# 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

- 4 Downloads

## Abstract

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.

## Keywords

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

65M06 65M12 65M22 65Y20## Notes

### Acknowledgements

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.

## References

- 1.Aronson DG, Weinberger HF (1978) Multidimensional nonlinear diffusion arising in population genetics. Adv Math 30:33–67MathSciNetCrossRefGoogle Scholar
- 2.Conte R (2003) Exact solutions of nonlinear partial differential equations by singularity analysis. Lecture notes in physics. Springer, Berlin, pp 1–83zbMATHGoogle Scholar
- 3.Dee GT, van Saarloos W (1988) Bistable systems with propagating fronts leading to pattern formation. Phys Rev Lett 60:2641–2644CrossRefGoogle Scholar
- 4.Hooper AP, Grimshaw R (1985) Nonlinear instability at the interface between two viscous fluids. Phys Fluids 28:37–45CrossRefGoogle Scholar
- 5.Hornreich RM, Luban M, Shtrikman S (1975) Critical behaviour at the onset of
*k*-space instability at the λ line. Phys Rev Lett 35:1678–1681CrossRefGoogle Scholar - 6.Kuramoto Y, Tsuzuki T (1976) Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog Theory Phys 55:356–369CrossRefGoogle Scholar
- 7.Saprykin S, Demekhin EA, Kalliadasis S (2005) Two-dimensional wave dynamics in thin films. I. Stationary solitary pulses. Phys Fluids 17:117105MathSciNetCrossRefGoogle Scholar
- 8.Sivashinsky GI (1983) Instabilities, pattern-formation, and turbulence in flames. Annu Rev Fluid Mech 15:179–199CrossRefGoogle Scholar
- 9.Tatsumi T (1984) Irregularity, regularity and singularity of turbulence. Turbulence and chaotic phenomena in fluids. In: Proceedings of IUTAM, pp 1–10Google Scholar
- 10.Zhu G (1982) Experiments on director waves in nematic liquid crystals. Phys Rev Lett 49:1332–1335CrossRefGoogle Scholar
- 11.Dehghan M, Mohebbi A (2006) Multigrid solution of high order discretization for three-dimensional biharmonic equation with Dirichlet boundary conditions of second kind. Appl Math Comput 180:575–593MathSciNetzbMATHGoogle Scholar
- 12.Dehghan M, Mohebbi A (2008) Solution of the two dimensional second biharmonic equation with high-order accuracy. Kybernetes 37:1165–1179MathSciNetCrossRefGoogle Scholar
- 13.Illati M, Dehghan M (2018) Direct local boundary integral equation method for numerical solution of extended Fisher–Kolmogorov equation. Eng Comput 34:203–213CrossRefGoogle Scholar
- 14.Mohanty RK, Sharma S (2019) A new two-level implicit scheme based on cubic spline approximations for the 1D time-dependent quasilinear biharmonic problems. Eng Comput. https://doi.org/10.1007/s00366-019-00778-1 CrossRefGoogle Scholar
- 15.Danumjaya P, Pani AK (2005) Orthogonal cubic spline collocation method for the extended Fisher–Kolmogorov equation. J Comput Appl Math 174:101–117MathSciNetCrossRefGoogle Scholar
- 16.Ganaiea IA, Arora S, Kukreja VK (2016) Cubic Hermite collocation solution of Kuramoto–Sivashinsky equation. Int J Comput Math 93:223–235MathSciNetCrossRefGoogle Scholar
- 17.Khater AH, Temsah RS (2008) Numerical solutions of the generalized Kuramoto–Sivashinsky equation by Chebyshev spectral collocation methods. Comput Math Appl 56:1456–1472MathSciNetCrossRefGoogle Scholar
- 18.Xu Y, Shu CW (2006) Local discontinuous Galerkin methods for the Kuramoto–Sivashinsky equations and the Ito-type coupled KdV equations. Comput Methods Appl Mech Eng 195:3430–3447MathSciNetCrossRefGoogle Scholar
- 19.Fan E (2000) Extended tanh-function method and its applications to nonlinear equations. Phys Lett A 277:212–218MathSciNetCrossRefGoogle Scholar
- 20.Doss LJT, Nandini AP (2012) An
*H*^{1}-Galerkin mixed finite element method for the extended Fisher–Kolmogorov equation. Int J Numer Anal Model Ser B 3:460–485MathSciNetzbMATHGoogle Scholar - 21.Lai H, Ma C (2009) Lattice Boltzmann method for the generalized Kuramoto–Sivashinsky equation. Phys A 388:1405–1412MathSciNetCrossRefGoogle Scholar
- 22.Uddin M, Haq S, Siraj-ul-Islam S (2009) A mesh-free numerical method for solution of the family of Kuramoto–Sivashinsky equations. Appl Math Comput 212:458–469MathSciNetzbMATHGoogle Scholar
- 23.Lakestani M, Dehghan M (2012) Numerical solutions of the generalized Kuramoto–Sivashinsky equation using B-spline functions. Appl Math Model 36:605–617MathSciNetCrossRefGoogle Scholar
- 24.Mittal RC, Arora G (2010) Quintic B-spline collocation method for numerical solution of the Kuramoto–Sivashinsky equation. Commun Nonlinear Sci Numer Simul 15:2798–2808MathSciNetCrossRefGoogle Scholar
- 25.Rashidinia J, Jokar M (2017) Polynomial scaling functions for numerical solution of generalized Kuramoto–Sivashinsky equation. Appl Anal 96:293–306MathSciNetCrossRefGoogle Scholar
- 26.Mitchell AR (1969) Computational methods in partial differential equations. Wiley, New YorkzbMATHGoogle Scholar
- 27.Mohanty RK (2003) An accurate three spatial grid-point discretization of
*O*(*k*^{2}+*h*^{4}) for the numerical solution of one-space dimensional unsteady quasi-linear biharmonic problem of second kind. Appl Math Comput 140:1–14MathSciNetGoogle Scholar - 28.Mohanty RK, Kaur D (2017) Numerov type variable mesh approximations for 1D unsteady quasi-linear biharmonic problem: application to Kuramoto–Sivashinsky equation. Numer Algorithm 74:427–459MathSciNetCrossRefGoogle Scholar
- 29.Stephenson JW (1984) Single cell discretizations of order two and four for biharmonic problems. J Comput Phys 55:65–80MathSciNetCrossRefGoogle Scholar
- 30.Jain MK, Aziz T (1983) Numerical solution of stiff and convection-diffusion equations using adaptive spline function approximation. Appl Math Model 7:57–62MathSciNetCrossRefGoogle Scholar
- 31.Kadalbajoo MK, Patidar KC (2002) Tension spline for the numerical solution of singularly perturbed nonlinear boundary value problems. J Comput Appl Math 21:717–742zbMATHGoogle Scholar
- 32.Mohanty RK, Gopal V (2013) A fourth order finite difference method based on spline in tension approximation for the solution of one-space dimensional second order quasi-linear hyperbolic equations. Adv Differ Equ 2013:70MathSciNetCrossRefGoogle Scholar
- 33.Talwar J, Mohanty RK, Singh S (2016) A new algorithm based on spline in tension approximation for 1D quasilinear parabolic equations on a variable mesh. Int J Comput Math 93:1771–1786MathSciNetCrossRefGoogle Scholar
- 34.Mohanty RK, Sharma S (2017) High accuracy quasi-variable mesh method for the system of 1D quasi-linear parabolic partial differential equations based on off-step spline in compression approximations. Adv Differ Eqn 2017:212MathSciNetCrossRefGoogle Scholar
- 35.Mohanty RK, Sharma S (2018) A new two-level implicit scheme for the system of 1D quasi-linear parabolic partial differential equations using spline in compression approximations. Differ Equ Dyn Syst 27:327–356MathSciNetCrossRefGoogle Scholar
- 36.Hageman LA, Young DM (2004) Applied iterative methods. Dover Publications, New YorkzbMATHGoogle Scholar
- 37.Kelly CT (1995) Iterative methods for linear and nonlinear equations. SIAM Publications, PhiladelphiaCrossRefGoogle Scholar
- 38.Saad Y (2003) Iterative methods for sparse linear systems. SIAM, PublisherCrossRefGoogle Scholar