Quintic spline solution of linear sixth-order boundary value problems

Siddiqi, Shahid S.; Akram, Ghazala; Nazeer, Saima

doi:10.1016/j.amc.2006.11.178

Cited by 28 publications

(26 citation statements)

References 8 publications

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“…If we want the solution at the points, there is a need of interpolating these values. Also, spline functions of Hermite types has been used by many authors for solving these problems [6,[9][10][11][12] . The literature on the numerical solutions of initial value problems by using lacunary spline functions is not too much.…”

Section: Introductionmentioning

confidence: 99%

The Existence, Uniqueness and Error Bounds of Approximation Splines Interpolation for Solving Second-Order Initial Value Problems

Mm¹

2009

Journal of Mathematics and Statistics

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Problem statement: The lacunary interpolation problem, which we had investigated in this study, consisted in finding the six degree spline S(x) of deficiency four, interpolating data given on the function value and third and fifth order in the interval [0,1]. Also, an extra initial condition was prescribed on the first derivative. Other purpose of this construction was to solve the second order differential equations by two examples showed that the spline function being interpolated very well. The convergence analysis and the stability of the approximation solution were investigated and compared with the exact solution to demonstrate the prescribed lacunary spline (0, 3, 5) function interpolation. Approach: An approximation solution with spline interpolation functions of degree six and deficiency four was derived for solving initial value problems, with prescribed nonlinear endpoint conditions. Under suitable assumptions, the existences; uniqueness and the error bounds of the spline (0, 3, 5) function had been investigated; also the upper bounds of errors were obtained. Results: Numerical examples, showed that the presented spline function proved their effectiveness in solving the second order initial value problems. Also, we noted that, the better error bounds were obtained for a small step size h. Conclusion: In this study we treated for a first time a lacunary data (0,3,5) by constructing spline function of degree six which interpolated the lacunary data (0,3,5) and the constructed spline function applied to solve the second order initial value problems.

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Section: Introductionmentioning

confidence: 99%

The Existence, Uniqueness and Error Bounds of Approximation Splines Interpolation for Solving Second-Order Initial Value Problems

Mm¹

2009

Journal of Mathematics and Statistics

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“…As the models considered in applied sciences and engineering are nonlinear in nature, only seldom are analytical solutions available [2] . Siddiqi and Akram [7] used the quintic spline to find approximation solution of fourth order boundary-value problems. In several years various authors have been used spline functions for finding an approximate solution of initial value problem including as [3,4] studies the for third and fourth order boundary value problem.…”

Section: Intoductionmentioning

confidence: 99%

On Sixtic Lacunary Spline Solutions of Second Order Initial Value Problem

Faraj¹

2009

J. of Mathematics and Statistics

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Problem statement:The lacunary problem, which we had investigated in this study, consider in finding the spline function of degree six S(x) of deficiency four, interpolating data given on the function value and third, forth order in the interval [0,1]. Also, on the extra initial condition was prescribed on the first derivative. Other purpose of this construction was to solve the second order initial value problem by one example showed that the spline function being interpolation very well compare to [1]. The convergence analysis and the stability of approximation solution were investigated and compared with the exact solution to demonstrate the prescribed lacunary spline (0,3,4) function interpolation. Approach: An approximation with spline functions of degree six and deficiency four is developed for solving initial value problems, with prescribed nonlinear endpoint conditions. Under suitable assumptions with applications showed this spline of the type (0,3,4) are existences, uniqueness and error bounds of the deficient of the solution. Result: Numerical example showed that the presented spline function their effectiveness in solving the second order initial value problem and also showed that our result more well to result in [1]. Also, we note that, the better error bounds were obtained for small step size h. Conclusion: In this study we showed that the lacunary data (0,3,4) are more well approximate to the given second order initial value problem compare with the lacunary data (0,3,5) used in [1]. Key words: Spline function, mathematical model, second orders differential equations INTODUCTIONThe lacunary interpolation problem we investigate in this paper consists in finding the sixtic spline S(x) of deficiency four, interpolating data given on the function and its Cauchy's problem for second derivative at a given set of nodes of the interval [0,1]. The initial value problems play an important role in mathematical physics. As the models considered in applied sciences and engineering are nonlinear in nature, only seldom are analytical solutions available [2] . Siddiqi and Akram [7] used the quintic spline to find approximation solution of fourth order boundary-value problems. In several years various authors have been used spline functions for finding an approximate solution of initial value problem including as [3,4] studies the for third and fourth order boundary value problem.Consider the second order initial value problem: 0 0 0 0

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“…Akram and Siddiqi [8] solved the boundary value problem of type (1)- (2) with non-polynomial spline technique. Siddiqi et al [9] solved the same boundary value problems using quintic splines. Also Siddiqi and Akram [10] used septic splines to solve the boundary value problems of type (1)-(2).…”

Section: Introductionmentioning

confidence: 99%

Cubic B-Spline Collocation Method for Sixth Order Boundary Value Problems

2017

International Journal of Scientific and Innovative Mathematical

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Quintic spline solution of linear sixth-order boundary value problems

Cited by 28 publications

References 8 publications

The Existence, Uniqueness and Error Bounds of Approximation Splines Interpolation for Solving Second-Order Initial Value Problems

The Existence, Uniqueness and Error Bounds of Approximation Splines Interpolation for Solving Second-Order Initial Value Problems

On Sixtic Lacunary Spline Solutions of Second Order Initial Value Problem

Cubic B-Spline Collocation Method for Sixth Order Boundary Value Problems

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