturbation method (HPM). Some statistical moments are computed for the different methods using mathematica 5. Comparisons are illustrated through figures for different case-studies. [Copyright &y& Elsevier]

Abstract: In this paper, we apply a new modified homotopy perturbation method to find exact solutions for nonlinear integral equations. A reliable modification of the homotopy perturbation method is proposed, and the modified method is employed to solve the nonlinear integral equations; the results are compared with those obtained by the original homotopy perturbation method. Some examples are given to illustrate the ability and reliability of the modified method. The results reveal that the modified method is very simple and effective. [Copyright &y& Elsevier]

Abstract: In this paper we construct three new test problems, called Models A, B and C, whose solutions have two-dimensional boundary layers. Approximate analytic solutions are found for these problems, which converge rapidly as the number of terms in their expansion increases. The approximations are valid for in practical computations. Surprisingly, the algorithm for Model A can be carried out even for . Model C has a simple exact solution. These three new accurate and approximate analytic solutions with two-dimensional boundary layers may be more useful for testing numerical methods than those in [Z.C. Li, H.Y. Hu, C.H. Hsu, S. Wang, Particular solutions of singularly perturbed partial differential equations with constant coefficients in rectangular domains, I. Convergence analysis, J. Comput. Appl. Math. 166 (2004) 181–208] in the sense that the series solutions from the former converge much faster than those of the latter when is small. [Copyright &y& Elsevier]

Abstract: A radial basis function approximation is typically a linear combination of shifts of a radially symmetric function, possibly augmented by a polynomial of suitable degree, that is, it takes the formIn the mid 1980s, Micchelli, building on pioneering work of Schoenberg in the 1930s and 1940s, provided simple sufficient conditions on ƒ that imply radial basis functions can interpolate scattered data. However, when the data density varies locally, several authors, such as Hon and Kansa [1], have suggested scaling the translates. In other words, it can be advantageous to replace the Euclidean norm by some more general distance functional Δ(·,·), ), that isThis distance functional A need not be a metric, but we shall require that Δ be symmetric and satisfy Δ (χ, χ) = 0, for all χ ∈ ℝ d . Unfortunately, the Micchelli-Schoenberg theory does not obviously apply in this more general setting, but some papers have observed that interpolation is well defined if the distance functional is a sufficiently small perturbation of the Euclidean norm. However, in this study we follow a different approach which returns to the roots of Schoenberg''s work. Specifically, we use Schoenberg''s classification of Euclidean distance matrices to provide a simple technique which, given a suggested distance functional Δ, calculates a perturbed distance functional Δ for which the underlying interpolation matrix is invertible, when the function θ is strictly positive definite (i.e., a Mercer kernel) or strictly conditionally positive (or negative) definite of order one. As a simple by-product of this method, we can also apply the Narcowich-Ward [2] norm estimate results easily, since the minimum distance between points is now under our control via Δ. [Copyright &y& Elsevier]

This paper deals with simulations of a phase-change problem related to an extended heat conduction problem. ''Under physically reasonable hypotheses, effects of several values of the relaxation times are considered and the evolution of the moving interface is discussed. Moreover, a contribution to the analysis of the singular perturbation problem that arises when the small parameters that characterize the model tend to zero, is given by means of numerical mathematics. [Copyright &y& Elsevier]