Self-adjoint Singular Perturbation Research Articles

Computational Approach for Fourth-Order Self-Adjoint Singularly Perturbed Boundary Value Problems via Non-polynomial Quintic Spline

This paper proposes a non-polynomial quintic spline method for approximate solutions of fourth-order self-adjoint singular perturbation boundary value problems, which comprises the polynomial part of degree three and two non-polynomial terms, i.e., sine and cosine. It is explicitly shown that the free parameter \(\tau\) of the non-polynomial part can be used to raise the order of accuracy of the scheme. The method has been proved for the second and fourth-order convergence. The proposed scheme is tested on two examples. The experimental results are compared with the existing method.

Fourth-order stable central difference method for self-adjoint singular perturbation problems

In this paper, fourth order stable central difference method is presented for solving self-adjoint singular perturbation problems for small values of perturbation  parameter, e . First, the given differential equation was reduced to its conventional form and then it was transformed into linear system of algebraic equations in the form of a three-term recurrence relation, which can easily be solved by using Thomas Algorithm. To validate the applicability of the method, four model examples have been solved for different values of perturbation parameter and mesh sizes. The numerical results are tabulated and compared with some of the previous findings reported in the literature and it is observed that the present method is more efficient. Graphs are also depicted in support of the numerical results. Both theoretical error bounds and numerical rate of  convergence have been established for the method. Key Words/Phrases: Singular perturbation; stable central difference method;  self-adjoint.

Reliable finite element methods for self-adjoint singular perturbation problems

It is well known that the standard finite element method based on the space Vh of continuous piecewise linear functions is not reliable in solving singular perturbation problems. It is also known that the solution of a two-point boundary-value singular perturbation problem admits a decomposition into a regular part and a finite linear combination of explicit singular functions. Taking into account this decomposition, first, we design a finite element method (which we call singular function method) where the space of trial and test functions is the direct sum of Vhand the space spanned by these singular functions. The second method, based on the finite element discretization on a suitably refined mesh, is referred to as mesh refinement method. Both of these methods are proved to be e-uniformly convergent. Numerical examples which confirm the theory are presented.

To the article “Initial-Value Technique for Self-Adjoint Singular Perturbation Boundary Value Problems” by H. K. Mishra, M. Kumar, and P. Singh, Vol. 20, No. 2, pp. 207–217, April–June, 2009

To the article “Initial-Value Technique for Self-Adjoint Singular Perturbation Boundary Value Problems” by H. K. Mishra, M. Kumar, and P. Singh, Vol. 20, No. 2, pp. 207–217, April–June, 2009

Initial-value technique for self-adjoint singular perturbation boundary value problems

We have developed an initial-value technique for self-adjoint singularly perturbed two-point boundary value problems. The original problem is reduced to its normal form, and the reduced problem is converted into first-order initial-value problems. These initial-value problems are solved by the cubic spline method. Numerical illustrations are given at the end to demonstrate the efficiency of our method. Graphs are also depicted in support of the results.

Wavelet optimized finite difference method using interpolating wavelets for self-adjoint singularly perturbed problems

We design a wavelet optimized finite difference (WOFD) scheme for solving self-adjoint singularly perturbed boundary value problems. The method is based on an interpolating wavelet transform using polynomial interpolation on dyadic grids. Small dissipation of the solution is captured significantly using an adaptive grid. The adaptive feature is performed automatically by thresholding the wavelet coefficients. Numerical examples have been solved and compared with non-standard finite difference schemes in [J.M.S. Lubuma, K.C. Patidar, Uniformly convergent non-standard finite difference methods for self-adjoint singular perturbation problems, J. Comput. Appl. Math. 191 (2006) 228–238]. The proposed method outperforms the non-standard finite difference for studying singular perturbation problems for small dissipations (very small ϵ ) and effective grid generation. Therefore, the proposed method is better for studying the more challenging cases of singularly perturbed problems.

Towards the implementation of the singular function method for singular perturbation problems

The numerical solution of singular perturbation problems (SPPs) is delicate because the perturbation parameter ε and the mesh size h cannot vary independently of one another. In the extended abstract [J.M.-S. Lubuma, K.C. Patidar, Finite element methods for self-adjoint singular perturbation problems, in: T.E. Simos, G. Maroulis (Eds.), ICCMSE 2005: Advances in Computational Methods in Sciences and Engineering, Lecture Series on Computer and Computational Sciences, vol. 4, VSP International Science Publishers, The Netherlands, 2005, pp. 344–347; J.M.-S. Lubuma, K.C. Patidar, Reliable Finite Element Methods for Self-adjoint Singular Perturbation Problems, University of Pretoria Technical Report UPWT 2005/11], the authors proposed the singular function method (SFM) to solve a self-adjoint singular perturbation problem. The SFM is a variant of the finite element method (FEM), where the space of trial and test functions is enriched by the singular functions of the SPP. We proved in the above mentioned work that the SFM is ε-uniformly convergent of optimal order in appropriate norms. However, the numerical implementation of the SFM, based on numerical integration, did not provide satisfactory results. In this paper, we present an alternative way of implementing the SFM. That is, we avoid numerical integration by making use of the explicit form of the singular functions in the computations of the stiffness matrix and the load vector. We obtain improved results. These results are compared with those obtained with numerical integration. The new results confirm theoretical order of convergence.

On Richardson extrapolation for fitted operator finite difference methods

Abstract Recently, there has been a great interest towards the higher order methods for singularly perturbed problems. As compared to their lower order counterparts, they provide better accuracy with fewer mesh points. Construction and/or implementation of direct higher order methods is usually very complicated. Thus a natural choice is to use some convergence acceleration techniques, e.g. Richardson extrapolation. However, as we see in this article, such techniques do not perform equally well on all type of methods. To investigate this, we consider two fitted operator finite difference methods (FOFDMs) developed by Patidar [K.C. Patidar, High order fitted operator numerical method for self-adjoint singular perturbation problems Appl. Math. Comput. 171(1) (2005) 547–566] and Lubuma and Patidar [J. Lubuma, K.C. Patidar, Uniformly convergent non-standard finite difference methods for self-adjoint singular perturbation problems, J. Comput. Appl. Math. 191 (2006) 228–238], referred to as FOFDM-I and FOFDM-II, respectively. The FOFDM-I is fourth and second order accurate for moderate and smaller values of e, respectively. Unfortunately, Richardson extrapolation does not improve the order of this method. The FOFDM-II is second order uniformly convergent and we show that its order can be improved up to four by using Richardson extrapolation. Both the methods are analyzed for convergence and comparative numerical results supporting theoretical estimates are provided.

A parallel approach for self-adjoint singular perturbation problems using Numerov's scheme

In this paper, we considered singularly perturbed self-adjoint boundary-value problems and proposed a computational technique based on Numerov's scheme, which is also suitable for parallel computing. The whole domain is divided into three non-overlapping subdomains, and corresponding subproblems are obtained by using zeroth-order approximations of the solution at the boundaries of these subproblems. The subproblems corresponding to boundary-layer regions are solved using Numerov's method after the introduction of suitable stretching variables and the solution of the reduced problem is taken as an approximate solution in the outer region. A numerical example is provided to show the efficiency and accuracy of the technique.

Geometric mesh FDM for self-adjoint singular perturbation boundary value problems

A numerical method based on finite difference method with variable mesh is given for second order singularly perturbed self-adjoint two point boundary value problems. The original problem is reduced to its normal form and the reduced problem is solved by FDM taking variable mesh(geometric mesh). The maximum absolute errors max i | y ( x i ) - y i | , for different values of parameter ϵ, number of points N, and the mesh ratio r, for three examples have been given in tables to support the efficiency of the method.

Uniformly convergent non-standard finite difference methods for self-adjoint singular perturbation problems

We design non-standard finite difference schemes for self-adjoint singularly perturbed two-point boundary value problems. Essential physical properties (e.g., dissipativity) of the solutions of such problems are captured in the schemes by an appropriate renormalization of the denominator of the discrete derivative. The schemes are analyzed for ε -uniform convergence. Several numerical examples are given to support the predicted theory.

A computational method for self-adjoint singular perturbation problems using quintic spline

Singularly perturbed self-adjoint boundary-value problems are considered in this article. A difference scheme based on quintic spline is proposed. This scheme is applied to the subproblems obtained from the given problem by dividing the whole domain into nonoverlapping subdomains. The proposed scheme is of fourth-order convergent and more suitable for parallel computers. Stability and convergence of the method are discussed. Numerical examples are provided to show the efficiency and accuracy.

High order fitted operator numerical method for self-adjoint singular perturbation problems

We consider self-adjoint singularly perturbed two-point boundary value problems in conservation form. Highest possible order of uniform convergence for such problems achieved hitherto, via fitted operator methods, was one (see, e.g., [Doolan et al. Uniform numerical methods for problems with initial and boundary layers, Boole Press, Dublin, 1980], p. 121]). Reducing the original problem into the normal form and then using the theory of inverse monotone matrices, a fitted operator finite difference method is derived via the standard Numerov's method. The scheme thus derived is fourth order accurate for moderate values of the perturbation parameter @e whereas for very small values of this parameter the method is ''@e-uniformly convergent with order two''. Numerical examples are given in support of the theory.

Spline Approximation Method for Solving Self-Adjoint Singular Perturbation Problems on Non-Uniform Grids

A numerical method based on cubic spline with adaptive grid is given for the self-adjoint singularly perturbed two point boundary value problems. The scheme derived in this method is second order accurate. Numerical examples are given to support the predicted theory.

Tension Spline for the Solution of Self-adjoint Singular Perturbation Problems

A numerical method based on spline in tension is given for the self-adjoint singularly perturbed two point boundary value problems. The schemes derived in this method are second order accurate. One numerical example is given to support the predicted theory.

Improvement of numerical solution of selfadjoint singular perturbation problems by incorporation of asymptotic approximations

A numerical method for singularly perturbed two-point boundary-value problems for second-order ordinary differential equations without a first derivative term arising in the study of chemical catalysis and Michaelis-Menten process in biology is proposed. In this method (“Booster method”), an asymptotic approximation is incorporated into a suitable finite-difference scheme to improve the numerical solution. Uniform error estimates are derived for this method when implemented in known difference schemes. An improvement by a factor of ɛ n+1 can be obtained (where ɛ is the small parameter and n is the order of the asymptotic approximation) for a small amount of extra work. Numerical results are presented in support of this claim.

Exponentially fitted spline approximation method for solving selfadjoint singular perturbation problems

A numerical method based on cubic spline with exponential fitting factor is given for the selfadjoint singularly perturbed two-point boundary value problems. The scheme derived in this method is second-order accurate. Numerical examples are given to support the predicted theory.