An asymptotic hybrid difference scheme for singularly perturbed third and fourth order ordinary differential equations with discontinuous source term

  • Authors:

  • V. Shanthi;N. Ramanujam

  • Affiliations:

  • Department of Mathematics, National Institute of Technology, Tiruchirappalli, Tamilnadu, India and Department of Mathematics, Bharathidasan University, Tiruchirappalli, Tamilnadu, India;Department of Mathematics, National Institute of Technology, Tiruchirappalli, Tamilnadu, India and Department of Mathematics, Bharathidasan University, Tiruchirappalli, Tamilnadu, India

  • Venue:
  • Neural, Parallel & Scientific Computations
  • Year:
  • 2008

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Abstract

We consider Singularly perturbed Boundary-Value Problems (BVPs) for third and fourth order Ordinary Differential Equations (ODEs) with a discontinuous source term and a small positive parameter multiplying the highest derivative. Because of the type of Boundary Conditions (BCs) imposed on these equations these problems can be transformed into weakly coupled systems. In this system, the first equations does not have the small parameter but the second contains it. In this paper a computational method named as "An asymptotic hybrid finite difference scheme" for solving these systems is presented. In this method we first find an zero order asymptotic approximation to the solution and then the system is decoupled by replacing the first component of the solution by this approximation in the second equation. Then the second equation is independently solved by a hybrid finite difference method. Numerical experiments support our theoretical results.