Nonpolynomial sextic spline method for the solution along with convergence of linear special case fifth-order two-point boundary value problems

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Abstract

Nonpolynomial sextic spline functions are used to approximate the solution of linear special case fifth-order boundary value problems. Since, presently, the convergence of spline solution of fifth-order boundary value problem is not found in the literature, therefore the convergence of the method is determined which is found to be of fifth order. The convergence of the method is the extension of that developed by Usmani and Sakai [Riaz A. Usmani, Manabu Sakai, Quartic spline solution of the third-order boundary value problems involving third-order differential equations, J. Math. Phys. Sci. 18 (4) (1984) 365–380] for the solution of third-order boundary value problem. The usefulness of the method is illustrated by four examples and confirm the convergence analysis of the method developed.

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 A-1).

The following boundary value problem is to be solved:y(5)(x)+f(x)y(x)=g(x),axb,y(a)=α0,y(b)=α1,y(1)(a)=γ0,y(1)(b)=γ1,y(3)(a)=δ0,where αi, γi, i=0,1 and δ0 are finite real constants and the functions f(x) and g(x) are continuous on [a,b]. The spline functions, developed will have the formTn=Span{1,x,x2,x3,x4,cos(kx),sin(kx)},where 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 [a,b] is divided into n+1 equal subintervals using the grid points xi=a+ih,

i=0,1,,n+1, where h=b-an+1. Consider the following restriction Si of S to each subinterval [xi,xi+1],i=0,1,,n,Si(x)=aicosk(x-xi)+bisink(x-xi)+ci(x-xi)4+di(x-xi)3+ei(x-xi)2+pi(x-xi)+qi.Letyi=Si(xi),mi=Si(1)(xi),Mi=Si(2)(xi),li=Si(5)(xi),,i=0,1,n.Assuming y(x) 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 Y=[y1,y2,,yn]T and Y=[y(x1),y(x2),,y(xn)]T then the method developed leads to the following matrix form(i)MY=C+T,(ii)MY=C,(iii)ME=T,where M=(A+h5BF), C=[c1,c2,,cn]T, T=[t1,t2,,tn]T, and A, B, F are n×n matrices.

AlsoA=1681265621200229922012638375118001263839511200126383551252012638312139001263830510105101510105115101051015101050021256126168,B=100011159523612638300

Convergence analysis

To determine the convergence of the method,

A-1 is required to be calculated. Assuming A-1=(aij) and multiplying the ith row of A-1, with each column of the matrix A, give the following equations:-168ai1+2299220126383ai2-5ai3+ai4=0,126ai1-7511800126383ai2+10ai3-5ai4+ai5=0,-56ai1+9511200126383ai2-10ai3+10ai4-5ai5+ai6=0,212ai1-5512520126383ai2-5ai3-10ai4+10ai5-5ai6+ai7=0,1213900126383ai2+ai3+5ai4-10ai5+10ai6-5ai7+ai8=0,-aij-2+5aij-1-10aij+10aij+1-5aij+2+aij+3=0,j=6,7,,i-1,-aij-2+5aij-1-10aij+10

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 x[0,1], the following boundary value problem is considered:y(5)(x)+y(x)sinx=cosx(1+sinx)+sinx(sinx-1),y(0)=1,y(1)=cos(1)+sin(1),y(1)(0)=1,y(1)(1)=cos(1)-sin(1),y(3)(0)=-1.The analytic solution of the problem (5.1) isy(x)=cos(x)+sin(x).

Example 2

For x[1,2], the following boundary value problem is considered:y(5)(x)+xy(x)=120cos(x)x5-20cos(x)x3+cos(x)x-120sin(x)x6+60sin(x

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 O(h7) along with the consistent end

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