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Procedures are developed for constructing asymptotic solutions for certain nonlinear singularly-perturbed vector two-point boundary value problems having boundary layers at one or both end points. The asymptotic approximations are generated numerically and can either be used as is or to furnish a two-point boundary value code (e.g. COLSYS) with an initial approximation and a nonuniform computational mesh. The procedures are applied to several examples involving the deformation of nonlinear elastic beams.

Procedures are developed for constructing asymptotic solutions for certain nonlinear singularly-perturbed vector two-point boundary value, problems having boundary layers at one or both end points. The asymptotic approximations are generated numerically and can either be used as is or to furnish a two-point boundary value code (e.g. COLSYS) with an initial approximation and a nonuniform computational mesh. The procedures are applied to several examples involving the deformation of nonlinear elastic beams. (Author)

A boundary value problem for a fractional power $0 < \varepsilon < 1$ of the second-order elliptic operator is considered. The boundary value problem is singularly perturbed when $\varepsilon \rightarrow 0$. It is solved numerically using a time-dependent problem for a pseudo-parabolic equation. For the auxiliary Cauchy problem, the standard two-level schemes with weights are applied. The numerical results are presented for a model two-dimen\-sional boundary value problem with a fractional power of an elliptic operator. Our work focuses on the solution of the boundary value problem with $0 < \varepsilon \ll 1$.

Book Source: Digital Library of India Item 2015.198162 dc.contributor.author: Arumugam. K dc.date.accessioned: 2015-07-08T12:28:27Z dc.date.available: 2015-07-08T12:28:27Z dc.date.digitalpublicationdate: 2005-08-27 dc.identifier.barcode: 5990010099996 dc.identifier.origpath: /rawdataupload/upload/0099/996 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/198162 dc.description.scannerno: 14 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 105 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher: Indian Institute Of Technology Kanpur dc.rights: Out_of_copyright dc.source.library: Indian Institute Of Technology Kanpur dc.subject.classification: Department Of Mathematics dc.title: A Numerical Treatment Of Singularly Perturbed Boundary Value Problems Of Elliptic Type Using The Boundary Element Method

This work develops an epsilon-uniform finite element method for singularly perturbed boundary value problems. A surprising and remarkable observation is illustrated: By moving one node arbitrarily in between its adjacent nodes, the new finite element solution always intersect with original one at fixed point. Using this fact, an effective epsilon-uniform approximation out of boundary is proposed by adding one point only in the grid that contains boundary layer. The thickness of boundary layer is not necessary to be known from priori estimation. Numerical results are carried out and compared to Shishkin mesh for demonstration purpose.

Collocation methods using both cubic polynomials and splines in tension are developed for second order linear singularly-perturbed two-point boundary value problems. Rules are developed for selecting tension parameters and collocation points. The methods converge outside of boundary layer regions without the necessity of using a fine discretization. Numerical examples comparing the methods are presented. (Author)

Many high-order discretization methods for the solution of two-point boundary value problems for systems of ordinary differential equations are already circulated. However, these methods can behave quite poorly in case the solution has large derivatives, unless severe restrictions on the mesh are imposed. The reason for these restrictions is mainly a stability problem. In this paper a strongly A-stable difference method based on polynomial collocation is developed for a class of qusilinear, singularly perturbed, two-point boundary value problems. Many problems of practical interest asre included in this class, for instance the nonlinear deformation of thin beams or one-dimensional models of carrier transport in semiconductor devices. The method combines the advantages of having the same stability properties as the lower order methods which are used already, with the high order of convergence of collocation methods. It is shown that the number of gridpoints (and therefore the amount of computing time and required storage) is of the same order of magnitude as the one required for solving unperturbed problems.

In this paper, a class of second-order singularly perturbed interior layer problems is examined. A nonpolynomial mixed spline is used to develop the tridiagonal scheme. The developed method is second as well as fourth- order accurate based on the parameters. Error analysis is also carried out. The method is shown to converge point-wise to the true solution with higher accuracy. Linear and nonlinear second-order singularly perturbed boundary value problems have been solved by the presented method. Five numerical illustrations are given to demonstrate the applicability of the proposed method. Absolute errors are given in tables, which show that our method is more efficient than previously existing methods.

In this paper, a class of second-order singularly perturbed interior layer problems is examined. A nonpolynomial mixed spline is used to develop the tridiagonal scheme. The developed method is second as well as fourth- order accurate based on the parameters. Error analysis is also carried out. The method is shown to converge point-wise to the true solution with higher accuracy. Linear and nonlinear second-order singularly perturbed boundary value problems have been solved by the presented method. Five numerical illustrations are given to demonstrate the applicability of the proposed method. Absolute errors are given in tables, which show that our method is more efficient than previously existing methods.

Singularly perturbed robin type boundary value problems with discontinuous source terms applicable in geophysical fluid are considered. Due to the discontinuity, interior layers appear in the solution. To fit the interior and boundary layers, a fitted nonstandard numerical method is constructed. To treat the robin boundary condition, we use a finite difference formula. The stability and parameter uniform convergence of the proposed method is proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, ε , and mesh size, h . The numerical result is tabulated, and it is observed that the present method is more accurate and uniformly convergent with order of convergence of O ( h ) .

Singularly perturbed robin type boundary value problems with discontinuous source terms applicable in geophysical fluid are considered. Due to the discontinuity, interior layers appear in the solution. To fit the interior and boundary layers, a fitted nonstandard numerical method is constructed. To treat the robin boundary condition, we use a finite difference formula. The stability and parameter uniform convergence of the proposed method is proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, ε , and mesh size, h . The numerical result is tabulated, and it is observed that the present method is more accurate and uniformly convergent with order of convergence of O ( h ) .

Book Source: Digital Library of India Item 2015.195127 dc.contributor.author: Rajesh Kumar Bawa dc.date.accessioned: 2015-07-08T05:38:39Z dc.date.available: 2015-07-08T05:38:39Z dc.date.digitalpublicationdate: 2005-08-20 dc.identifier.barcode: 1990010086891 dc.identifier.origpath: /rawdataupload/upload/0086/891 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/195127 dc.description.scannerno: 14 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 175 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher: Indian Institute Of Technology Kanpur dc.rights: Out_of_copyright dc.source.library: Indian Institute Of Technology Kanpur dc.subject.classification: Technology dc.subject.classification: Engineering. Technology In General dc.subject.classification: Mechanical Engineering In General. Nuclear Technology. Electrical Engineering. Machinery dc.subject.keywords: An Aspect Of The Dynamics Of Mechanical Face Seals dc.title: Numerical Solutions Of Singularly Perturbed Initial And Boundary Value Problems Using Splines