Abstract
We propose a numerical Taylor's Decomposition method to compute approximate eigenvalues and eigenfunctions for regular Sturm-Liouville eigenvalue problem and nonlinear Euler buckling problem very accurately for relatively large step sizes. For regular Sturm-Liouville problem, the technique is illustrated with three examples and the numerical results show that the approximate eigenvalues are obtained with high-order accuracy without using any correction, and they are compared with the results of other methods. The numerical results of Euler Buckling problem are compared with theoretical aspects, and it is seen that they agree with each other.
Highlights
We investigate the computation of eigenvalues of regular Sturm-Liouville eigenvalue problems:−yxrxyx λy x, 0 ≤ x0 < x < xn, 1.1 y x0 y xn 0, where r x ∈ Cp q x0, xn and p, q ∈ N and Euler Buckling problem:y λ sin y 0, 1.2 y 0 0, y 1 0.Regular Sturm-Liouville problems arise in many applications, and many methods are available for their numerical solution Pryce 1 .Abstract and Applied AnalysisWe examine an elementary, classical problem buckling of an end-loaded rod which possesses a completely soluble continuous model in the form of a nonlinear, secondorder boundary value problem as described in elsewhere 2–5
We observe that there is only trivial initial condition for 0 ≤ λ ≤ π2, there is one nontrivial initial condition from 2.54 for π2 < λ ≤ 4π2, there are n nontrivial initial conditions for n2π2 < λ ≤ n 1 2π2. These results show that the numerical results obtained using Taylor’s decomposition method agree with the theoretical results of Euler buckling problem given in 2
We have described Taylor’s Decomposition method for regular SturmLiouville eigenvalue problems with Dirichlet and Neumann boundary conditions to obtain approximate eigenvalues and eigenfunctions and for Euler Buckling Problem to obtain y0 = 3.0718 y0 = 2.3413 y0 = 0.3236
Summary
We investigate the computation of eigenvalues of regular Sturm-Liouville eigenvalue problems:. 2. Application and Error Analysis of Taylor’s Decomposition Method for Regular Sturm-Liouville Eigenvalue Problems. Application of Taylor’s Decomposition on Two Points for Regular Sturm-Liouville Eigenvalue Problems. To find the corresponding eigenfunctions of the regular Sturm-Liouville eigenvalue problem 2.1 , we substitute the eigenvalue to 2.1 and we solve the obtained boundary value problem by Taylor’s decomposition method on two points xk−1 and xk with the uniform grid 0, 1 h for p q. Solving the 2n×2n homogeneous system, we obtain approximate values of the eigenfunction and its derivative of 1.1 at the point x xk
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