Imposing boundary conditions in Sinc method using highest derivative approximation

Sinc approximate methods are often used to solve complex boundary value problems such as problems on unbounded domains or problems with endpoint singularities. A recent implementation of the Sinc method [Li, C. and Wu, X., Numerical solution of differential equations using Sinc method based on the interpolation of the highest derivatives, Applied Mathematical Modeling 31 (1) 2007 1–9] in which Sinc basis functions are used to approximate the highest derivative in the governing equation of the boundary value problem is evaluated for structural mechanics applications in which interlaminar stresses are desired. We suggest an alternative approach for specifying the boundary conditions, and we compare the numerical results for analysis of a laminated composite Timoshenko beam, implementing both Li and Wu’s approach and our alternative approach for applying the boundary conditions. For the Timoshenko beam problem, we obtain accurate results using both approaches, including transverse shear stress by integration of the 3D equilibrium equations of elasticity. The beam results indicate our approach is less dependent on the selection of the Sinc mesh size than Li and Wu’s SIHD. We also apply SIHD to analyze a classical laminated composite plate. For the plate example, we experience difficulty in obtaining a complete system of equations using Li and Wu’s approach. For our approach, we suggest that additional necessary information may be obtained by applying the derivatives of the boundary conditions on each edge. Using this technique, we obtain accurate results for deflection and stresses, including interlaminar stresses by integration of the 3D equilibrium equations of elasticity. Our results for both the beam and the plate problems indicate that this approach is easily implemented, has a high level of accuracy, and good convergence properties.     

A subdivision collocation method for solving two point boundary value problems of order three

In this paper, we propose a method for the numerical solution of self adjoint singularly perturbed third order boundary value problems in which the highest order derivative is multiplied by a small parameter $\varepsilon$. In this method, first we introduce the derivatives of two scale relations satisfied by the subdivision schemes. After that we use these derivatives to construct the subdivision collocation method for the numerical solution of singularly perturbed boundary value problems. Convergence of the subdivision collocation method is also discussed. Numerical examples are presented to illustrate the proposed method.     

The Sinc–Legendre collocation method for a class of fractional convection–diffusion equations with variable coefficients

This paper deals with the numerical solution of classes of fractional convection–diffusion equations with variable coefficients. The fractional derivatives are described based on the Caputo sense. Our approach is based on the collocation techniques. The method consists of reducing the problem to the solution of linear algebraic equations by expanding the required approximate solution as the elements of shifted Legendre polynomials in time and the Sinc functions in space with unknown coefficients. The properties of Sinc functions and shifted Legendre polynomials are then utilized to evaluate the unknown coefficients. Several examples are given and the numerical results are shown to demonstrate the efficiency of the newly proposed method.     

Piecewise polynomial collocation for linear boundary value problems of fractional differential equations

We consider a class of boundary value problems for linear multi-term fractional differential equations which involve Caputo-type fractional derivatives. Using an integral equation reformulation of the boundary value problem, some regularity properties of the exact solution are derived. Based on these properties, the numerical solution of boundary value problems by piecewise polynomial collocation methods is discussed. In particular, we study the attainable order of convergence of proposed algorithms and show how the convergence rate depends on the choice of the grid and collocation points. Theoretical results are verified by two numerical examples.     

Laguerre collocation solutions to boundary layer type problems

In this paper a Laguerre collocation type method based on usual Laguerre functions is designed in order to solve high order nonlinear boundary value problems as well as eigenvalue problems, on semi-infinite domain. The method is first applied to Falkner–Skan boundary value problem. The solution along with its first two derivatives are computed inside the boundary layer on a fine grid which cluster towards the fixed boundary. Then the method is used to solve a generalized eigenvalue problem which arise in the study of the stability of the Ekman boundary layer. The method provides reliable numerical approximations, is robust and easy implementable. It introduces the boundary condition at infinity without any truncation of the domain. A particular attention is payed to the treatment of boundary conditions at origin. The dependence of the set of solutions to Falkner–Skan problem on the parameter embedded in the system is reproduced correctly. For Ekman eigenvalue problem, the critical Reynolds number which assure the linear stability is computed and compared with existing results. The leftmost part of the spectrum is validated using QZ as well as some Jacobi–Davidson type methods.     

An adaptive boundary collocation method for linear PDEs

The article proposes an adaptive algorithm based on a boundary collocation method for linear PDEs satisfying the maximal principle with possibly nonlinear boundary conditions. Given the error tolerance and an initial number of terms in the solution expansion, the algorithm computes expansion coefficients by collocation of boundary conditions and evaluates the maximum absolute error on the boundary. If error exceeds the error tolerance, additional expansion terms and boundary collocation points are added and the process repeated until the tolerance is satisfied. The performance of the algorithm is illustrated by an example of the potential flow past a cylinder placed between parallel walls. © 1995 John Wiley & Sons, Inc.     

Quartic spline method for solving fourth order obstacle boundary value problems

In this paper, we use uniform quartic polynomial splines to develop a new method, which is used for computing approximations to the solution and its first, second as well as third derivatives for a system of fourth order boundary value problems associated with obstacle, unilateral and contact problems. It is shown that the present method is of order two and gives approximations which are better than those produced by other collocation and finite difference methods. Numerical examples are presented to illustrate the applicability of the new method.     

On the solution a of second order singularly-perturbed boundary value problem by the Sinc-Galerkin method

Sinc methods are now recognized as an efficient numerical method for problems whose solutions may have singularities, or infinite domains, or boundary layers. This work deals with the Sinc-Galerkin method for solving second order singularly perturbed boundary value problems. The method is then tested on linear and nonlinear examples and a comparison with spline method and finite element scheme is made. It is shown that the Sinc-Galerkin method yields better results.     

The fully Sinc‐Galerkin method for time‐dependent boundary conditions

The fully Sinc‐Galerkin method is developed for a family of complex‐valued partial differential equations with time‐dependent boundary conditions. The Sinc‐Galerkin discrete system is formulated and represented by a Kronecker product form of those equations. The numerical solution is efficiently calculated and the method exhibits an exponential convergence rate. Several examples, some with a real‐valued solution and some with a complex‐valued solution, are used to demonstrate the performance of this method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Numerical solution of differential equations using Sinc method based on the interpolation of the highest derivatives

In general, we will use the numerical differentiation when dealing with the differential equations. Thus the differential equations can be transformed into algebraic equations and then we can get the numerical solutions. But as we all have known, the numerical differentiation process is very sensitive to even a small level of errors. In contrast it is expected that on average the numerical integration process is much less sensitive to errors. In this paper, based on the Sinc method we provide a new method using Sinc method incorporated with the double exponential transformation based on the interpolation of the highest derivatives (SIHD) for the differential equations. The error in the approximation of the solution is shown to converge at an exponential rate. The numerical results show that compared with the exiting results, our method is of high accuracy, of good convergence with little computational efforts. It is easy to treat nonhomogeneous mixed boundary condition for our method, which is unlike the traditional Sinc method.     

On the solution a of second order singularly-perturbed boundary value problem by the Sinc-Galerkin method

Sinc methods are now recognized as an efficient numerical method for problems whose solutions may have singularities, or infinite domains, or boundary layers. This work deals with the Sinc-Galerkin method for solving second order singularly perturbed boundary value problems. The method is then tested on linear and nonlinear examples and a comparison with spline method and finite element scheme is made. It is shown that the Sinc-Galerkin method yields better results.Received: January 3, 2003; revised: July 14, 2003     

Application of orthogonal collocation on finite elements for solving non-linear boundary value problems

A simple, convenient and easy approach to solve non-linear boundary value problems (BVP) using orthogonal collocation on finite elements (OCFE) is presented. The algorithm is the conjunction of finite element method (FEM) and orthogonal collocation method (OCM). The stability of the numerical results is checked by a novel algorithm which not only justifies the stability of the results but also checks the convergence of the method. The method is applied to the non-symmetric boundary value problems having Dirichlet’s and mixed Robbin’s boundary conditions.     

Approximation of boundary values in the boundary element method

Various techniques may be applied to the approximation of the unknown boundary functions involved in the boundary element method (BEM). Several techniques have been examined numerically to find the most efficient. Techniques considered were: Lagrangian polynomials of the zeroth, first and second orders; spline functions; and the novel weighted minimization technique used successfully in the finite difference method (FDM) for arbitrarily irregular meshes. All these approaches have been used in the BEM for the numerical analysis of plates with various boundary conditions.Both coarse and fine grids on the boundary have been assumed. Maximal errors of the deflections of each plate and the bending moments have been found and the effective computer CPU times determined.Analysis of the results showed that, for the same computer time, the greatest accuracy was obtained by the weighted FDM approach. In the case of the Lagrange approximation, higher order polynomials have proved more efficient. The spline technique yielded more accurate results, but with a higher CPU time.Two discretization approaches have been investigated: the least-squares technique and the collocation method. Despite the fact that the simultaneous algebraic equations obtained were not symmetric, the collocation approach has been confirmed as clearly superior to the least-squares technique, because of the amount of computation time used.     

On the convergence of the collocation method for nonlinear boundary integral equations

Recently, Galerkin and collocation methods have been analysed for some nonlinear boundary integral equations. For the collocation method it has been assumed that the nonlinearity is asymptotically linear. In this paper we remove this restriction. We shall prove the convergence of the collocation method for nonlinear boundary integral equations, when the nonlinearity has a polynomial growth condition. In addition to this the optimal order error estimates follow in L^q(Γ)-norm.     

An efficient numerical approach to solve a class of variable‐order fractional integro‐partial differential equations

The main purpose of this work is to investigate an initial boundary value problem related to a suitable class of variable order fractional integro‐partial differential equations with a weakly singular kernel. To discretize the problem in the time direction, a finite difference method will be used. Then, the Sinc‐collocation approach combined with the double exponential transformation is employed to solve the problem in each time level. The proposed numerical algorithm is completely described and the convergence analysis of the numerical solution is presented. Finally, some illustrative examples are given to demonstrate the pertinent features of the proposed algorithm.     

The use of a stable boundary analogue of the collocation method to approximate the solution of some problems in mechanics

To solve boundary-value problems for elliptic equations, the boundary analogue of the method of least squares is replaced by a boundary analogue of the collocation method. The change is made using a discrete representation of the scalar product in the spaces of functions which, in the case of a smooth boundary, are integrable with their square over the boundary of the region, and of functions which, in the case of a piecewise-linear boundary, are integrable with their square, when weighted, over the boundary of the region. The method used to choose the collocation points which ensure the collocation method to be stable is justified for the case of the Dirichlet problem.     

The use of Sinc-collocation method for solving multi-point boundary value problems

Multi-point boundary value problems have received considerable interest in the mathematical applications in different areas of science and engineering. In this work, our goal is to obtain numerically the approximate solution of these problems by using the Sinc-collocation method. Some properties of the Sinc-collocation method required for our subsequent development are given and are utilized to reduce the computation of solution of multi-point boundary value problems to some algebraic equations. It is well known that the Sinc procedure converges to the solution at an exponential rate. Numerical examples are included to demonstrate the validity and applicability of the new technique.     

Numerical study of Fisher's reaction–diffusion equation by the Sinc collocation method

Fisher's equation, which describes a balance between linear diffusion and nonlinear reaction or multiplication, is studied numerically by the Sinc collocation method. The derivatives and integrals are replaced by the necessary matrices, and a system of algebraic equations is obtained to approximate solution of the problem. The error in the approximation of the solution is shown to converge at an exponential rate. Numerical examples are given to illustrate the accuracy and the implementation of the method, the results show that any local initial disturbance can propagate with a constant limiting speed when time becomes sufficiently large. Both the limiting wave fronts and the limiting speed are independent of the initial values.     

The performance of numerical methods for elliptic problems with mixed boundary conditions

We consider solving linear, second order, elliptic partial differential equations with boundary conditions of types Dirichlet (DIR), mixed (MIX), and nearly Neumann (Neu) by using software modules that implement five numerical methods (one finite element and four finite differences). They represent both the new generation of improved methods and the traditional ones; they are: Hermite collocation plus band Gauss elimination (HC), ordinary finite differences plus band Gauss elimination (5P), ordinary finite differences with Dyaknov iteration (DY), DY with Richardson extrapolation to achieve fourth order convergence (D4), and ordinary finite differences with multigrid iteration (MG). We carry out a performance evaluation in which we measure the grid size and the computer time needed to achieve three significant digits of accuracy in the solution. We compute the changes in these two measures as we change boundary condition types from DIR to MIX and MIX to NEU and then test the following hypotheses: (i) the performance of all the modules is degraded by introducing the derivative terms into the boundary conditions; (ii) finite element collocation (HC) is least affected; (iii) the fourth order modules (HC and D4) are less affected than the other second order modules; and (iv) the traditional 5-point finite differences (5P) are most affected. We establish these hypotheses with high levels of confidence by using several sample problems. The most significant conclusion is that a high order collocation method is preferred for problems with general operators and derivatives in the boundary conditions. We also establish with considerable confidence that these modules have the following rankings in absolute comparative time performance: MG (best), HC and D4, DY, and 5P (worst).     

On spline collocation methods for boundary integral equations in the plane

Nowadays boundary elemen; methods belong to the most popular numerical methods for solving elliptic boundary value problems. They consist in the reduction of the problem to equivalent integral equations (or certain generalizations) on the boundary Γ of the given domain and the approximate solution of these boundary equations. For the numerical treatment the boundary surface is decomposed into a finite number of segments and the unknown functions are approximated by corresponding finite elements and usually determined by collocation and Galerkin procedures. One finds the least difficulties in the theoretical foundation of the convergence of Galerkin methods for certain classes of equations, whereas the convergence of collocation methods, which are mostly used in numerical computations, has yet been proved only for special equations and methods. In the present paper we analyse spline collocation methods on uniform meshes with variable collocation points for one-dimensional pseudodifferential equations on a closed curve with convolutional principal parts, which encompass many classes of boundary integral equations in the plane. We give necessary and sufficient conditions for convergence and prove asymptotic error estimates. In particular we generalize some results on nodal and midpoint collocation obtained in [2], [7] and [8]. The paper is organized as follows. In Section 1 we formulate the problems and the results, Section 2 deals with spline interpolation in periodic Sobolev spaces, and in Section 3 we prove the convergence theorems for the considered collocation methods.