A central problem of numerical analysis is the approximate evaluation of integrals or derivatives of functions. In more generality, this is the approximate evaluation of a linear functional defined on a space of functions. Users often just have values of a function u at scattered points x1, . . . ,xN in the domain Ω of u, and then the value λ (u) of a linear functional λ must be approximated vi...

The application of nonlinear schemes like dual time stepping as preconditioners in matrix-free Newton-Krylov-solvers is considered and analyzed. We provide a novel formulation of the left preconditioned operator that says it is in fact linear in the matrix-free sense, but changes the Newton scheme. This allows to get some insight in the convergence properties of these schemes which are demonstr...

This paper deals with a special family of implicit Runge–Kutta formulas of orders 2, 4 and 6. These methods are of Gauss type; i.e., they are based on Gauss quadrature formulas of orders 2, 4 and 6, respectively. However, the methods under discussion have only explicit internal stages that lead to cheap practical implementation. Some of the stage values calculated in a step of the numerical int...

Abstract. Differential equations involving derivatives of non-integer order have shown to be adequate models for various physical phenomena in areas like damping laws, diffusion processes, etc. A small number of algorithms for the numerical solution of these equations has been suggested, but mainly without any error estimates. In this paper, we propose an implicit algorithm for the approximate ...

and Applied Analysis 3 The class of Runge-Kutta methods with CQ formula has been applied to delay-integro-differential equations by many authors (c.f. [18, 19]). For the CQ formula (9), we usually adopt the repeated trapezoidal rule, the repeated Simpson’s rule, or the repeated Newton-cotes rule, and so forth, denote η = max{?̃? 0 , ?̃? 1 , . . . , ?̃? m }. It should be pointed out that the adopte...

the spline collocation method  is employed to solve a system of linear and nonlinear fredholm and volterra integro-differential equations. the solutions are collocated by cubic b-spline and the integrand is approximated by the newton-cotes formula. we obtain the unique solution for linear and nonlinear system $(nn+3n)times(nn+3n)$ of integro-differential equations. this approximation reduces th...

One 3-D transient temperature field model for a thin uniform plate caused by moving laser heat source is described in present study. The with power density Gaussian-distribution form considered when finite-thin heated. By using the separate variable method and Newton-Cotes method, semi-analytical solution of conduction equation finite obtained. Numerical results show that effect distribution, s...

We construct locally supported basis functions which are biorthogonal to conforming nodal finite element basis functions of degree p in one dimension. In contrast to earlier approaches, these basis functions have the same support as the nodal finite element basis functions and reproduce the conforming finite element space of degree p − 1. Working with Gauß–Lobatto nodes, we find an interesting ...

The time integration of semilinear parabolic problems by exponential methods of different kinds is considered. A new algorithm for the implementation of these methods is proposed. The algorithm evaluates the operators required by the exponential methods by means of a quadrature formula that converges like O(e−cK/ ln K), with K the number of quadrature nodes. The algorithm allows also the evalua...

Purpose – The purpose of this work is to study the flow of mixed convection and mass transfer of a steady laminar boundary layer about an isothermal vertical flat plate embedded in a non-Darcian porous medium in the presence of chemical reaction and viscous dissipation effects. Design/methodology/approach – The governing partial differential equations are converted into ordinary differential eq...