Spline Collocation Method for Solving Boundary Value Problems

1989

Collocation methods based on quintic splines are fonnulated and analyzed for the secondorder two-point boundary value problems with mixed boundary conditions. These methods determine quintic spline approximation to the solution of the boundary value problem. by forcing lhe approximating solution to satisfy the given operator equation, or a perturbed one at the nodes, the boundary conditions and auxiliary end conditions. The methods that are based on the initial operator equation produce not optimal approximation. as compared to lhe corresponding interpolation procedures. This paper derives appropriate perturbations of lhe initial differential equation, such. that the application of the collocation procedure leads 10 optimal approximating schemes. The theoretical behavior of the method has been verified numerically on a variety of benchmark problems found in !.he literature. L INTRODUCTION In !.his paper we consider a collocation of the solul..ion u of the second order two-point boWl...

International Journal for Numerical Methods in Engineering, 1988

A new collocation method based on quadratic splines is presented for second order two-point boundary value problems. First, O(h4) approximations to the first and second derivative of a function are derived using a quadratic-spline interpolant of u. Then these approximations are used to define an O(h4) perturbation of the given boundary value problem. Second, the perturbed problem is used to define a collocation approximation at interval midpoints for which an optimal O(h3 -j ) global estimate for the jth derivative of the error is derived. Further, O(h4-j) error bounds for the jth derivative are obtained for certain superconvergence points. It should be observed that standard collocation at midpoints gives O(h2 -J ) bounds. Results from numerical experiments are reported that verify the theoretical behaviour of the method.

2013

Collocation methods based on quintic splines are fonnulated and analyzed for the secondorder two-point boundary value problems with mixed boundary conditions. These methods determine quintic spline approximation to the solution of the boundary value problem. by forcing lhe approximating solution to satisfy the given operator equation, or a perturbed one at the nodes, the boundary conditions and auxiliary end conditions. The methods that are based on the initial operator equation produce not optimal approximation. as compared to lhe corresponding interpolation procedures. This paper derives appropriate perturbations of lhe initial differential equation, such. that the application of the collocation procedure leads 10 optimal approximating schemes. The theoretical behavior of the method has been verified numerically on a variety of benchmark problems found in !.he literature. L INTRODUCTION In !.his paper we consider a collocation of the solul..ion u of the second order two-point boWl...

2010

Abstract: Collocation method with sixth degree B-splines as basis functions has been developed to solve a fifth order special case boundary value problem. To get an accurate solution by the collocation method with sixth degree B-splines, the original sixth degree B-splines are redefined into a new set of basis functions which in number match with the number of collocation points. The method is tested for solving both linear and nonlinear boundary value problems. The proposed method is giving better results when compared with the methods available in literature.

International Journal of Computing Science and Mathematics, 2016

In this paper, an over-determined, global collocation method based upon B-spline basis functions is presented for solving boundary value problems in complex domains. The method was truly meshless approach, hence simple and efficient to programme. In the method, any governing equations were discretised by global B-spline approximation as the B-spline interpolants. As the interpolating B-spline basis functions were chosen, the present method also posed the Kronecker delta property allowing boundary conditions to be incorporated efficiently. The present method showed high accuracy for elliptic partial differential equations in arbitrary domain with Neumann boundary conditions. For coupled Poisson problems with complex Neumann boundary conditions, the boundary collocation approach was adopted and applied in a simple and less costly manner to further improve the accuracy and stability. Applications from elasticity problems were given to demonstrate the efficacy and capability of the present method. In addition, the relation between accuracy and stability for the method was better justified by the new effective condition number given in literature.

International Journal for Computational Methods in Engineering Science and Mechanics, 2019

Objective: A higher order numerical scheme for two-point boundary interface problem with Dirichlet and Neumann boundary condition on two different sides is propounded. Methods: Orthogonal cubic spline collocation techniques have been used (OSC) for the two-point interface boundary value problem. To approximate the solution a piecewise Hermite cubic basis functions have been used. Findings: Remarkable features of the OSC are accounted for the numerous applications, theoretical clarity, and convenient execution. The stability and efficiency of orthogonal spline collocation methods over B-splines have made the former more preferable than the latter. As against finite element methods, determining the approximate solution and the coefficients of stiffness matrices and mass is relatively fast as the evaluation of integrals is not a requirement. The systematic incorporation of boundary and interface conditions in OSC adds to the list of advantages of preferring this method. Novelty: As against the existing methodologies it becomes clear from our findings that OSC is dominantly computationally superior. A computational treatment has been implemented on the two-point interface boundary value problem with super-convergent results of derivative at the nodal points, being the noteworthy finding of the study.

Journal of American …, 2010

Second and fourth order convergent methods based on Quartic nonpolynomial spline function are presented for the numerical solution of a third order two-point boundary value problem. The proposed approach gives better approximations than existing polynomial spline and finite difference methods and has a lower computational cost. Convergence analysis of the proposed method is discussed; two numerical examples are included to illustrate the efficiency of the method.

2014

A finite element method involving collocation method with sextic B-splines as basis functions has been developed to solve eighth order boundary value problems. The sixth order, seventh order and eighth order derivatives for the dependent variable are approximated by the central differences of fifth order derivatives. The basis functions are redefined into a new set of basis functions which in number match with the number of collocated points selected in the space variable domain. The proposed method is tested on several linear and non-linear boundary value problems. The solution of a non-linear boundary value problem has been obtained as the limit of a sequence of solutions of linear boundary value problems generated by quasilinearization technique. Numerical results obtained by the present method are in good agreement with the exact solutions available in the literature.

European Journal of Pure and Applied Mathematics, 2021

In this research, second order linear two-point boundary value problems are treated using new method based on hybrid cubic B-spline. The values of the free parameter,Gamma , are chosen via optimization. The value of the free parameter plays an important role in giving accurate results. Optimization of this parameter is carried out. This method is tested on four examples and a comparison with cubic B-spline, trigonometric cubic B-spline and extended cubic B-spline methods has been carried out. The examples suggest that this method produces more accurate results than the other three methods. The numerical results are presented to illustrate the efficiency of our method.

Applied and Computational Mechanics, 2020

Several applications of computational science and engineering, including population dynamics, optimal control, and physics, reduce to the study of a system of second-order boundary value problems. To achieve the improved solution of these problems, an efficient numerical method is developed by using spline functions. A non-polynomial cubic spline-based method is proposed for the first time to solve a linear system of second-order differential equations. Convergence and stability of the proposed method are also investigated. A mathematical procedure is described in detail, and several examples are solved with numerical and graphical illustrations. It is shown that our method yields improved results when compared to the results available in the literature.

2013

A finite element method involving collocation method with quintic B-splines as basis functions has been developed to solve tenth order boundary value problems. The fifth order, sixth order, seventh order, eighth order, ninth order and tenth order derivatives for the dependent variable are approximated by the central differences of fourth order derivatives. The basis functions are redefined into a new set of basis functions which in number match with the number of selected collocated points in the space variable domain. The proposed method is tested on several linear and non-linear boundary value problems. The solution of a non-linear boundary value problem has been obtained as the limit of a sequence of solutions of linear boundary value problems generated by quasilinearization technique. Numerical results obtained by the present method are in good agreement with the exact solutions available in the literature.

Global Journal of Research In Engineering, 2012

A finite element method involving collocation method with quintic B-splines as basis functions have been developed to solve sixth order boundary value problems. The sixth order and fifth order derivatives for the dependent variable are approximated by the central differences of fourth order derivatives. The basis functions are redefined into a new set of basis functions which in number match with the number of collocated points selected in the space variable domain. The proposed method is tested on several linear and non-linear boundary value problems. The solution of a non-linear boundary value problem has been obtained as the limit of a sequence of solutions of linear boundary value problems generated by quasilinearization technique. Numerical results obtained by the present method are in good agreement with the exact solutions or numerical solutions available in the literature.

This paper is an extension of Mamadu and Ojobor (2017) were the efficiency of the collocation method was considered based on the type of basis function in developing the scheme. Here, we investigate the convergence of the method as applied to second order boundary value problems (BVPs) at the various collocation points: Gauss-Lobatto (G-L), Gauss-Chebychev (G-C) and Gauss-Radau (G – R) collocation points. Also, the class of Chebychev polynomials of the first kind have been adopted as basis function. We have employed Maple 18 software in our analysis and computations. Introduction Let the generalized form of a differential equation be given as í µí°¿ í µí±¦ í µí±¥ = í µí±” í µí±¥ , í µí¼‘ í µí±¦ í µí±Ž 1 = í µí±Ž, í µí¼ í µí±¦ í µí±Ž 2 = í µí±, (1.0) where í µí»¼, í µí¼‘ and í µí¼ are considered as differential operators. Differential equation are often applied in the construction and development of most mathematical models such as predictive control in AP monitor (Hedengen et. al., 2014), temperature distribution in cylindrical conductor (Fortini et. al., 2008), dynamic optimization (cizinar et. al., 2015), etc. Modeling is the bridge between the subject and real-life situations for students realization. Differential equations model real-life situations, and provide the real-life answers with the help of computer calculations and graphics. Investigation into methods for solving these problems has been on the increase in recent years. Obviously, many methods (analytical or numerical methods) have been developed and implemented by many researchers. Of these methods, the numerical methods seem to be more popular than their analytic counterpart due to their adequate approximation of the analytic solution in a rapid converging series. Popular numerical methods include; the Tau method (Adeniyi, 2004), orthogonal collocation method (Mamadu and Ojobor, 2017), Tau-Collocation method (Mamadu and Njoseh, 2016), Variation iteration decomposition method (Ojobor and Mamadu, 2017), Elzaki transform method (Mamadu and Njoseh, 2017), Power series approximation method (Njoseh and Mamadu, 2016a), Modified power series approximation method (Njoseh and Mamadu, 2017), etc. However, the collocation method remains one of the best numerical method due to its level of simplicity and accuracy. Moreover, the efficiency of the method is dependent on the class of basis function and the collocation point adopted in constructing the scheme. There exist different types of basis functions that can be adopted to construct the scheme such as; canonical polynomials, Chebychev polynomials, Bernoulli polynomials, Lagrange polynomials (Fox and Pascal, 1968; Lanczos, 1938). And, the different collocations that can be adopted include; the equally spaced points; Gauss-lobotto points, Gauss-Chebychev points, Gauss-Radau point, etc. These points improves better than one another in terms of convergence.

Applied Mathematics and Computation, 2006

A quadratic non-polynomial spline functions based method is developed to find approximations solution to a system of second-order boundary-value problems associated with obstacle, unilateral, and contact problems. The present approach has less computational cost and gives better approximations than those produced by other collocation, finite-difference and spline methods. Convergence analysis of the method is discussed. A numerical example is given to illustrate practical usefulness of the new method.

European Journal of Pure and Applied Mathematics

This paper introduces a novel trigonometric B-spline collocation method for solving a specific class of second-order boundary value problems. The study showcases the method’s practicality and effectiveness through various numerical examples. Furthermore, it evaluates the technique’s performance by calculating maximum errors for different step sizes in the spatial domain. The paper also conducts a comparative analysis with alternative methods, demonstrating the superior accuracy of the trigonometric B-spline approach.