Eigenvalues and eigenfunctions of fourth-order sturm-liouville problems using Bernoulli series with Chebychev collocation points

A collocation method based on Bernoulli polynomial is developed to compute the eigenvalues and eigenfunctions of some known fourth-order Sturm-Liouville problems. Properties of Bernoulli matrix method are presented to convert the problem into a system of linear algebraic equations. Error estimation is introduced. The eigenfunctions are calculated for the test problems. A comparison is made with other relevant studies.

[1]  Daniel Lesnic,et al.  An efficient method for computing eigenelements of Sturm-Liouville fourth-order boundary value problems , 2006, Appl. Math. Comput..

[2]  Kira V. Khmelnytskaya,et al.  Spectral parameter power series for fourth-order Sturm-Liouville problems , 2012, Appl. Math. Comput..

[3]  S. Bazm,et al.  Bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations , 2015, J. Comput. Appl. Math..

[4]  Khosrow Maleknejad,et al.  Numerical solution of Volterra type integral equation of the first kind with wavelet basis , 2007, Appl. Math. Comput..

[5]  Khosrow Maleknejad,et al.  Modification of Block Pulse Functions and their application to solve numerically Volterra integral equation of the first kind , 2011 .

[6]  Ibrahim Çelik,et al.  Approximate computation of eigenvalues with Chebyshev collocation method , 2005, Appl. Math. Comput..

[7]  K. Boubaker,et al.  Differential quadrature method (DQM) and Boubaker Polynomials Expansion Scheme (BPES) for efficient computation of the eigenvalues of fourth-order Sturm–Liouville problems , 2012 .

[8]  Amin Boumenir,et al.  Sampling for the fourth-order Sturm-Liouville differential operator , 2003 .

[9]  Bilal Chanane,et al.  Fliess series approach to the computation of the eigenvalues of fourth-order Sturm-Liouville problems , 2002, Appl. Math. Lett..

[10]  A numerical scheme for solving nonhomogeneous time-dependent problems , 2006 .

[11]  Mehmet Sezer,et al.  A Bernoulli polynomial approach with residual correction for solving mixed linear Fredholm integro-differential-difference equations , 2013 .

[12]  Marco Marletta,et al.  Algorithm 775: the code SLEUTH for solving fourth-order Sturm-Liouville problems , 1997, TOMS.

[13]  Emran Tohidi,et al.  A New Matrix Approach For Solving Second-Order Linear Matrix Partial Differential Equations , 2016 .

[14]  H. Mirzaei.  A family of isospectral fourth order Sturm Liouville problems and equivalent beam equations , 2018 .

[15]  M. El-Gamel,et al.  Two very accurate and efficient methods for computing eigenvalues of Sturm–Liouville problems , 2013 .

[16]  M. El-Gamel,et al.  Sinc-Galerkin solution to the clamped plate eigenvalue problem , 2017 .

[17]  David Kennedy,et al.  Adaptive finite element method for eigensolutions of regular second and fourth order Sturm-Liouville problems via the element energy projection technique , 2017 .

[19]  M. El-Gamel,et al.  Bernoulli polynomial and the numerical solution of high-order boundary value problems , 2019, Mathematics in Natural Science.

[20]  Necas Jindrich.  Les Méthodes directes en théorie des équations elliptiques , 2017 .

[21]  Mohamed El-Gamel,et al.  An Efficient Technique for Finding the Eigenvalues of Fourth-Order Sturm-Liouville Problems , 2012 .

[22]  A. Bhrawy,et al.  A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation , 2013 .

[23]  Esmail Babolian,et al.  Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions , 2008 .

[24]  M. El-Gamel,et al.  An efficient technique for finding the eigenvalues and the eigenelements of fourth-order Sturm-Liouville problems , 2017 .

[25]  Anna Napoli,et al.  Solutions of linear second order initial value problems by using Bernoulli polynomials , 2016 .

[26]  Zhi Shi,et al.  Application of Haar wavelet method to eigenvalue problems of high order differential equations , 2012 .

[27]  H. Mirzaei.  Computing the eigenvalues of fourth order Sturm-Liouville problems with Lie Group method , 2017 .

[28]  Alan L. Andrew.  Twenty years of asymptotic correction for eigenvalue computation , 2000 .

[29]  Mortaza Gachpazan,et al.  A New Tau Method for Solving Nonlinear Lane-Emden Type Equations via Bernoulli Operational Matrix of Differentiation , 2013, J. Appl. Math..

[30]  W. Adel.  A fast and efficient scheme for solving a class of nonlinear Lienard’s equations , 2020 .

[31]  Marco Marletta,et al.  Oscillation theory and numerical solution of fourth-order Sturm—Liouville problems , 1995 .

[32]  Leon Greenberg,et al.  An oscillation method for fourth-order, self-adjoint, two-point boundary value problems with nonlinear eigenvalues , 1991 .

[33]  Bilal Chanane.  Accurate solutions of fourth order Sturm-Liouville problems , 2010, J. Comput. Appl. Math..

[34]  Emran Tohidi,et al.  A new Bernoulli matrix method for solving second order linear partial differential equations with the convergence analysis , 2013, Appl. Math. Comput..

[35]  Christine Böckmann,et al.  Matrix methods for computing eigenvalues of Sturm-Liouville problems of order four , 2013, J. Comput. Appl. Math..

[36]  M. El-Gamel,et al.  B-spline and singular higher-order boundary value problems , 2016 .

[37]  B. Chanane.  Eigenvalues of fourth order Sturm-Liouville problems using Fliess series , 1998 .

[38]  Fazlollah Soleymani,et al.  A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals , 2012, Appl. Math. Comput..

[39]  Zulqurnain Sabir,et al.  Solving a new design of nonlinear second-order Lane–Emden pantograph delay differential model via Bernoulli collocation method , 2020 .

[40]  Jian Chen,et al.  A simple approach for determining the eigenvalues of the fourth-order Sturm-Liouville problem with variable coefficients , 2013, Appl. Math. Lett..

[41]  Xian‐Fang Li,et al.  Exact frequency equations of free vibration of exponentially functionally graded beams , 2013 .

[42]  S. Abbasbandy,et al.  A new application of the homotopy analysis method: Solving the Sturm–Liouville problems , 2011 .