The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations

Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear Volterra integral equations with algebraic or logarithmic singularities in their kernels.

     

Rational Chebyshev collocation method for solving higher‐order linear ordinary differential equations

A collocation method to find an approximate solution of higher‐order linear ordinary differential equation with variable coefficients under the mixed conditions is proposed. This method is based on the rational Chebyshev (RC) Tau method and Taylor‐Chebyshev collocation methods. The solution is obtained in terms of RC functions. Also, illustrative examples are included to demonstrate the validity and applicability of the technique, and performed on the computer using a program written in maple9. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1130–1142, 2011     

二阶线性常微分方程的两点边值问题的泰勒展开式解法

本文用泰勒展开公式求解二阶线性常微分方程的两点边值问题.首先将两点边值问题化为一个F redho lm积分方程,进一步通过泰勒展开公式化F redho lm积分方程为线性方程组,利用G ramm er法则可求得问题的近似解.     

A survey of numerical techniques for solving singularly perturbed ordinary differential equations

This survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory. Starting from Prandtl's work a large amount of work has been done in the area of singular perturbations. This paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984–2000. The work done in this area during the period 1905–1984 has already been surveyed by the first author of this paper, see [Appl. Math. Comput. 30 (1989) 223] for details. Due to the space constraints we have covered only singularly perturbed one-dimensional problems.     

Existence and convergence of solutions of singular boundary value problems for second order ordinary differential equations and applications

In this paper,we consider the singular boundary value problems for second order quasilinear ordinary differential equations and prove existence and convergence on second order perturbation terms.The result is applied to solve the Riemann problem for 2×2 hyperbolic conservation laws,which is a partial differential equation arising in applied mathematical area.     

How to solve the polynomial ordinary differential equations

This paper discussed how to solve the polynomial ordinary differential equations. At first, we construct the theory of the linear equations about the unknown one variable functions with constant coefficients. Secondly, we use this theory to convert the polynomial ordinary differential equations into the simultaneous first order linear ordinary differential equations with constant coefficients and quadratic equations. Thirdly, we work out the general solution of the polynomial ordinary differential equations which is no longer concerned with the differential. Finally, we discuss the necessary and sufficient condition of the existence of the solution.     

A new HPM for ordinary differential equations

In this paper, we introduce a new version of the homotopy perturbation method (NHPM) that efficiently solves linear and non‐linear ordinary differential equations. Several examples, including Euler‐Lagrange, Bernoulli and Ricatti differential equations, are given to demonstrate the efficiency of the new method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Moving collocation methods for time fractional differential equations and simulation of blowup

A moving collocation method is proposed and implemented to solve time fractional differential equations. The method is derived by writing the fractional differential equation into a form of time difference equation. The method is stable and has a third-order convergence in space and first-order convergence in time for either linear or nonlinear equations. In addition, the method is used to simulate the blowup in the nonlinear equations.     

Two-step almost collocation methods for ordinary differential equations

A new class of two-step Runge-Kutta methods for the numerical solution of ordinary differential equations is proposed. These methods are obtained using the collocation approach by relaxing some of the collocation conditions to obtain methods with desirable stability properties. Local error estimation for these methods is also discussed.     

Solutions of 2nd-order linear differential equations subject to Dirichlet boundary conditions in a Bernstein polynomial basis

An algorithm for approximating solutions to 2nd-order linear differential equations with polynomial coefficients in B-polynomials (Bernstein polynomial basis) subject to Dirichlet conditions is introduced. The algorithm expands the desired solution in terms of B-polynomials over a closed interval [0, 1] and then makes use of the orthonormal relation of B-polynomials with its dual basis to determine the expansion coefficients to construct a solution. Matrix formulation is used throughout the entire procedure. However, accuracy and efficiency are dependent on the size of the set of B-polynomials, and the procedure is much simpler compared to orthogonal polynomials for solving differential equations. The current procedure is implemented to solve five linear equations and one first-order nonlinear equation, and excellent agreement is found between the exact and approximate solutions. In addition, the algorithm improves the accuracy and efficiency of the traditional methods for solving differential equations that rely on much more complicated numerical techniques. This procedure has great potential to be implemented in more complex systems where there are no exact solutions available except approximations.     

Adaptive partitioning techniques for ordinary differential equations

Many stiff systems of ordinary differential equations (ODEs) modeling practical problems can be partitioned into loosely coupled subsystems. In this paper the objective of the partitioning is to permit the numerical integration of one time step to be performed as the solution of a sequence of small subproblems. This reduces the computational complexity compared to solving one large system and permits efficient parallel execution under appropriate conditions. The subsystems are integrated using methods based on low order backward differentiation formulas.This paper presents an adaptive partitioning algorithm based on a classical graph algorithm and techniques for the efficient evaluation of the error introduced by the partitioning.The power of the adaptive partitioning algorithm is demonstrated by a real world example, a variable step-size integration algorithm which solves a system of ODEs originating from chemical reaction kinetics. The computational savings are substantial. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65L06, 65Y05     

Uniqueness and existence results for ordinary differential equations

We establish some uniqueness and existence results for first-order ordinary differential equations with constant-signed discontinuous nonlinear parts. Several examples are given to illustrate the applicability of our work.     

Global bifurcation theorem for a class of boundary conditions for ordinary differential equations of second order

In this paper we deal with boundary value problems where l : C¹([a, b], ?^k) → ?^k × ?^k is continuous, μ ≤ 0 and φ is a Caratheodory map. We define the class S of maps l, for which a global bifurcation theorem holds for the problem (+), with φ(t, x, y, λ) = λ(|x₁|, …, |x^k|) + o(|x| + |y|). We show that the class S contains Sturm‐Liouville boundary conditions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Families of methods for ordinary differential equations based on trigonometric polynomials

We consider the construction of methods based on trigonometric polynomials for the initial value problems whose solutions are known to be periodic. It is assumed that the frequency w can be estimated in advance. The resulting methods depend on a parameter ν = hw, where h is the step size, and reduce to classical multistep methods if ν → 0. Gautschi [4] developed Adams and Störmer type methods. In our paper we construct Nyström's and Milne-Simpson's type methods. Numerical experiments show that these methods are not sensitive to changes in w, but require the Jacobian matrix to have purely imaginary eigenvalues.     

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.      

An efficient spectral method for ordinary differential equations with rational function coefficients

We present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients. We employ expansions in terms of a large class of orthogonal polynomial families, including all the classical orthogonal polynomials. These families obey a simple 3-term recurrence relation for differentiation, which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse. The inverse is an integration operator for the family, and it is simply the tridiagonal coefficient matrix for the recurrence. Since in these families convolution operators (i.e., matrix representations of multiplication by a function) are banded for polynomials, we are able to obtain a banded representation for linear differential operators with rational coefficients. This leads to a method of solution of initial or boundary value problems that, besides having an operation count that scales linearly with the order of truncation , is computationally well conditioned. Among the applications considered is the use of rational maps for the resolution of sharp interior layers.