A new two-level implicit scheme based on cubic spline approximations for the 1D time-dependent quasilinear biharmonic pr
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ORIGINAL ARTICLE
A new two-level implicit scheme based on cubic spline approximations for the 1D time-dependent quasilinear biharmonic problems R. K. Mohanty1
•
Sachin Sharma2
Received: 8 September 2018 / Accepted: 14 May 2019 Springer-Verlag London Ltd., part of Springer Nature 2019
Abstract In this article, we present a new two-level implicit cubic spline numerical method of accuracy 2 in time and 4 in spatial direction for the numerical solution of 1D time-dependent quasilinear biharmonic equation subject to appropriate initial and natural boundary conditions prescribed. The easiness of the proposed numerical method lies in their 3-point discretization in which we use two points x ðh=2Þ and a central point ‘x’ in spatial direction. Using the continuity of the first-order derivative of cubic spline function, we derive the fourth-order accurate numerical method for the time-dependent biharmonic equation on a uniform mesh. The stability consideration of the proposed method is discussed using a model linear problem. The proposed cubic spline method successfully implements on generalized Kuramoto–Sivashinsky and extended Fisher–Kolmogorov equations. From the numerical experiments, we obtain better computational results compared to the results discussed in earlier research work. Keywords Cubic spline function Quasilinear biharmonic equations Kuramoto–Sivashinsky equation Fisher– Kolmogorov equation Newton’s iterative method Mathematics Subject Classifications 65M06 65M12 65M22 65Y20
1 Introduction
o2 u ¼ v; ðx; tÞ 2 X; ox2
Consider the fourth-order time-dependent nonlinear biharmonic equation of the form:
Aðx; tÞ
o4 u ou ¼ f ðx; t; u; ux ; uxx ; uxxx Þ; Aðx; tÞ 4 þ ox ot
a\x\b; t [ 0: ð1Þ
The solution space is defined by X fðx; tÞja\x\b; t [ 0g. The equation above may be represented in a coupled form as:
& R. K. Mohanty [email protected] 1
Department of Applied Mathematics, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi 110021, India
2
Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, New Delhi 110007, India
ð2Þ
o2 v ou ¼ þ f ðx; t; u; ux ; v; vx Þ 2 ox ot gðx; t; u; v; ux ; vx ; ut Þ; ðx; tÞ 2 X:
ð3Þ
The values of initial and boundary conditions are given by uðx; 0Þ ¼ u0 ðxÞ;
a x b;
uða; tÞ ¼ a0 ðtÞ; uðb; tÞ ¼ b0 ðtÞ;
ð4Þ ð5Þ
t [ 0;
uxx ða; tÞ ¼ vða; tÞ ¼ a1 ðtÞ; uxx ðb; tÞ ¼ vðb; tÞ ¼ b1 ðtÞ; t [ 0;
ð6Þ
where u0 ; a0 ; b0 ; a1 and b1 are differentiable functions with sufficient smoothness in the solution region X. In this study, we develop a numerical method to find the behavior of time-dependent biharmonic problems in onedimensional model equations. Such kind of equations arises in various mathematical models of physical phenomenon in science and engineering such as flame front instability [1], stress waves in fragmented porous media [2], solitary pulses in a falling film [3], unstable drift waves
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Engineering with Computers
in plasmas [4], reaction–diffusion sys
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