A new high-resolution two-level implicit method based on non-polynomial spline in tension approximations for time-dependent quasi-linear biharmonic equations with engineering applications

  • R. K. MohantyEmail author
  • Sachin Sharma
Original Article


In this present work, a new two-level implicit non-polynomial spline in tension method is proposed for the numerical solution of the 1D unsteady quasi-linear biharmonic problem. Using the continuity of the first-order derivative of the spline in tension function, a fourth-order accurate implicit finite-difference method is developed in this manuscript. By considering the linear model biharmonic problem, the given implicit spline method is unconditionally stable. Since the proposed method is based on half-step grid points, so it can be directly applied to 1D singular biharmonic problems without alteration in the scheme. Finally, the numerical experiments of the various biharmonic equation such as the generalized Kuramoto–Sivashinsky equation, extended Fisher–Kolmogorov equation, and 1D linear singular biharmonic problems are carried out to show the efficacy, accuracy, and reliability of the method. From the computational experiments, improved numerical results obtained as compared to the results obtained in earlier research work.


Time-dependent biharmonic equations Spline in tension function Finite-difference method Kuramoto–Sivashinsky equation 

Mathematics Subject Classification

65M06 65M12 65M22 65Y20 



This research work is supported by CSIR-SRF, Grant No: 09/045(1161)/2012-EMR-I. The authors thank the reviewers for their valuable suggestions, which substantially improved the standard of the paper.


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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2020

Authors and Affiliations

  1. 1.Department of Applied MathematicsSouth Asian UniversityChanakyapuri, New DelhiIndia
  2. 2.Department of Mathematics, Faculty of Mathematical SciencesUniversity of DelhiNew DelhiIndia

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