Discrete cubic spline method for second-order boundary value problems

In this paper, we use discrete cubic spline based on central differences to obtain approximate solution of a second-order boundary value problem. It is shown that the method is of order 4 if a parameter takes a specific value, else it is of order 2. Two numerical examples are included to illustrate our method as well as to compare the performance with other numerical methods proposed in the literature.     

On sign-changing solutions for a second-order integral boundary value problem

In this paper, we obtain the existence of sign-changing solutions for nonlinear second-order differential equations with integral boundary value conditions, by applying a new fixed point theorem in ordered Banach spaces, with the lattice structure derived by Liu and Sun. Our results improve on those in the literature.     

A class of methods based on a septic non-polynomial spline function for the solution of sixth-order two-point boundary value problems

Two new second- and fourth-order methods based on a septic non-polynomial spline function for the numerical solution of sixth-order two-point boundary value problems are presented. The spline function is used to derive some consistency relations for computing approximations to the solution of this problem. The proposed approach gives better approximations than existing polynomial spline and finite difference methods up to order four and has a lower computational cost. Convergence analysis of these two methods is discussed. Three numerical examples are included to illustrate the practical use of our methods as well as their accuracy compared with existing spline function methods.     

On the use of spline functions of even degree for the numerical solution of the delay differential equations

One considers and investigates the notion of natural spline functions of even degree, satisfying given derivative-interpolating conditions on simple knots. By using such spline functions we shall develop some theory and algorithms for the numerical solution of a class of delay differential equations. It will be shown that such kind of spline functions are very suitable for the numerical treatment of delay differential equations with initial conditions.     

On a method for solving a local boundary problem for a nonlinear stationary system with perturbations in the class of piecewise constant controls

The wide class of nonlinear stationary systems of ordinary differential equations taking into account restrictions on control and external perturbation is considered. An algorithm for constructing a discrete control function that guarantees the transfer of the systems from the initial state to the origin and an arbitrary neighborhood of the origin is proposed. A constructive sufficient condition of the Kalman type, in which the specified translation is possible, is obtained. The problem of robot‐manipulator control is considered and its numerical simulation is carried out.     

A fast and accurate computational technique for efficient numerical solution of nonlinear singular boundary value problems

This paper is concerned with design and implementation of a computational technique for the efficient solution of a class of singular boundary value problems. The method is based on a modified homotopy analysis method. The method is illustrated by six examples, two of which arise in chemical engineering: the first problem arises in the study of thermal explosions, while the second problem arises in the study of heat and mass transfer within the porous catalyst particles. Numerical results reveal that our method provides better results as compared to some existing methods. Furthermore, it is a powerful tool for dealing with different types of problems with strong nonlinearity.     

Numerical solution of optimal control problem for a model of tumour growth with drug application

In this study, a combination of spectral and fixed point methods is used to solve an optimal control problem for a model of tumour growth. The growth of tumour is modelled using three first-order hyperbolic equations describing the evolution of cells and two second-order parabolic equations describing the diffusion of nutrient and drug concentration. In the optimal control problem, four control variables are employed to control the concentration of nutrient and drug on the boundary and inside the tumour. Since the problem is nonlinear, applying the fixed point method, in each step of iteration, the problem is changed to a linear one and the parabolic equations are solved using the spectral method. The convergence and stability of method are proven. Some examples are considered to illustrate the efficiency of method. Finally, some figures are provided to reflect the effects of control on the densities of tumourcells.     

The Dirichlet problem for the Laplacian with discontinuous boundary data in a 2D multiply connected exterior domain

The Dirichlet problem for Laplacian in a planar multiply connected exterior domain bounded by smooth closed curves is considered in case, when the boundary data is piecewise continuous, i.e. it may have jumps in certain points of the boundary. It is assumed that the solution to the problem may be not continuous at the same points. The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, the integral representation for a classical solution is obtained. The problem is reduced to a uniquely solvable Fredholm integral equation of the second kind and of index zero. It is shown that a weak solution to the problem does not exist typically, though the classical solution exists.     

An efficient method for solving elliptic boundary element problems with application to the tokamak vacuum problem

A method for regularizing ill-posed Neumann Poisson-type problems based on applying operator transformations is presented. This method can be implemented in the context of the finite element method to compute the solution to inhomogeneous Neumann boundary conditions; it allows to overcome cases where the Neumann problem otherwise admits an infinite number of solutions. As a test application, we solve the Grad–Shafranov boundary problem in a toroidally symmetric geometry. Solving the regularized Neumann response problem is found to be several orders of magnitudes more efficient than solving the Dirichlet problem, which makes the approach competitive with the boundary element method without the need to derive a Green function. In the context of the boundary element method, the operator transformation technique can also be applied to obtain the response of the Grad–Shafranov equation from the toroidal Laplace n=1 response matrix using a simple matrix transformation.     

A numerical method for multipoint boundary value problems with application to a restricted three body problem

A type of parallel shooting method is proposed for the solution of nonlinear multipoint boundary value problems. It extends the usual quasilinearization method and a previous shooting method developed for such problems, and reduces to usual multiple shooting techniques for two point boundary value problems. The effectiveness of the method for stiff problems is illustrated by an application to the problem of finding periodic solutions of a restricted three body problem with given Jacobian constant and unknown period.     

A new method for computing a solution of the Cauchy problem with polynomial data for the system of crystal optics

A new analytical method for solving an initial value problem (IVP) for the system of crystal optics with polynomial data and a polynomial inhomogeneous term is suggested. The found solution of the IVP is a polynomial. Theoretical and computational analysis of polynomial solutions and their comparison with non-polynomial solutions corresponding to smooth data are given. The applicability of polynomial solutions to physical processes is discussed. An implementation of this method has been made by symbolic computations in Maple 10.     

The Dirichlet problem for a 3D elliptic equation with two singular coefficients

In the present work, we investigate the Dirichlet problem for a three-dimensional (3D) elliptic equation with two singular coefficients. We find four fundamental solutions of the equation, containing hypergeometric functions of Appell. Then using an “a-b-c” method, the uniqueness for the solution of the Dirichlet problem is proved. Applying a method of Green’s function, we are able to find the solution of the problem in an explicit form. Moreover, decomposition formulas, formulas of differentiation and some adjacent relations for Appell’s hypergeometric functions were used in order to find the explicit solution for the formulated problem.     

Milne, a routine for the numerical solution of Milne's problem

The routine Milne provides accurate numerical values for the classical Milne's problem of neutron transport for the planar one speed and isotropic scattering case. The solution is based on the Case eigen-function formalism. The relevant X functions are evaluated accurately by the Double Exponential quadrature. The calculated quantities are the extrapolation distance and the scalar and the angular fluxes. Also, the H function needed in astrophysical calculations is evaluated as a byproduct.

Program summary

Program title: MilneCatalogue identifier: AEGS_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGS_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 701No. of bytes in distributed program, including test data, etc.: 6845Distribution format: tar.gzProgramming language: Fortran 77Computer: PC under Linux or WindowsOperating system: Ubuntu 8.04 (Kernel version 2.6.24-16-generic), Windows-XPClassification: 4.11, 21.1, 21.2Nature of problem: The X functions are integral expressions. The convergence of these regular and Cauchy Principal Value integrals are impaired by the singularities of the integrand in the complex plane. The DE quadrature scheme tackles these singularities in a robust manner compared to the standard Gauss quadrature.Running time: The test included in the distribution takes a few seconds to run.     

Numerical computation of the eigenvalues of a discontinuous Dirac system using the sinc method with error analysis

In this paper, we apply a regularized sinc method to compute the eigenvalues of a discontinuous regular Dirac system with transmission conditions at the point of discontinuity. The regularized technique allows us to insert some parameters to the well-known sinc method, strengthening the existing technique, and to avoid the aliasing error. The error analysis is established considering both the truncation and amplitude errors associated with the sampling theorem. Numerical examples together with tables and illustrative figures are given.     

The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure

In this article, we propose a new approach for solving an initial–boundary value problem with a non-classic condition for the one-dimensional wave equation. Our approach depends mainly on Adomian's technique. We will deal here with new type of nonlocal boundary value problems that are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The decomposition method of G. Adomian can be an effective scheme to obtain the analytical and approximate solutions. This new approach provides immediate and visible symbolic terms of analytic solution as well as numerical approximate solution to both linear and nonlinear problems without linearization. The Adomian's method establishes symbolic and approximate solutions by using the decomposition procedure. This technique is useful for obtaining both analytical and numerical approximations of linear and nonlinear differential equations and it is also quite straightforward to write computer code. In comparison to traditional procedures, the series-based technique of the Adomian decomposition technique is shown to evaluate solutions accurately and efficiently. The method is very reliable and effective that provides the solution in terms of rapid convergent series. Several examples are tested to support our study.     

Generalized Tikhonov method for the final value problem of time-fractional diffusion equation

This paper investigates the final value problem for a time-fractional diffusion equation. We determine the initial data from a noisy final data. A generalized Tikhonov regularization method is proposed to deal with this ill-posed problem, and then the convergence rates is derived by the a-priori and a-posteriori choice rules of regularization parameter. A numerical example is presented to show that this method is stable and feasible.     

A model with a solution algorithm for the cash transportation vehicle routing and scheduling problem

Cash transportation vehicle routing and scheduling are essential for security carriers to minimize their operating costs and ensure safe cash conveyance. In real operations, to increase cash conveyance safety, there must be significant variation in daily cash transportation vehicle routes and schedules, making such vehicle routes and schedules difficult to formulate. However, for convenient planning purposes, security carriers normally plan such routes and schedules based on personal experience, without considering variations in routes and schedules from a system perspective. As a result, the obtained routes and schedules are neither safe nor efficient for transporting cash. In this study, a model is developed where the time–space network technique is utilized to formulate the potential movements of cash transportation vehicles among all demand points in the dimensions of time and space. This model incorporates a new concept of similarity of time and space for routing and scheduling, which is expected to help security carriers formulate more flexible routing and scheduling strategies. This is helpful to reduce the risk of robbery. Mathematically, the model is formulated as an integer multiple-commodity network flow problem. A solution algorithm, based on a problem decomposition/collapsing technique, coupled with the use of a mathematical programming software, is developed to efficiently solve the problem. The case study results show that our model and solution algorithm could be useful references for security carriers in actual practice.     

A new type of solution method for the generalized linear complementarity problem over a polyhedral cone

This paper addresses the generalized linear complementarity problem （GLCP） over a polyhedral cone. To solve the problem, we first equivalently convert the problem into an affine variational inequalities problem over a closed polyhedral cone, and then propose a new type of method to solve the GLCP based on the error bound estimation. The global and R-linear convergence rate is established. The numerical experiments show the efficiency of the method.     

Approximate analytic solution of the Dirichlet problems for Laplace’s equation in planar domains by a perturbation method

In this paper, we propose a regular perturbation method to obtain approximate analytic solutions of exterior and interior Dirichlet problems for Laplace’s equation in planar domains. This method, starting from a geometrical perturbation of these planar domains, reduces our problems to a family of classical Dirichlet problems for Laplace’s equation in a circle. Numerical examples are given and comparisons are made with the solutions obtained by other approximation methods.