| Citation: |
A.N. Nirmala, S. Kumbinarasaiah. NUMERICAL APPROACH FOR THE HUNTER SAXTON EQUATION ARISING IN LIQUID CRYSTAL MODEL THROUGH COCKTAIL PARTY GRAPHS CLIQUE POLYNOMIAL[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 2037-2062. doi: 10.11948/20230114
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NUMERICAL APPROACH FOR THE HUNTER SAXTON EQUATION ARISING IN LIQUID CRYSTAL MODEL THROUGH COCKTAIL PARTY GRAPHS CLIQUE POLYNOMIAL
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Abstract
In this paper, a well-known nematic liquid crystal model, the Hunter Saxton equation, is solved by the new graph theoretic polynomial approach. At first, we extracted the clique polynomials from the cocktail party graph (CPG) and generated the generalized operational matrix of integration through the clique polynomials of CPG. Then, an effective computational technique called the cocktail party graphs clique polynomial collocation method (CCCM) is developed to obtain an approximate numerical solution for the nonlinear Hunter-Saxton equation (HSE). The operational matrix of CPG reduces the HSE into an algebraic system of nonlinear equations that makes the solution relatively superficial. The Newton-Raphson method solves these nonlinear algebraic equations to obtain the clique polynomial solution for HSE. The efficiency of the CCCM is illustrated by examining two numerical examples. The solution of the HSE is presented through figures and tables for different values of x,t, and N. The convergence analysis, tabulated results of numerical comparison of absolute errors of CCCM with the recent numerical methods, and error norms projected that, CCCM is considerably efficacious on the computational ground for higher accuracy and convergence of numerical solutions.
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References
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A.N. Nirmala, S. Kumbinarasaiah. NUMERICAL APPROACH FOR THE HUNTER SAXTON EQUATION ARISING IN LIQUID CRYSTAL MODEL THROUGH COCKTAIL PARTY GRAPHS CLIQUE POLYNOMIAL[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 2037-2062. doi: 10.11948/20230114
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Figure 1.
Graph
$ G_{1} $ -
Figure 2.
Graph
$ G_{2} $ -
Figure 3.
Graph
$ G_{3} $ -
Figure 4.
Graphical representation of the Exact, CCCM solution along with its absolute error at N = 4 (Example 5.1).
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Figure 5.
Graphical representation of the Exact, CCCM solution along with its absolute error at N = 10 (Example 5.1).
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Figure 6.
Graphical representation of the Exact, CCCM solution at fixed
$ x=0.1,0.2,0.3,0.4,0.5 $ $ t \in [0,1], \Delta t=0.1 $ $ N=4 $ -
Figure 7.
Graphical representation of the Exact CCCM solution at
$ N=10 $ $ t=0.1,0.2,0.3,0.4,0.5 $ $ x \in [0,1], \Delta x=0.1 $ -
Figure 8.
Graphical representation of Absolute errors of Example 5.1 at
$ N =3 $ $ t \in [0,1], \Delta t=0.1 $ $ x=0.1,0.2,0.3,0.4,0.5 $ -
Figure 9.
Graphical representation of comparison of absolute errors of various numerical methods for Example 5.1 at
$ x \in [0,1], \Delta x=0.1, t=0.1,N=3 $ -
Figure 10.
Graphical representation of the Exact, CCCM solution obtained at
$ N =4, t=0.1, x \in [0,1], \Delta x=0.1 $ -
Figure 11.
Graphical representation of the Exact, and CCCM solution at
$ x=0.1,0.2,0.3,0.4,0.5. (N =4) $ -
Figure 12.
Graphical representation of the Exact, CCCM solution at
$ t=0.1,0.2,0.3,0.4,0.5. $ $ N=4 $ -
Figure 13.
Graphical representation of absolute errors for Example 5.2 at
$ x \in [0,1] $ $ t=0.1,0.01,0.001, N=4 $ -
Figure 14.
Graphical representation of comparison of absolute errors of CCCM, Fibonacci wavelet method(FWM), Haar wavelet method(HWM)), B-spline method(BSM) at
$ t=0.1 , N=2, x \in [0,1] $