Mono-implicit Runge–Kutta formulae for the numerical solution of second order nonlinear two-point boundary value problems

Mono-implicit Runge–Kutta (MIRK) formulae are widely used for the numerical solution of first order systems of nonlinear two-point boundary value problems. In order to avoid costly matrix multiplications, MIRK formulae are usually implemented in a deferred correction framework and this is the basis of the well known boundary value code TWPBVP. However, many two-point boundary value problems occur naturally as second (or higher) order equations or systems and for such problems there are significant savings in computational effort to be made if the MIRK methods are tailored for these higher order forms. In this paper, we describe MIRK algorithms for second order equations and report numerical results that illustrate the substantial savings that are possible particularly for second order systems of equations where the first derivative is absent.     

On initial boundary value problems for variants of the Hunter–Saxton equation

The Hunter?CSaxton equation serves as a mathematical model for orientation waves in a nematic liquid crystal. The present paper discusses a modified variant of this equation, coming up in the study of critical points for the speed of orientation waves, as well as a two-component extension. We establish well-posedness and blow-up results for some initial boundary value problems for the modified Hunter?CSaxton equation and the two-component Hunter?CSaxton system.     

A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation

In this paper, we present a numerical method for solving Volterra integral equations of the second kind (VK2), first kind (VK1) and even singular type of these equations. The proposed method is based on approximating unknown function with Bernstein’s approximation. This method using simple computation with quite acceptable approximate solution. Furthermore we get an estimation of error bound for this method. For showing efficiency of this method we use several examples.     

A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints

A Lavrentiev type regularization technique for solving elliptic boundary control problems with pointwise state constraints is considered. The main concept behind this regularization is to look for controls in the range of the adjoint control-to-state mapping. After investigating the analysis of the method, a semismooth Newton method based on the optimality conditions is presented. The theoretical results are confirmed by numerical tests. Moreover, they are validated by comparing the regularization technique with standard numerical codes based on the discretize-then-optimize concept. The authors acknowledge support through DFG Research Center “Mathematics for Key Technologies” (FZT 86) in Berlin.     

Decay mode solution of nonlinear boundary–initial value problems for the cylindrical (spherical) Boussinesq–Burgers equations

The major target of this paper is to construct new nonlinear boundary–initial value problems for Boussinesq–Burgers Equations, and derive the solutions of these nonlinear boundary–initial value problems by the simplified homogeneous balance method. The nonlinear transformation and its inversion between the Boussinesq–Burgers Equations and the linear heat conduction equation are firstly derived; then a new nonlinear boundary–initial value problem for the Boussinesq–Burgers equations with variable damping on the half infinite straight line is put forward for the first time, and the solution of this nonlinear boundary–initial value problem is obtained, especially, the decay mode solution of nonlinear boundary–initial value problem for the cylindrical (spherical) Boussinesq–Burgers equations is obtained.     

On the solvability of initial boundary value problems for nonlinear time-dependent Schrödinger equations

In this paper, we study the initial boundary value problems for a nonlinear time-dependent Schrödinger equation with Dirichlet and Neumann boundary conditions, respectively. We prove the existence and uniqueness of solutions of the initial boundary value problems by using Galerkin’s method.     

Life-span of classical solutions to one dimensional initial–boundary value problems for general quasilinear wave equations

This paper is devoted to studying the initial–boundary value problem for one dimensional general quasilinear wave equations $u_{tt}?u_{xx}=b(u,Du)u_{xx}+2a_0(u,Du)u_{tx}+F(u,Du)$ on exterior domain. We obtain the sharp lower bound of the life-span of classical solutions to the initial–boundary value problem with small initial data and zero boundary data for one dimensional general quasilinear wave equations.     

A bilinear approach to the pooling problem†

In this paper we present an algorithm for the pooling problem in refinery optimization based on a bilinear programming approach. The pooling problem occurs frequently in process optimization problems, especially refinery planning models. The main difficulty is that pooling causes an inherent nonlinearity in the otherwise linear models. We shall define the problem by formulating an aggregate mathematical model of a refinery, comment on solution methods for pooling problems that have been presented in the literature, and develop a new method based on convex approximations of the bilinear terms. The method is illustrated on numerical examples     

Multiple positive solutions of four point boundary value problems for finite difference equations†

In this paper, we consider the following second-order four-point boundary-value problems Δ² u(k ? 1)+f(k,u(k), Δu(k)) = 0,k ∈ {1,2,…,T}, u(0) = au(l ₁), u(T+1) = bu(l ₂). We give conditions on f to ensure the existence of at least three positive solutions of the given problem by applying a new fixed-point theorem of functional type in a cone. The emphasis is put on the nonlinear term involved with the first-order difference.     

On boundary value problems for the ϕ-Laplacian

For the φ-Laplacian, we consider a boundary value problem with functional boundary conditions. The Dirichlet problem is a special case of this problem.     

Sinc-Galerkin method for numerical solution of the Bratu’s problems

Study of the performance of the Galerkin method using sinc basis functions for solving Bratu’s problem is presented. Error analysis of the presented method is given. The method is applied to two test examples. By considering the maximum absolute errors in the solutions at the sinc grid points are tabulated in tables for different choices of step size. We conclude that the Sinc-Galerkin method converges to the exact solution rapidly, with order, $O(\exp{(-c \sqrt{n}}))$ accuracy, where c is independent of n.     

Sample solutions of stochastic boundary value problems ∗

We prove existence theorems for nonlinear stochastic Sturm-Liouville problems which improve results from [4]. In the simplest case this is done by means of a known result about measurable selections of multivalued maps and a new fixed point theorem for stochastic nonlinear operators which is more realistic than existing ones     

Modified Runge–Kutta Verner methods for the numerical solution of initial and boundary-value problems with engineering applications

Two new modified Runge–Kutta methods with minimal phase-lag are developed for the numerical solution of Ordinary Differential Equations with engineering applications. These methods are based on the well-known Runge–Kutta method of Verner RK6(5)9b (see J.H. Verner, some Runge–Kutta formula pairs, SIAM J. Numer. Anal 28 (1991) 496–511) of order six. Numerical and theoretical results in some problems of the plate deflection theory show that this new approach is more efficient compared with the well-known classical sixth order Runge–Kutta Verner method.     

A reliable approach to the solution of Navier–Stokes equations

In this paper we will demonstrate an affective approach of solving Navier–Stokes equations by using a very reliable transformation method known as the Cole–Hopf transformation, which reduces the problem from nonlinear into a linear differential equation which, in turn, can be solved effectively.