The Chebyshev collocation method for finding the eigenvalues of fourth-order Sturm-Liouville problems

In this work, we have suggested that the Chebyshev collocation method can be employed for detecting the eigenvalues of fourth-order Sturm-Liouville problems. Two examples are presented subsequently. Numerical eventuates indicate that the present method is accurate. Introduction The boundary value problems for ordinary differential equations have a notable role theoretically. Also, they have diverse applications. A great number of physical, biological and chemical phenomena, can be explained through using boundary value problems. In this paper, Chebyshev collocationmethod is used to acquire the solutions for the subsequent fourth order nonsingular Sturm-Liouville problems (q0(x)y ′′(x))′′ + (q1(x)y ′(x))′ + (μv(x)− q2(x))y(x) = 0, a < x < b, (1) or y = F (y(x), y′(x), y′′(x), y′′′(x), μ) (2) or y + p3(x)y ′′′(x) + p2(x)y ′′(x) + p1(x)y ′(x) + (μw(x)− r(x))y(x) = 0 (3) with the four linearly independent homogeneous boundary conditions


Introduction
The boundary value problems for ordinary differential equations have a notable role theoretically. Also, they have diverse applications. A great number of physical, biological and chemical phenomena, can be explained through using boundary value problems. In this paper, Chebyshev collocation method is used to acquire the solutions for the subsequent fourth order nonsingular Sturm-Liouville problems (q 0 (x)y ′′ (x)) ′′ + (q 1 (x)y ′ (x)) ′ + (µv(x) − q 2 (x))y(x) = 0, a < x < b, (1) or y (4) = F (y(x), y ′ (x), y ′′ (x), y ′′′ (x), µ) (2) or y (4) with the four linearly independent homogeneous boundary conditions It is significant to note that y(x) is an accurate solution of (3) and (4). It is not simple to find solutions for fourth order Sturm-Liouville or there are no accurate solutions of Sturm-Liouville problems. There can be various approximate methods. Eq.(1) is mostly defined as the metamorphose of an elastic beam under several boundary conditions [1][2][3][4][5][6][7]. For instance, if we use problem (1) in Lidstone boundary value conditions (y(a) = y(b) = y ′′ (a) = y ′′ (b) = 0), it is applied to model such phenomena as the deflection of elastic beam. Many studies have been conducted on the collocation method in recent writing about the numerical solution of boundary value problem [8][9][10][11]. elik [12][13][14] studied the corrected collocation method to give an approximate calculation of eigenvalues of Sturm-Liouville and periodic Sturm-Liouville problems by simplified Chebyshev series.

Chebyshev collocation method
Fourth-order nonsingular Sturm-Liouville problems with variable coefficients are of the form where q 2 (x), q 1 (x), q 0 (x) and w(x) are piecewise continuous functions with q 0 (x), w(x) ≥ 0. The most popular homogeneity boundary condition is Because finite range a ≤ t ≤ b can be converted to the basic range −1 ≤ x ≤ 1 with the change of variable w = (b−a)x+b+a 2 , sans losing generality, we presume −1 ≤ x ≤ 1. In this method, for ( 3) we offer an evident solution of the form where T i (x) i = 0, 1, ..., N are the Chebyshev polynomials and a i are indeterminate parameters. The suggested solution (5) can be explained in the matrix form as where Collocation points may be taken as which denominates the turning points. If we change the Chebyshev collocation points in (3), the subsequent matrix expression can be obtained: where, and The kth derivative of (5) with respect to x can be written as can be written in the matrix form as or matrix equation where . . . . . . . . .
The relevance between the Chebyshev coefficient matrix A of y(x) and Chebyshev coefficient matrix can be illustrated as

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where if N is even. Accordingly, the matrix demonstration of (8) and the matrix equation (6) can be expressed in order as Moreover, the boundary conditions can be written in the matrix form as By using the subsequent display: 2, 3, ..., N − 1, j = 0, 1, 2, ..., N, Matrix equation can be displayed as where The rank of the matrix W is N + 1. This is a set of equations for the uncertain parameter a k i . This set of equations has a non obvious solution only if the determinant of the coefficients matrix vanishes. This gives an equation of degree N − 1 in µ and has N − 1 roots which are the first N − 1 numerical eigenvalues of the main problem.

Numerical results
After applying the procedure described in previous section, we produce two numerical results of the fourth order Sturm-Liouville problem in this section. Example 1. Assume the following fourth order eigenvalue problem subject to We can choose the trial solution as where T i (x), i = 0, 1, ..., N are the chebyshev polynomial and a i are non specified parameters. Through applying the chebyshev collocation method, points can be obtained as

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If we change the Chebyshev collocation points in (12), we will get the expression below or it could be written in the matrix form The fourth order derivative of (14) with respect to x may be written as where a We can choose the trial solution as where T i (x), i = 0, 1, ..., N are presumed the Chebyshev polynomial and a i are undetermined parameters. We apply the same method as we did in Example 1. Therefore, the first four approximate eigenvalues of the (15-17) are given in Table 1.