A Nyström method for solving 2sth order boundary value problems

A Nyström method is proposed for solving Fredholm integral equations equivalent to special boundary value problems of order 2s. The stability and the convergence of the proposed procedure is proved. Some numerical examples are provided in order to illustrate the accuracy of the method.     

Associate symmetries: A novel procedure for finding contact symmetries

A new method for finding contact symmetries is proposed for both ordinary and partial differential equations. Symmetries more general than Lie point are often difficult to find owing to an increased dependency of the infinitesimal functions on differential quantities. As a consequence, the invariant surface condition is often unable to be “split” into a reasonably sized set of determining equations, if at all. The problem of solving such a system of determining equations is here reduced to the problem of finding its own point symmetries and thus subsequent similarity solutions to these equations. These solutions will (in general) correspond to some subset of symmetries of the original differential equations. For this reason, we have termed such symmetries associate symmetries. We use this novel method of associate symmetries to determine new contact symmetries for a non-linear PDE and a second order ODE which could not previously be found using computer algebra packages; such symmetries for the latter are particularly difficult to find. We also consider a differential equation with known contact symmetries in order to illustrate that the associate symmetry procedure may, in some cases, be able to retrieve all such symmetries.     

Laguerre polynomial approach for solving linear delay difference equations

This paper presents a numerical method for the approximate solution of mth-order linear delay difference equations with variable coefficients under the mixed conditions in terms of Laguerre polynomials. The aim of this article is to present an efficient numerical procedure for solving mth-order linear delay difference equations with variable coefficients. Our method depends mainly on a Laguerre series expansion approach. This method transforms linear delay difference equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equation. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system Maple.     

High accuracy cubic spline approximation for two dimensional quasi-linear elliptic boundary value problems

We report a new 9 point compact discretization of order two in y- and order four in x-directions, based on cubic spline approximation, for the solution of two dimensional quasi-linear elliptic partial differential equations. We describe the complete derivation procedure of the method in details and also discuss how our discretization is able to handle Poisson’s equation in polar coordinates. The convergence analysis of the proposed cubic spline approximation for the nonlinear elliptic equation is discussed in details and we have shown under appropriate conditions the proposed method converges. Some physical examples and their numerical results are provided to justify the advantages of the proposed method.     

An optimal Steffensen-type family for solving nonlinear equations

In this paper, a general family of Steffensen-type methods with optimal order of convergence for solving nonlinear equations is constructed by using Newton’s iteration for the direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub [H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Math. 21 (1974) 634-651] that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2^m−1. Its error equations and asymptotic convergence constants are obtained. Finally, it is compared with the related methods for solving nonlinear equations in the numerical examples.     

Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nth-order linear differential equations

Some efficient and accurate algorithms based on the ultraspherical-Galerkin method are developed and implemented for solving 2nth-order linear differential equations in one variable subject to homogeneous and nonhomogeneous boundary conditions using a spectral discretization. We extend the proposed algorithms to solve the two-dimensional 2nth-order differential equations. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to linear systems with specially structured matrices that can be efficiently inverted, hence greatly reducing the cost and roundoff errors.     

An adaptive multiphase approach for large unconditional and conditional p-median problems

A multiphase approach that incorporates demand points aggregation, Variable Neighbourhood Search (VNS) and an exact method is proposed for the solution of large-scale unconditional and conditional p-median problems. The method consists of four phases. In the first phase several aggregated problems are solved with a “Local Search with Shaking” procedure to generate promising facility sites which are then used to solve a reduced problem in Phase 2 using VNS or an exact method. The new solution is then fed into an iterative learning process which tackles the aggregated problem (Phase 3). Phase 4 is a post optimisation phase applied to the original (disaggregated) problem. For the p-median problem, the method is tested on three types of datasets which consist of up to 89,600 demand points. The first two datasets are the BIRCH and the TSP datasets whereas the third is our newly geometrically constructed dataset that has guaranteed optimal solutions. The computational experiments show that the proposed approach produces very competitive results. The proposed approach is also adapted to cater for the conditional p-median problem with interesting results.     

Convergent series solution of nonlinear equations

The author's decomposition method [1] provides a new, efficient computational procedure for solving large classes of nonlinear (and/or stochastic) equations. These include differential equations containing polynomial, exponential, and trigonometric terms, negative or irrational powers, and product nonlinearities [2]. Also included are partial differential equations [3], delay-differential equations [4], algebraic equations [5], and matrix equations [6] which describe physical systems. Essentially the method provides a systematic computational procedure for equations containing any nonlinear terms of physical significance. The procedure depends on calculation of the author's A_n, a finite set of polynomials [1,13] in terms of which the nonlinearities can be expressed. This paper shows important properties of the A_n which ensure an accurate and computable convergent solution by the author's decomposition method [1]. Since the nonlinearities and/or stochasticity which can be handled are quite general, the results are potentially extremely useful for applications and make a number of common approximations such as linearization, unnecessary.     

High accuracy cubic spline finite difference approximation for the solution of one-space dimensional non-linear wave equations

In this paper, we propose a new three-level implicit nine point compact cubic spline finite difference formulation of order two in time and four in space directions, based on cubic spline approximation in x-direction and finite difference approximation in t-direction for the numerical solution of one-space dimensional second order non-linear hyperbolic partial differential equations. We describe the mathematical formulation procedure in details and also discuss how our formulation is able to handle wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable. Numerical results are provided to justify the usefulness of the proposed method.     

A simple method and its applications to nonlinear partial differential equations

In this paper, a simple method is proposed for constructing more general exact solutions of nonlinear partial differential equations. We choose the Camassa and Holm-Degasperis and Procesi equation and the generalized b family equations to illustrate the validity and advantages of the method. As a result, many new and more general exact solutions are obtained. Some previous results are extended.     

Analytical approximation of weakly nonlinear continuous systems using renormalization group method

A direct method based on renormalization group method (RGM) is proposed for determining the analytical approximation of weakly nonlinear continuous systems. To demonstrate the application of the method, we use it to analyze some examples. First, we analyze the vibration of a beam resting on a nonlinear elastic foundation with distributed quadratic and cubic nonlinearities in the cases of primary and subharmonic resonances of the nth mode. We apply the RGM to the discretized governing equation and also directly to the governing partial differential equations (PDE). The results are in full agreement with those previously obtained with multiple scales method. Second, we obtain higher order approximation for free vibrations of a beam resting on a nonlinear elastic foundation with distributed cubic nonlinearities. The method is applied to the discretized governing equation as well as directly to the governing PDE. The proposed method is capable of producing directly higher order approximation of weakly nonlinear continuous systems. It is shown that the higher order approximation of discretization and direct methods are not in general equal. Finally, we analyze the previous problem in the case that the governing differential equation expressed in complex-variable form. The results of second order form and complex-variable form are not in agreement. We observe that in use of RGM in higher order approximation of continuous systems, the equations must not be treated in second order form.     

A non-uniform basis order for the discontinuous Galerkin method of the acoustic and elastic wave equations

Here, we solve the time-dependent acoustic and elastic wave equations using the discontinuous Galerkin method for spatial discretization and the low-storage Runge-Kutta and Crank-Nicolson methods for time integration. The aim of the present paper is to study how to choose the order of polynomial basis functions for each element in the computational mesh to obtain a predetermined relative error. In this work, the formula 2p+1≈κhk, which connects the polynomial basis order p, mesh parameter h, wave number k, and free parameter κ, is studied. The aim is to obtain a simple selection method for the order of the basis functions so that a relatively constant error level of the solution can be achieved. The method is examined using numerical experiments. The results of the experiments indicate that this method is a promising approach for approximating the degree of the basis functions for an arbitrarily sized element. However, in certain model problems we show the failure of the proposed selection scheme. In such a case, the method provides an initial basis for a more general p-adaptive discontinuous Galerkin method.     

The Cauchy problems for evolutionary pseudo-differential equations over p-adic field and the wavelet theory

We solve the Cauchy problems for p-adic linear and semi-linear evolutionary pseudo-differential equations (the time-variable t∈R and the space-variable ). Among the equations under consideration there are the heat type equation and the Schrödinger type equations (linear and nonlinear). To solve these problems, we develop the “variable separation method” (an analog of the classical Fourier method) which reduces solving evolutionary pseudo-differential equations to solving ordinary differential equations with respect to real variable t. The problem of stabilization for solutions of the Cauchy problems as t→∞ is also studied. These results give significant advance in the theory of p-adic pseudo-differential equations and can be used in applications.     

Space-time discontinuous galerkin method for maxwell equations in dispersive media

In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the underlying problem. Unconditional L²-stability and error estimate of order O(τ^r+1+h^k+1/2) are obtained when polynomials of degree at most r and k are used for the temporal discretization and spatial discretization respectively. 2-D and 3-D numerical examples are given to validate the theoretical results. Moreover, numerical results show an ultra-convergence of order 2r+1 in temporal variable t.     

A second order finite difference-spectral method for space fractional diffusion equations

A high order finite difference-spectral method is derived for solving space fractional diffusion equations,by combining the second order finite difference method in time and the spectral Galerkin method in space.The stability and error estimates of the temporal semidiscrete scheme are rigorously discussed,and the convergence order of the proposed method is proved to be O(τ2+Nα-m)in L2-norm,whereτ,N,αand m are the time step size,polynomial degree,fractional derivative index and regularity of the exact solution,respectively.Numerical experiments are carried out to demonstrate the theoretical analysis.     

A simple augmented ∊-constraint method for multi-objective mathematical integer programming problems

A simple augmented ?-constraint (SAUGMECON) method is put forward to generate all non-dominated solutions of multi-objective integer programming (MOIP) problems. The SAUGMECON method is a variant of the augmented ?-constraint (AUGMECON) method proposed in 2009 and improved in 2013 by Mavrotas et al. However, with the SAUGMECON method, all non-dominated solutions can be found much more efficiently thanks to our innovations to algorithm acceleration. These innovative acceleration mechanisms include: (1) an extension to the acceleration algorithm with early exit and (2) an addition of an acceleration algorithm with bouncing steps. The same numerical example in Lokman and Köksalan (2012) is used to illustrate workings of the method. Then comparisons of computational performance among the method proposed by  and , the method developed by Lokman and Köksalan (2012) and the SAUGMECON method are made by solving randomly generated general MOIP problem instances as well as special MOIP problem instances such as the MOKP and MOSP problem instances presented in Table 4 in Lokman and Köksalan (2012). The experimental results show that the SAUGMECON method performs the best among these methods. More importantly, the advantage of the SAUGMECON method over the method proposed by Lokman and Köksalan (2012) turns out to be increasingly more prominent as the number of objectives increases.     

Continuous block implicit hybrid one-step methods for ordinary and delay differential equations

A class of high order continuous block implicit hybrid one-step methods has been proposed to solve numerically initial value problems for ordinary and delay differential equations. The convergence and Aω-stability of the continuous block implicit hybrid methods for ordinary differential equations are studied. Alternative form of continuous extension is constructed such that the block implicit hybrid one-step methods can be used to solve delay differential equations and have same convergence order as for ordinary differential equations. Some numerical experiments are conducted to illustrate the efficiency of the continuous methods.     

Boundary equations in the finite transfer method for solving differential equation systems

The finite transfer method is going to be used to solve a p system of linear ordinary differential equations. The complete problem is extended by adding the p boundary equations involved. It is chosen a fourth order scheme to obtain finite transfer expressions. A recurrence strategy is used in these equations and permits one to relate different points in the domain where boundary equations are defined. Finally a 2p algebraic system of equations is noted and solved. To show the efficiency and accuracy, the method is applied to determine the structural behavior of a bending beam with different supports and to solve a differential equation of second degree with different boundary conditions.     

An adaptive Huber method for non-linear systems of weakly singular second kind Volterra integral equations

Numerical methods for systems of weakly singular Volterra integral equations are rarely considered in the literature, especially if the equations involve non-linear dependencies between unknowns and their integrals. In the present work an adaptive Huber method for such systems is proposed, by extending the method previously formulated for single weakly singular second kind Volterra equations. The method is tested on example systems of integral equations involving integrals with kernels K(t, τ) = (t − τ)^−1/2, K(t, τ) = exp[−λ(t − τ)](t − τ)^−1/2 (where λ > 0), and K(t, τ) = 1. The magnitude of the errors, and practical accuracy orders, observed for IE systems, are comparable to those for single IEs. In cases when the solution vector is not differentiable at t = 0, the estimation of errors at t = 0 is found somewhat less reliable for IE systems, than it was for single IEs. The stability of the IE systems solved appears to be sufficient, in practice, for the numerical stability of the method.     

A class of Steffensen type methods with optimal order of convergence

In this paper, a family of Steffensen type methods of fourth-order convergence for solving nonlinear smooth equations is suggested. In the proposed methods, a linear combination of divided differences is used to get a better approximation to the derivative of the given function. Each derivative-free member of the family requires only three evaluations of the given function per iteration. Therefore, this class of methods has efficiency index equal to 1.587. Kung and Traub conjectured that the order of convergence of any multipoint method without memory cannot exceed the bound 2^d-1, where d is the number of functional evaluations per step. The new class of methods agrees with this conjecture for the case d=3. Numerical examples are made to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare with other ones.