Boundary Characteristic Orthogonal Polynomials-Based Galerkin and Least Square Methods for Solving Bagley–Torvik Equations

  • Rajarama Mohan Jena
  • S. ChakravertyEmail author
Conference paper
Part of the Lecture Notes in Mechanical Engineering book series (LNME)


In this paper, efficient numerical methods for solving Bagley–Torvik (B-T) equations with variable coefficients and three-point boundary value conditions are considered. This model is considered as a viscoelastic behavior of geological strata, metal, and glasses using fractional differential equations. Many viscoelastic materials are proposed in which derivatives of fractional-order replace the usual time derivatives of integer order. An application of such a model is the prediction of the transient response of frequency-dependent materials. As such the titled problem is challenging to solve using the efficient method(s). The fractional derivative is described in the Caputo sense. First, a linearly independent set such as \( \left\{ {1,x,x^{2} ,x^{3} , \ldots } \right\} \) is converted to Boundary Characteristic Orthogonal Polynomials (BCOPS) by Gram–Schmidt Orthogonalization process then these are used in the Galerkin and Least Square methods to reduce B-T Equations to the linear or nonlinear system of algebraic equations. Example problems are addressed to show the powerfulness and efficacy of the method.


Bagley–Torvik equation Caputo fractional derivatives Characteristic orthogonal polynomials Galerkin method Least square method 



The first author acknowledges the Department of Science and Technology, Govt. of India for providing INSPIRE fellowship (IF170207) to carry out the present work.


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© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of Technology RourkelaRourkelaIndia

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