JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (10): 30-39.doi: 10.6040/j.issn.1671-9352.0.2023.112

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Differential quadrature method for solving the generalized Burgers-Fisher equations

ALI Adil(),RAHMAN Kaysar*()   

  1. College of Mathematics and System Science, Xinjiang University, Urumqi 830046, Xinjiang, China
  • Received:2023-03-13 Online:2024-10-20 Published:2024-10-10
  • Contact: RAHMAN Kaysar E-mail:adili0515@163.com;kaysar106@xju.edu.cn

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Abstract:

In this paper, a high accuracy numerical scheme is constructed for the generalized Burgers-Fisher equation with Dirichlet boundary or Neumann boundary conditions. Firstly, the Lagrange interpolation polynomial differential quadrature method with uniform grid and Chebyshev-Gauss-Lobatto grid is used in space, and the third-order strong stability-preserving Runge-Kutta scheme is used in time. Secondly, the stability of the scheme is analyzed by using the matrix method. Finally, two numerical examples with different boundary conditions are calculated, and the results are compared with other numerical methods to verify the effectiveness of the proposed scheme.

Key words: generalized Burgers-Fisher equation, differential quadrature method, Chebyshev-Gauss-Lobatto grid, strong stability-preserving Runge-Kutta scheme

CLC Number: 

  • O241

Fig.1

Eigenvalues of matrix A with α=1, δ=1, N at different values of N"

Table 1

Absolute errors for various values of x, t and α with β=δ=1, N=11, Δt=0.000 1 (using CGL grid)"

x t α=-1 α=0.1 α=1
x4 0.3 2.4536e-14 2.2982e-14 2.2649e-14
0.5 2.4647e-14 2.4314e-14 2.3981e-14
0.9 4.1855e-14 2.3759e-14 2.2538e-14
x7 0.3 3.2196e-14 3.1974e-14 3.3418e-14
0.5 3.1641e-14 3.4639e-14 3.6748e-14
0.9 5.4068e-14 3.3307e-14 3.4306e-14
x10 0.3 3.3307e-15 3.5527e-15 4.0523e-15
0.5 3.5527e-15 3.4417e-15 3.9968e-15
0.9 5.3291e-15 3.9968e-15 3.4417e-15

Table 2

Absolute errors for various values of x, t and β with α=0.1, δ=1, N=11, Δt=0.000 1 (using CGL grid)"

x t β=-0.25 β=0.1 β=0.5
x4 0.3 2.1538e-14 2.2649e-14 2.4425e-14
0.5 2.2427e-14 2.3870e-14 2.4758e-14
0.9 2.2260e-14 2.5313e-14 2.4536e-14
x7 0.3 3.0587e-14 3.0753e-14 3.3418e-14
0.5 3.1530e-14 3.6304e-14 3.5305e-14
0.9 3.1142e-14 3.5638e-14 3.5749e-14
x10 0.3 3.2196e-15 3.2196e-15 3.2196e-15
0.5 3.3862e-15 4.1078e-15 3.7748e-15
0.9 3.3862e-15 3.8858e-15 3.6637e-15

Table 3

Absolute errors for various values of x, t and δ with α=1, δ=1, N=11, Δt=0.000 1 (using CGL grid)"

x t δ=2 δ=4 δ=8
x4 0.3 3.0531e-14 2.1871e-14 6.3061e-14
0.5 3.5638e-14 1.5099e-14 8.9928e-15
0.9 3.3640e-14 2.3981e-14 2.2315e-14
x7 0.3 3.9968e-14 3.3973e-14 1.0414e-13
0.5 5.5955e-14 2.5091e-14 2.1760e-14
0.9 5.2514e-14 3.7303e-14 3.3307e-14
x10 0.3 3.6637e-15 5.5511e-16 1.3212e-14
0.5 4.4409e-15 4.7740e-15 2.9976e-15
0.9 6.8834e-15 4.4409e-15 4.3299e-15

Table 4

Absolute errors for various values of x, t and with α=1, B=0, δ=0, N=11, Δt=0.000 1 (using uniform grid)"

x t VIM[12] ADM[6] CFD6[3] 本文格式
0.1 1 1.92e-14 1.92e-14 5.77e-15 1.17e-15
0.5 9.73e-14 9.73e-14 4.19e-14 6.10e-14
0.9 1.75e-13 1.75e-13 1.29e-14 4.89e-15
0.1 10 1.63e-12 1.63e-12 8.49e-15 1.16e-14
0.5 9.44e-12 9.44e-12 4.26e-14 4.92e-14
0.9 1.73e-11 1.73e-11 1.82e-14 1.37e-14
0.1 50 8.14e-12 8.14e-12 2.11e-15 2.22e-16
0.5 2.03e-10 2.03e-10 2.46e-14 1.24e-14
0.9 3.99e-10 3.99e-10 8.05e-15 2.22e-16

Table 5

Comparison of L∞ errors using the two grids respectively with α=1, δ=1, N=11, Δt=0.000 1"

tΔt=0.000 5 Δt=0.000 1 Δt=0.000 05
均匀网格 CGL网格 均匀网格 CGL网格 均匀网格 CGL网格
0.2 2.7311e-14 2.8311e-14   7.6272e-14 3.3751e-14   1.0614e-13 6.7613e-14
0.5 2.9887e-13 2.8277e-13 1.1069e-13 5.8842e-14 1.4544e-13 1.2157e-13
1.0 4.8095e-13 4.8606e-13 6.7057e-14 5.4068e-14 1.2712e-13 1.2046e-13
1.5 2.7700e-13 2.7789e-13 5.8842e-14 5.6621e-14 1.2257e-13 1.1458e-13

Fig.2

Comparison of the numerical and exact solutions for α=1, β=1, δ=4, N=11, Δt=0.000 1 with Dirichlet boundary condition(using CGL grid)"

Table 6

Absolute errors for various values of x, t and α with β=δ=1, N=11, Δt=0.000 1 (using CGL grid)"

x t α=-1 α=0.1 α=1
x4 0.3 1.0358e-13 1.0458e-13 1.0725e-13
0.5 1.6331e-13 1.6676e-13 1.7586e-13
0.9 3.1242e-13 2.7667e-13 3.0032e-13
x7 0.3 9.5146e-14 9.7256e-14 9.4702e-14
0.5 1.5454e-13 1.5754e-13 1.6664e-13
0.9 3.0898e-13 2.6912e-13 2.9321e-13
x10 0.3 9.1926e-14 9.5035e-14 9.0372e-14
0.5 1.5177e-13 1.5632e-13 1.6365e-13
0.9 3.0875e-13 2.6779e-13 2.9288e-13

Table 7

Absolute errors for various values of x, t and β with α=0.1, δ=1, N=11, Δt=0.000 1 (using CGL grid)"

x t β=-0.25 β=0.1 β=0.5
x4 0.3 9.6922e-14 9.8810e-14 1.0580e-13
0.5 1.5826e-13 1.6720e-13 1.7264e-13
0.9 2.7855e-13 3.0265e-13 3.0109e-13
x7 0.3 9.1094e-14 9.2260e-14 9.8588e-14
0.5 1.5193e-13 1.6120e-13 1.6631e-13
0.9 2.7156e-13 2.9665e-13 2.9332e-13
x10 0.3 8.9373e-14 8.9539e-14 9.7589e-14
0.5 1.5005e-13 1.5943e-13 1.6276e-13
0.9 2.6951e-13 2.9532e-13 2.9121e-13

Table 8

Absolute errors for various values of x, t and δ with α=1, β=1, N=11, Δt=0.000 1 (using CGL grid)"

x t δ=2 δ=4 δ=8
x4 0.3 1.3445e-13 8.4599e-14 1.5543e-13
0.5 2.0317e-13 1.2434e-13 5.6288e-14
0.9 3.2008e-13 1.3511e-13 7.0166e-14
x7 0.3 1.0625e-13 1.0825e-13 1.8574e-13
0.5 2.0517e-13 1.0592e-13 8.1490e-14
0.9 3.0731e-13 1.2257e-13 6.0618e-14
x10 0.3 9.6145e-14 1.2945e-13 1.9840e-13
0.5 2.1338e-13 9.7811e-14 9.1926e-14
0.9 3.0398e-13 1.1935e-13 5.7288e-14

Fig.3

Comparison of the numerical and eact solutions for α=1, β=1, δ=8, N=11, Δt=0.000 1 with Neumann boundary condition (using CGL grid)"

1 FISHER R A . The wave of advance of advantageous genes[J]. Annals of Eugenics, 1937, 7 (4): 355- 369.
doi: 10.1111/j.1469-1809.1937.tb02153.x
2 MENDOZA J , MURIEL C . New exact solutions for a generalized Burgers-Fisher equation[J]. Chaos, Solitons & Fractals, 2021, 152, 101- 109.
3 SARI M , GÊRARSLAN G , DAǦİ . A compact finite difference method for the solution of the generalized Burgers-Fisher equation[J]. Numerical Methods for Partial Differential Equations: An International Journal, 2010, 26 (1): 125- 134.
doi: 10.1002/num.20421
4 武莉莉. 求解一类非线性偏微分方程的高精度紧致差分方法[J]. 西北师范大学学报(自然科学版), 2021, 57 (3): 26- 31.
WU Lili . A high-order compact difference method for solving a class of nonlinear partial differential equations[J]. Journal of Northwest Normal University(Natural Science), 2021, 57 (3): 26- 31.
5 KAYA D , EL-SAYED S M . A numerical simulation and explicit solutions of the generalized Burgers-Fisher equation[J]. Applied Mathematics and Computation, 2004, 152 (2): 403- 413.
doi: 10.1016/S0096-3003(03)00565-4
6 ISMAIL H N A , RASLAN K , ABD RABBOH A A . Adomian decomposition method for Burger's-Huxley and Burger's-Fisher equations[J]. Applied Mathematics and Computation, 2004, 159 (1): 291- 301.
doi: 10.1016/j.amc.2003.10.050
7 ISMAIL H N A , ABD RABBOH A A . A restrictive Padé approximation for the solution of the generalized Fisher and Burger-Fisher equations[J]. Applied Mathematics and Computation, 2004, 154 (1): 203- 210.
doi: 10.1016/S0096-3003(03)00703-3
8 BRATSOS A G , KHALIQ A Q M . An exponential time differencing method of lines for Burgers-Fisher and coupled Burgers equations[J]. Journal of Computational and Applied Mathematics, 2019, 356, 182- 197.
doi: 10.1016/j.cam.2019.01.028
9 WASIM I , ABBAS M , AMIN M . Hybrid B-spline collocation method for solving the generalized Burgers-Fisher and Burgers-Huxley equations[J]. Mathematical Problems in Engineering, 2018, 2018, 1- 18.
10 WAZWAZ A M . The tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations[J]. Applied Mathematics and Computation, 2005, 169 (1): 321- 338.
doi: 10.1016/j.amc.2004.09.054
11 WAZWAZ A M . The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations[J]. Applied Mathematics and Computation, 2007, 188 (2): 1467- 1475.
doi: 10.1016/j.amc.2006.11.013
12 MOGHIMI M , HEJAZI F S A . Variational iteration method for solving generalized Burger-Fisher and Burger equations[J]. Chaos, Solitons & Fractals, 2007, 33 (5): 1756- 1761.
13 FAHMY E S . Travelling wave solutions for some time-delayed equations through factorizations[J]. Chaos, Solitons & Fractals, 2008, 38 (4): 1209- 1216.
14 GOLBABAI A , JAVIDI M . A spectral domain decomposition approach for the generalized Burger's-Fisher equation[J]. Chaos, Solitons & Fractals, 2009, 39 (1): 385- 392.
15 LI X , WANG D , SAEED T . Multi-scale numerical approach to the polymer filling process in the weld line region[J]. Facta Universitatis, Series: Mechanical Engineering, 2022, 20 (2): 363- 380.
doi: 10.22190/FUME220131021L
16 周家全, 孙应德, 张永胜. Burgers方程的非协调特征有限元方法[J]. 山东大学学报(理学版), 2012, 47 (12): 103- 108.
ZHOU Jiaquan , SUN Yingde , ZHANG Yongsheng . A nonconforming characteristic finite element method for Burgers equations[J]. Journal of Shandong University(Natural Science), 2012, 47 (12): 103- 108.
17 BELLMAN R , KASHEF B G , CASTI J . Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations[J]. Journal of Computational Physics, 1972, 10 (1): 40- 52.
doi: 10.1016/0021-9991(72)90089-7
18 BERT C W , MALIK M . Differential quadrature method in computational mechanics: a review[J]. Applied Mechanics Reviews, 1996, 49 (1): 1- 28.
doi: 10.1115/1.3101882
19 JIWARI R , SINGH S , KUMAR A . Numerical simulation to capture the pattern formation of coupled reaction-diffusion models[J]. Chaos, Solitons & Fractals, 2017, 103, 422- 439.
20 ARORA G , JOSHI V . A computational approach for one and two dimensional Fisher's equation using quadrature technique[J]. American Journal of Mathematical and Management Sciences, 2021, 40 (2): 145- 162.
doi: 10.1080/01966324.2021.1933660
21 SHU C . Differential quadrature and its application in engineering[M]. New York: Springer, 2000.
22 KORKMAZ A , DAǦ İ . Shock wave simulations using sinc differential quadrature method[J]. Engineering Computations, 2011, 28 (6): 654- 674.
doi: 10.1108/02644401111154619
23 KORKMAZ A , DAǦ İ . A differential quadrature algorithm for simulations of nonlinear Schrödinger equation[J]. Computers & Mathematics with Applications, 2008, 56 (9): 2222- 2234.
24 SARI M . Differential quadrature solutions of the generalized Burgers-Fisher equation with a strong stability preserving high-order time integration[J]. Mathematical and Computational Applications, 2011, 16 (2): 477- 486.
doi: 10.3390/mca16020477
25 SARI M , GVRARSLAN G . Numerical solutions of the generalized Burgers-Huxley equation by a differential quadrature method[J]. Mathematical Problems in Engineering, 2009, 2009, 1- 11.
26 JIWARI R , PANDIT S , MITTAL R C . A differential quadrature algorithm to solve the two dimensional linear hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions[J]. Applied Mathematics and Computation, 2012, 218 (13): 7279- 7294.
doi: 10.1016/j.amc.2012.01.006
27 GOTTLIEB S , SHU C W , TADMOR E . Strong stability-preserving high-order time discretization methods[J]. SIAM Review, 2001, 43 (1): 89- 112.
doi: 10.1137/S003614450036757X
28 TOMASIELLO S . Stability and accuracy of the iterative differential quadrature method[J]. International Journal for Numerical Methods in Engineering, 2003, 58 (9): 1277- 1296.
doi: 10.1002/nme.815
29 TOMASIELLO S . Numerical stability of DQ solutions of wave problems[J]. Numerical Algorithms, 2011, 57 (3): 289- 312.
doi: 10.1007/s11075-010-9429-2
30 JAIN M K . Numerical solution of differential equations[M]. 2nd ed New York: Wiley, 1983.
31 WANG K L , HE C H . A remark on Wang's fractal variational principle[J]. Fractals, 2019, 27 (8): 195- 198.
[1] GUO Peng, ZHANG Lei, WANG Xiao-yun, SUN Xiao-wei. [J]. J4, 2012, 47(12): 115-120.
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