Spectral collocation methods for nonlinear weakly singular Volterra integro‐differential equations

In this paper, a spectral collocation approximation is proposed for neutral and nonlinear weakly singular Volterra integro‐differential equations (VIDEs) with non‐smooth solutions. We use some suitable variable transformations to change the original equation into a new equation, so that the solution of the resulting equation possesses better regularity, and the the Jacobi orthogonal polynomial theory can be applied conveniently. Under reasonable assumptions on the nonlinearity, we carry out a rigorous error analysis in L^∞ norm and weighted L² norm. To perform the numerical simulations, some test examples (linear and nonlinear) are considered with nonsmooth solutions, and numerical results are presented. Further more, the comparative study of the proposed methods with some existing numerical methods is provided.     

Discrete time waveform relaxation method for stochastic delay differential equations

We propose in this paper the discrete time waveform relaxation method for the stochastic delay differential equations and prove that it is convergent in the mean square sense. In addition, the results obtained are supported by numerical experiments.     

Quadratic spline collocation methods for elliptic partial differential equations

We consider Quadratic Spline Collocation (QSC) methods for linear second order elliptic Partial Differential Equations (PDEs). The standard formulation of these methods leads to non-optimal approximations. In order to derive optimal QSC approximations, high order perturbations of the PDE problem are generated. These perturbations can be applied either to the PDE problem operators or to the right sides, thus leading to two different formulations of optimal QSC methods. The convergence properties of the QSC methods are studied. OptimalO(h ^3–j ) global error estimates for thejth partial derivative are obtained for a certain class of problems. Moreover,O(h ^4–j ) error bounds for thejth partial derivative are obtained at certain sets of points. Results from numerical experiments verify the theoretical behaviour of the QSC methods. Performance results also show that the QSC methods are very effective from the computational point of view. They have been implemented efficiently on parallel machines.This research was supported in part by David Ross Foundation (U.S.A) and NSERC (Natural Sciences and Engineering Research Council of Canada).     

On collocation methods for delay differential and Volterra integral equations with proportional delay

To compute long term integrations for the pantograph differential equation with proportional delay qt, 0 < q ⩽ 1: y′(t) = ay(t) + by(qt) + f(t), y(0) = y ₀, we offer two kinds of numerical methods using special mesh distributions, that is, a rational approximant with ‘quasi-uniform meshes’ (see E. Ishiwata and Y. Muroya [Appl. Math. Comput., 2007, 187: 741-747]) and a Gauss collocation method with ‘quasi-constrained meshes’. If we apply these meshes to rational approximant and Gauss collocation method, respectively, then we obtain useful numerical methods of order p ^* = 2m for computing long term integrations. Numerical investigations for these methods are also presented.      

A parareal waveform relaxation algorithm for semi-linear parabolic partial differential equations

We report a new parallel iterative algorithm for semi-linear parabolic partial differential equations (PDEs) by combining a kind of waveform relaxation (WR) techniques into the classical parareal algorithm. The parallelism can be simultaneously exploited by WR and parareal in different directions. We provide sharp error estimations for the new algorithm on bounded time domain and on unbounded time domain, respectively. The iterations of the parareal and the WR are balanced to optimize the performance of the algorithm. Furthermore, the speedup and the parallel efficiency of the new approach are analyzed. Numerical experiments are carried out to verify the effectiveness of the theoretic work.     

Stability of collocation methods for delay differential equations with vanishing delays

We analyze the asymptotic stability of collocation solutions in spaces of globally continuous piecewise polynomials on uniform meshes for linear delay differential equations with vanishing proportional delay qt (0<q<1) (pantograph DDEs). It is shown that if the collocation points are such that the analogous collocation solution for ODEs is A-stable, then this asymptotic behaviour is inherited by the collocation solution for the pantograph DDE.     

Spline collocation methods for nonlinear Volterra integral equations with unknown delay

In this paper we introduce and study polynomial spline collocation methods for systems of Volterra integral equations with unknown lower integral limit arising in mathematical economics. Their discretization leads to implicit Runge-Kutta-type methods. The global convergence and local superconvergence properties of these methods are proved, and the theory is illustrated by a numerical example arising in the application of such equations in certain mathematical models of liquidation.     

Unconditionally stable difference methods for delay partial differential equations

This paper is concerned with the numerical solution of parabolic partial differential equations with time-delay. We focus in particular on the delay dependent stability analysis of difference methods that use a non-constrained mesh, i.e., the time step-size is not required to be a submultiple of the delay. We prove that the fully discrete system unconditionally preserves the delay dependent asymptotic stability of the linear test problem under consideration, when the following discretization is used: a variant of the classical second-order central differences to approximate the diffusion operator, a linear interpolation to approximate the delay argument, and, finally, the trapezoidal rule or the second-order backward differentiation formula to discretize the time derivative. We end the paper with some numerical experiments that confirm the theoretical results.     

Implicit ad hoc methods for nonlinear partial differential equations

Several special methods including implicit separation of variables, explicit and implicit generalized traveling waves are introduced and employed to obtain solutions for nonlinear equations. Certain nonlinear wave propagation problems are shown to yield to implicit separation while generalized traveling wave concepts are applied in diffusion, fluid mechanics and wave propagation.     

Nonstationary monotone iterative methods for nonlinear partial differential equations

A simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations. With the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system. Picard, Gauss–Seidel, and Jacobi monotone iterative methods are presented and analyzed for the adaptive solutions. The adaptive meshes are generated by the 1-irregular mesh refinement scheme which together with the M-matrix of the finite element stiffness matrix lead to existence–uniqueness–comparison theorems with simple upper and lower solutions as initial iterates. Some numerical examples, including a test problem with known analytical solution, are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties. Numerical results of simulations on a MOSFET with the gate length down to 34 nm are also given.     

Flow invariance for solutions to nonlinear nonautonomous partial differential delay equations

We investigate the problem of existence and flow invariance of mild solutions to nonautonomous partial differential delay equations , t?s, us=φ, where B(t) is a family of nonlinear multivalued, α-accretive operators with D(B(t)) possibly depending on t, and the operators F(t,.) being defined—and Lipschitz continuous—possibly only on “thin” subsets of the initial history space E. The results are applied to population dynamics models. We also study the asymptotic behavior of solutions to this equation. Our analysis will be based on the evolution operator associated to the equation in the initial history space E.     

Pseudospectral collocation methods for fourth-order differential equations

A multi-domain pseudospectral collocation scheme for the approximationof linear fourth-order differential equations in one and twodimensions is presented. A complete analysis of the scheme isprovided and error estimates are proved for the one-dimensionalproblem. An efficient preconditioner based on a low-order finite-differenceapproximation to the same differential operator is proposed.The extension of the method to the biharmonic equation in twodimensions is discussed and results are presented for a problemdefined in a non-rectangular domain. Present address: Department of Mathematics, Tarbiat ModarresUniversity, Tehran, iran     

Hermite collocation solution of partial differential equations via preconditioned Krylov methods

We are concerned with the numerical solution of partial differential equations (PDEs) in two spatial dimensions discretized via Hermite collocation. To efficiently solve the resulting systems of linear algebraic equations, we choose a Krylov subspace method. We implement two such methods: Bi‐CGSTAB [1] and GMRES [2]. In addition, we utilize two different preconditioners: one based on the Gauss–Seidel method with a block red‐black ordering (RBGS); the other based upon a block incomplete LU factorization (ILU). Our results suggest that, at least in the context of Hermite collocation, the RBGS preconditioner is superior to the ILU preconditioner and that the Bi‐CGSTAB method is superior to GMRES. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:120–136, 2001     

Numerical methods for nonlinear partial differential equations of fractional order

In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when applied to partial differential equations of fractional order.     

Iterative methods for solving inverse problems for nonlinear partial differential equations

Inverse coefficient problems are considered for the mathematical models of sorption dynamics and heat conduction. Iterative methods proposed for solving these inverse problems transform a supplementary condition into an integral relationship containing the unknown coefficient. Combined with the original boundary-value problem, this integral relationship makes it possible to construct an iterative process. A priori representation of the unknown nonlinear coefficients in parametric form is not required. Results of computational experiments are reported.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 142–149.     

Stability and error analysis of one-leg methods for nonlinear delay differential equations

This paper is concerned with the numerical solution of delay differential equations (DDEs). We focus on the stability behaviour and error analysis of one-leg methods with respect to nonlinear DDEs. The new concepts of GR-stability, GAR-stability and weak GAR-stability are introduced. It is proved that a strongly A-stable one-leg method with linear interpolation is GAR-stable, and that an A-stable one-leg method with linear interpolation is GR-stable, weakly GAR-stable and D-convergent of order s, if it is consistent of order s in the classical sense.     

Existence and uniqueness of solutions for nonlinear impulsive partial differential equations with delay

In this paper, a system of nonlinear impulsive partial differential equations with delay is investigated by the method of upper–lower solutions. The existence-uniqueness result of this system and the comparison principle of the corresponding equations are obtained. An application is given to some model problems in ecology.     

Legendre–Gauss–Lobatto spectral collocation method for nonlinear delay differential equations

A Legendre–Gauss–Lobatto spectral collocation method is introduced for the numerical solutions of a class of nonlinear delay differential equations. An efficient algorithm is designed for the single‐step scheme and applied to the multiple‐domain case. As a theoretical result, we obtain a general convergence theorem for the single‐step case. Numerical results show that the suggested algorithm enjoys high‐order accuracy both in time and in the delayed argument and can be implemented in a robust and efficient manner. Copyright © 2013 John Wiley & Sons, Ltd.     

A new stable collocation method for solving a class of nonlinear fractional delay differential equations

Numerical Algorithms - In this paper, a stable collocation method for solving the nonlinear fractional delay differential equations is proposed by constructing a new set of multiscale orthonormal...