A high order B-spline collocation method for linear boundary value problems

Jator, S. N.; Sinkala, Zachariah

doi:10.1016/j.amc.2007.02.027

Cited by 15 publications

(12 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As the famous articles on splines claim finding suitable bound for ‖ B −1 ‖ ∞ of m th order B‐splines in terms of 2 j + m − 1, the number of collocation points is not an easy problem and the full analysis of this argument as far as we know has not been yet established. Although, as we checked numerically for m < 18, we noticed that when the maximum points of the B‐splines (Botella) or Greville points over [0,1] are used as interpolation points, the bound of ‖ B −1 ‖ ∞ depends only on m and does not depend on j for fixed m . Figure illustrates this situation just for m < 10.…”

Section: Some Definitions and Elementary Resultsmentioning

confidence: 99%

B‐spline spectral method for constrained fractional optimal control problems

Ejlali

Hosseini

Yousefi

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we present a direct B‐spline spectral collocation method to approximate the solutions of fractional optimal control problems with inequality constraints. We use the location of the maximum of B‐spline functions as collocation points, which leads to sparse and nonsingular matrix B whose entries are the values of B‐spline functions at the collocation points. In this method, both the control and Caputo fractional derivative of the state are approximated by B‐spline functions. The fractional integral of these functions is computed by the Cox‐de Boor recursion formula. The convergence of the method is investigated. Several numerical examples are considered to indicate the efficiency of the method.

show abstract

Section: Some Definitions and Elementary Resultsmentioning

confidence: 99%

B‐spline spectral method for constrained fractional optimal control problems

Ejlali

Hosseini

Yousefi

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…Although, the order of accuracy for collocation methods are known to be suboptimal, choosing appropriate interior points is critical to obtain optimal order of accuracy [20]. To solve the equations, each tubule and medulla were divided with P interior points (called collocation points) that were chosen as roots of an orthogonal Jacobi polynomial of P-th degree.…”

Section: Boundary Conditionsmentioning

confidence: 99%

“…The purpose of this work is to explain an efficient and stable method, based on orthogonal collocation scheme, to solve numerically the differential-algebraic equations (DAEs) arising in steady-state models of renal concentrating mechanism. Because of simplicity, inherent efficiency for applications to boundary value problems and optimal order accuracy for the error, the orthogonal collocation method is widely used [20]. In Newton-based methods, to minimize the truncation error which is introduced by converting differential equations to algebraic ones, the discretization number must be large.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Numerical Method and Parametric Study for Electrolyte Transport in the Renal Medulla

Saadatmand¹,

Abdekhodaie²,

Pishvaie³

et al. 2014

JBiSE

View full text Add to dashboard Cite

Mathematical models of the mamalian urine concentrating mechanism consist of a large system of coupled, nonlinear and stiff equations. An efficient numerical orthogonal collocation method was employed to solve the steady-state formulation of urine concentrating mechanism. This method was used to solve the stiff and high order equations of electrolyte transport in a central core, single nephron model of the renal outer medulla. The presented results were in good agreement with implicit finite difference method's results, but this new method was faster and more stable. Due to the greater stability and larger convergence domain of collocation method over Newton's method, a parametric study on concentrated urine was investigated. The results showed that this model was sensitive to non-ideal countercurrent exchange between medulla interstitium and vasa recta. Although distal tubule lies in the cortex interstitium, it affects on the inflow to the collecting duct. This study showed that the effect of changing membrane transport properties of distal tubule wall on properties of outflow from the outer medulla collecting duct was considerable.

show abstract

“…Over the years, many researchers have worked on boundary value problems by using different methods for numerical solutions [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. So far various numerical methods such as Cubic spline method [2], Iterative methods [3], Perturbed collocation method [4], Modified decomposition method [5], Decomposition method [6], Differential transform method [7,8], A higher order B-spline collocation method [9], Homotopy perturbation method [10], Variational iteration technique [11], Sinc-Galerkin method [12], Spline techniques [13][14][15][16], Galerkin method with quintic B-splines [17], B-spline collocation method [18], Cubic B-spline collocation method [19], Galerkin method with cubic B-splines [20] have been employed to solve fourth order boundary balue problems. So far, fourth order boundary value problems have not been solved by using Petrov-Galerkin method with cubic B-splines as basis functions and quintic B-splines as weight functions .…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Fourth Order Boundary Value Problems by Petrov-Galerkin Method with Cubic B-splines as basis Functions and Quintic B-Splines as Weight Functions

KasiViswanadham¹,

Reddy²

2014

IJCA

View full text Add to dashboard Cite

This paper deals with a finite element method involving Petrov-Galerkin method with cubic B-splines as basis functions and quintic B-splines as weight functions to solve a general fourth order boundary value problem with a particular case of boundary conditions. The basis functions are redefined into a new set of basis functions which vanish on the boundary where the Dirichlet type of boundary conditions are prescribed. The weight functions are also redefined into a new set of weight functions which in number match with the number of redefined basis functions. The proposed method was applied to solve several examples of fourth order linear and nonlinear boundary value problems. The obtained numerical results were found to be in good agreement with the exact solutions available in the literature.

show abstract

A high order B-spline collocation method for linear boundary value problems

Cited by 15 publications

References 9 publications

B‐spline spectral method for constrained fractional optimal control problems

B‐spline spectral method for constrained fractional optimal control problems

An Efficient Numerical Method and Parametric Study for Electrolyte Transport in the Renal Medulla

Numerical Solution of Fourth Order Boundary Value Problems by Petrov-Galerkin Method with Cubic B-splines as basis Functions and Quintic B-Splines as Weight Functions

Contact Info

Product

Resources

About