A family of third-order parallel splitting methods for parabolic partial differential equations

A family of numerical methods, based upon a new rational approximation to the matrix exponential function, is developed for solving parabolic partial differential equations. These methods are L-acceptable, third-order accurate in space and time, and do not require the use of complex arithmetic. In these methods second-order spatial derivatives are approximated by third-order finite-difference approximations.- Parallel algorithms are developed and tested on the one-dimensional heat equation, with constant coefficients, subject to homogeneous boundary conditions and time-dependent boundary conditions. These methods are also extended to two- and three-dimensional heat equations, with constant coefficients, subject to homogeneous boundary conditions.     

A collocation approach for solving systems of linear Volterra integral equations with variable coefficients

In this paper, a numerical method is introduced to solve a system of linear Volterra integral equations (VIEs). By using the Bessel polynomials and the collocation points, this method transforms the system of linear Volterra integral equations into the matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. This method gives an analytic solution when the exact solutions are polynomials. Numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with existing results. All of the numerical computations have been performed on computer using a program written in MATLAB v7.6.0 (R2008a).     

Construction and implementation of general linear methods for ordinary differential equations: A review

It it the purpose of this paper to review the results on the construction and implementation of diagonally implicit multistage integration methods for ordinary differential equations. The systematic approach to the construction of these methods with Runge-Kutta stability is described. The estimation of local discretization error for both explicit and implicit methods is discussed. The other implementations issues such as the construction of continuous extensions, stepsize and order changing strategy, and solving the systems of nonlinear equations which arise in implicit schemes are also addressed. The performance of experimental codes based on these methods is briefly discussed and compared with codes from Matlab ordinary differential equation (ODE) suite. The recent work on general linear methods with inherent Runge-Kutta stability is also briefly discussed.     

A class of collocation methods for numerical integration of initial value problems

For the numerical solution of initial value problems a general procedure to determine global integration methods is derived and studied. They are collocation methods which can be easily implemented and provide a high order accuracy. They further provide globally continuous differentiable solutions. Computation of the integrals which appear in the coefficients are generated by a recurrence formula and no integrals are involved in the calculation. Numerical experiments provide favorable comparisons with other existing methods.     

A collocation and Cartesian grid methods using new radial basis function to solve class of partial differential equations

In the present paper, a class of partial differential equation represented by Poisson's type problems are solved using a proposed Cartesian grid method and a collocation technique using a new radial basis function. The advantage of using this new radial basis function represented by overcoming singularity from the diagonal elements when thin plate radial basis function is used. The new function is a combination of both multiquadric and thin plate radial basis functions. The new radial basis function contains a control parameter ?, that takes one when evaluating the singular elements and equals zero elsewhere. Collocation of the approximate solution of the potential over the governing and boundary condition equations leads to a double linear system of equations. A proposed algebraic procedure is then developed to solve the double system. Examples of Poisson and Helmholtz equations are solved and the present results are compared with the their analytical solutions. A good agreement with analytical results is achieved.     

A practical chebyshev collocation method for differential equations

This paper describes a method for solving ordinary and partial differential equations in Chebyshev series. The main feature of the method, which is based on the collocation principle, (Lanczos [8]) is that it solves the problem of differentiating a Chebyshev series directly by the use of a stable recurrence relation. As a practical consequence the method is very simple and can easily be coded into a general-purpose program for solving some differential equations.     

Nonequidistant modified predictor-corrector methods for solving systems of differential equations

This paper shows that it is possible to develop nonequidistant predictor-corrector formulae with minimum error bounds for solving systems of differential equations such that the tedious difficulties which arise in practical applications can be overcome. General predictor-corrector formulae with variable steps are constructed. Explicit third order- and fourth order-two points formulae are derived. Also fourth order-three points formulae are represented. Two theorems are given. A flow chart for general nonequidistant predictor-corrector methods using automatic control for the step length is compactly represented for solving systems of differential equations. These methods are recommended to be used widely in practice because of many advantages.     

A class of explicit two-step superstable methods for second-order linear initial value problems

A class of explicit two-step superstable methods of fourth algebraic order for the numerical solution of second-order linear initial value problems is presented in this article. We need Taylor expansion at an internal grid point and collocation formulae for the derivatives of the solution to derive a method and then modify it into a class of methods having the desired stability properties. Computational results are presented to demonstrate the applicability of the methods to some standard problems.     

A family of Newton-type methods for solving nonlinear equations

A family of Newton-type methods free from second and higher order derivatives for solving nonlinear equations is presented. The order of the convergence of this family depends on a function. Under a condition on this function this family converge cubically and by imposing one condition more on this function one can obtain methods of order four. It has been shown that this family covers many of the available iterative methods. From this family two new iterative methods are obtained. Numerical experiments are also included.