Shahid S. Siddiqi

Spline solutions of linear sixth-order boundary-value problems

Linear, sixth-order boundary-value problems (special case) are solved, using polynomial splines of degree six. The spline function values at the midknots of the interpolation interval, and the corresponding values of the even-order derivatives are related through consistency relations. The algorithm developed approximates the solutions, and their higher-order derivatives, of differential equations. Two numerical illustrations are given to show the practical usefulness of the algorithm developed. It is observed that this algorithm is second-order convergent.

Solutions of 12th order boundary value problems using non-polynomial spline technique

Abstract Non-polynomial spline is used for the numerical solutions of the 12th order linear special case boundary value problems. The end conditions are derived for the definition of spline. Siddiqi and Twizell [S.S. Siddiqi, E.H. Twizell, Spline solutions of linear twelfth-order boundary value problems, Int. J. Comput. Math. 78 (1998) 345–362] presented the solutions of 12th order boundary value problems using 12th degree spline, where some unexpected results for the solution and its derivatives were obtained, near the boundaries of the interval. No such situation is observed in this method, near the boundaries of the interval and the results are better in the whole interval. Out of three, two examples compared with that considered by Siddiqi and Twizell (1998), show that the method developed in the paper is more efficient.

Spline solutions of linear eighth-order boundary-value problems

Abstract Linear, eighth-order boundary-value problems (special case) are solved, using polynomial splines of degree six. The spline function values at the midknots of the interpolation interval, and the corresponding values of the even-order derivatives are related through consistency relations. The algorithm developed approximates the solution, and their higher-order derivatives, of differential equations. Four numerical illustrations are given to show the practical usefulness of the algorithm developed. It is observed that this algorithm is second-order convergent.

Spline solutions of linear twelfth-order boundary-value problems

Abstract Linear, twelfth-order boundary-value problems (special case) are solved, using polynomial splines of degree 12. The spline function values at the midknots of the interpolation interval, and the corresponding values of the even-order derivatives are related through consistency relations. The algorithm developed approximates the solutions, and their higher-order derivatives, of differential equations. Two numerical illustrations are given to show the practical usefulness of the algorithm developed. It is observed that this algorithm is second-order convergent.

Solution of fifth order boundary value problems using nonpolynomial spline technique

Abstract Nonpolynomial spline is used for the numerical solutions of the fifth order linear special case boundary value problems. End conditions for the definition of spline are taken from Siddiqi and Akram [S.S. Siddiqi, Ghazala Akram, Sextic spline solutions of fifth order boundary value problems, Appl. Math. Lett., submitted for publication], consistent with the fifth order boundary value problem. Caglar et al. [H.N. Caglar, S.H. Caglar, E.H. Twizell, The numerical solution of fifth-order boundary-value problems with sixth-degree B-spline functions, Appl. Math. Lett. 12 (1999) 25–30] presented the solution of fifth order boundary value problem using sixth degree B-spline, is first order convergent. The method presented in this paper is observed to be a second order convergent. For the numerical illustration of the method developed, an example has also been considered.

Nonic spline solutions of eighth order boundary value problems

Nonic spline is used for the numerical solutions of the eighth order linear special case boundary value problems. The end conditions are derived for the definition of spline. The algorithm developed not only approximates the solutions, but their higher order derivatives as well. The method presented in this paper has also been proved to be second order convergent. Two examples compared with those considered by Inc et al. [M. Inc, D.J. Evans, An efficient approach to approximate solutions of eighth-order boundary-value problems, Int. J. Comput. Math. 81 (6) (2004) 685-692] and Siddiqi et al. [S.S. Siddiqi, E.H. Twizell, Spline solution of linear eighth-order boundary value problems, Comput. Methods Appl. Mech. Eng. 131 (1996) 309-325], show that the method developed in this paper is more efficient.

Solutions of tenth-order boundary value problems using eleventh degree spline

Abstract Numerical solutions of the tenth-order linear special case boundary value problems are obtained using eleventh degree spline. The end conditions consistent with the BVP, are also derived. Siddiqi and Twizell [S.S. Siddiqi, E.H. Twizell, Spline solutions of linear tenth order boundary value problems, Int. J. Comput. Math. 68 (1998) 345–362] presented the solutions of tenth-order boundary value problems using tenth degree spline, where some unexpected results for the solution and higher order derivatives were obtained near the boundaries of the interval. No such unexpected situation is observed in this method, near the boundaries of the interval and the results are better in the whole interval. The algorithm developed approximates the solutions, and their higher order derivatives. Numerical illustrations are tabulated to compare the errors with those considered by Siddiqi and Twizell [S.S. Siddiqi, E.H. Twizell, Spline solutions of linear tenth order boundary value problems, Int. J. Comput. Math. 68 (1998) 345–362] and the method is observed to be better.

Solution of eighth-order boundary value problems using the non-polynomial spline technique

The non-polynomial spline technique is used for the numerical solution of eighth-order linear special case boundary value problems. The method presented in this paper has also been proven to be second-order convergent. To compare the method developed in this paper with those developed by Inc and Evans, and Siddiqi and Twizell, two examples are considered and it is observed that our method is more efficient.

Quintic spline solution of linear sixth-order boundary value problems

Abstract There are few techniques available to numerically solve sixth-order boundary value problems with two point boundary conditions. In this paper, quintic spline method is developed for the numerical solutions of linear special case sixth-order boundary value problems. Two examples are considered for the numerical illustration of the method developed. The method is also compared with those developed by EL-Gamel et al. [Mohamed EL-Gamel, John R. Cannon, Ahmed I. Zayed, Sinc–Galerkin method for solving sixth-order BVPs, Math. Comput. 73 (247) (2003)] and Wazwaz [A. Wazwaz, The numerical solution of sixth-order boundary value problems by the modified decomposition method, Appl. Math. Comput. 118 (2001) 311–325]. The comparison shows that the quintic spline method is more efficient and effective tool and yields better results.

A ternary three-point scheme for curve designing

In this paper, a stationary ternary three-point approximating subdivision scheme is presented, which generates C 2 limiting curve and its limiting function has a support on [−3, 2]. The analysis of the scheme is shown using the Laurent polynomial method. Two examples are illustrated for comparison with the ternary three-point approximating subdivision scheme developed by Hassan and Dodgson [Ternary and three point univariate subdivision schemes, in Curve and Surface Fitting: Sant-Malo 2002, A. Cohen, J.-L. Merrien, and L.L. Schumaker, eds., Nashboro Press, Brentwood, 2003, pp. 199–208.], which show that the scheme presented in the paper is better.