A class of Birkhoff–Lagrange-collocation methods for high order boundary value problems

Costabile, F.; Napoli, Anna

doi:10.1016/j.apnum.2016.12.003

Cited by 16 publications

(14 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof From the results in [20] it follows that the boundary value problem (7a)-(7b) is equivalent to the Fredholm type integral equation…”

Section: The Lidstone-euler Boundary Value Problemmentioning

confidence: 99%

“…Only a limited number of this type of problems can be solved analytically. For this reason many researchers have proposed numerical solutions (see, for instance, [1,13,[19][20][21]33,34,43]). Different approaches have been considered, among which collocation, also with spline functions and Lagrange interpolation, Galerkin weighted residual, decomposition, variational techniques.…”

Section: The Numerical Solutionmentioning

confidence: 99%

“…Here we present a general method for higher order LEbvp, inspired to the idea in [20]. Next we consider in more detail the important case r = 2, that arises in many applications.…”

Section: The Numerical Solutionmentioning

confidence: 99%

See 2 more Smart Citations

Lidstone–Euler interpolation and related high even order boundary value problem

Costabile

Gualtieri

Napoli

2021

Calcolo

Self Cite

View full text Add to dashboard Cite

We consider the Lidstone–Euler interpolation problem and the associated Lidstone–Euler boundary value problem, in both theoretical and computational aspects. After a theorem of existence and uniqueness of the solution to the Lidstone–Euler boundary value problem, we present a numerical method for solving it. This method uses the extrapolated Bernstein polynomials and produces an approximating convergent polynomial sequence. Particularly, we consider the fourth-order case, arising in various physical models. Finally, we present some numerical examples and we compare the proposed method with a modified decomposition method for a tenth-order problem. The numerical results confirm the theoretical and computational ones.

show abstract

“…Proof From the results in [20] it follows that the boundary value problem (7a)-(7b) is equivalent to the Fredholm type integral equation…”

Section: The Lidstone-euler Boundary Value Problemmentioning

confidence: 99%

Section: The Numerical Solutionmentioning

confidence: 99%

See 1 more Smart Citation

Lidstone–Euler interpolation and related high even order boundary value problem

Costabile

Gualtieri

Napoli

2021

Calcolo

Self Cite

View full text Add to dashboard Cite

show abstract

“…Even finite difference methods produce acceptable results for many BVPs, their local order refinement (p-refinement) is not easy task due to the direct disconnection of the higher order formulations. Another kind of direct method is the collocation methods which are based on direct substitutions of approximate solutions and getting algebraic equations at collocation points [11][12][13][14][15][16]. These types of methods are suitable for local order refinement and can also be used in either local form or global form.…”

Section: Introductionmentioning

confidence: 99%

A new implicit-explicit local differential method for boundary value problems

Tunc¹,

Sarı²

2021

Turk J Math

View full text Add to dashboard Cite

In this study, an effective numerical method based on Taylor expansions is presented for boundary value problems. This method is arbitrary directional and called as implicit-explicit local differential transform method (IELDTM). With the completion of this study, a reliable numerical method is derived by optimizing the required degrees of freedom. It is shown that the order refinement procedure of the IELDTM does not affect the degrees of freedom. A priori error analysis of the current method is constructed and order conditions are presented in a detailed analysis. The theoretical order expectations are verified for nonlinear BVPs. Stability of the IELDTM is investigated by following the analysis of approximation matrices. To illustrate efficiency of the method, qualitative and quantitative results are presented for various challenging BVPs. It is tested that the current method is reliable and accurate for a broad range of problems even for strongly nonlinear BVPs. The produced results have revealed that the IELDTM is more accurate than the existing ones in literature.

show abstract

“…It is noteworthy that the main idea for such a construction is inspired from the design of well-conditioned collocation methods using the notion of the Birkhoff interpolation (cf. [7,19,32,48,56,57]). Here, we also estimate the interpolation error of this type of the generalised Birkhoff interpolation and demonstrate that the new approach can provide very accurate approximation to a class of time-fractional PDEs with smooth source terms.…”

Section: Introductionmentioning

confidence: 99%

A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Differential Equations

Liu

2019

Commun. Appl. Math. Comput.

View full text Add to dashboard Cite

In this paper, we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches, the nonstandard singular basis functions are defined from some generalised Birkhoff interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such, the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs, leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems.

show abstract

A class of Birkhoff–Lagrange-collocation methods for high order boundary value problems

Cited by 16 publications

References 24 publications

Lidstone–Euler interpolation and related high even order boundary value problem

Lidstone–Euler interpolation and related high even order boundary value problem

A new implicit-explicit local differential method for boundary value problems

A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Differential Equations

Contact Info

Product

Resources

About