Nonpolynomial sextic spline method for the solution along with convergence of linear special case fifth-order two-point boundary value problems
Introduction
The solutions of fifth-order boundary value problems are not very much found in the literature. These problems are generally arise in the mathematical modelling of viscoelastic flows [1], [2]. The conditions for the existence and uniqueness of the solution of such problems can be found in [3]. Caglar et al. [5] proposed the solution of fifth-order boundary value problems using sixth degree B-spline and the method is observed to be first-order convergent. Siddiqi and Ghazala [6] developed polynomial sextic spline method for the solution of fifth-order boundary value problems and the method is observed to be second-order convergent. Siddiqi and Ghazala [7] used nonpolynomial spline for the numerical solution of the fifth-order linear special case boundary value problems. Khan et al. [8] developed an algorithm for the solution of fifth-order boundary value problem using nonpolynomial sextic spline and the algorithm is claimed to be second and fourth-order convergent without proof (i.e. without determining ).
The following boundary value problem is to be solved:where αi, γi, and δ0 are finite real constants and the functions f(x) and g(x) are continuous on . The spline functions, developed will have the formwhere k is the frequency of the trigonometric part and k could be real or pure imaginary. The paper is organized in five sections. Using derivatives continuities at knots, the consistency relation in terms of values of spline and its fifth derivatives at knots along with consistent end conditions are determined in Section 2.
In Section 3, the nonpolynomial sextic spline solution approximating the analytic solution of the BVP (1.1) is determined. The error bound of the solution is determined in Section 4. In Section 5, four examples are considered for the usefulness of the method developed.
Section snippets
Preliminary results
To develop the nonpolynomial spline approximation to the fifth-order boundary value problem (1.1), the interval is divided into equal subintervals using the grid points ,
where . Consider the following restriction Si of S to each subinterval LetAssuming to be the exact solution of the BVP (1.1) and yi
Nonpolynomial spline solution
Nonpolynomial spline solution of the BVP (1.1) is obtained using (2.15), (2.16), (2.14), (2.17). If and then the method developed leads to the following matrix formwhere , C, , and A, B, F are matrices.
Also
Convergence analysis
To determine the convergence of the method,
is required to be calculated. Assuming and multiplying the ith row of with each column of the matrix A, give the following equations:
Numerical examples
Four examples are considered in this section to check the accuracy of the method and the results are summarized in Table 1, Table 2. Example 1 For , the following boundary value problem is considered:The analytic solution of the problem (5.1) is Example 2 For , the following boundary value problem is considered:
Conclusion
Nonpolynomial sextic spline has been developed for the approximate solution of linear special case fifth-order boundary value problem and the method is proved to be fifth-order convergent. Numerical examples confirm the order of the method developed. The boundary conditions are considered in terms of first and third derivatives. It may be mentioned that such boundary conditions have not been considered before. The order of method has also been improved to along with the consistent end
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