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Numerical solutions of the generalized Burgers-Huxley equation are obtained using a polynomial differential quadrature method with minimal computational effort. To achieve this, a combination of a polynomial-based differential quadrature method in space and a low-storage third-order total variation diminishing Runge-Kutta scheme in time has been used. The computed results with the use of this technique have been compared with the exact solution to show the required accuracy of it. Since the scheme is explicit, linearization is not needed and the approximate solution to the nonlinear equation is obtained easily. The effectiveness of this method is verified through illustrative examples. The present method is seen to be a very reliable alternative method to some existing techniques for such realistic problems.

Nonlinear partial differential equations are encountered in various fields of science. Generalized Burgers-Huxley equation being a nonlinear partial differential equation is of high importance for describing the interaction between reaction mechanisms, convection effects, and diffusion transports. Since there exists no general technique for finding analytical solutions of nonlinear diffusion equations so far, numerical solutions of nonlinear differential equations are of great importance in physical problems.

There are many researchers
who used various numerical techniques to
obtain numerical solution of the Burgers-Huxley equation.
Wang et al. [

To the best knowledge of the authors, the idea of the differential
quadrature method (DQM), where approximations of the spatial derivatives have
been based on a polynomial of high degree, has not been implemented for the
problems in physical phenomena represented by the generalized
Burgers-Huxley equation so far. The DQM is an efficient
discretization technique in solving initial and/or boundary value problems
accurately using a considerably small number of grid points. Bellman et al. [

In the DQM,
derivatives of a function with respect to a coordinate direction are expressed
as linear weighted sums of all the functional values at all grid points along
that direction. The weighting coefficients in that weighting sum are determined
using test functions. Among the many kinds of test functions, the Lagrange
interpolation polynomial is widely used since it has no limitation on the
choice of the grid points. This leads to polynomial-based differential
quadrature (PDQ) method which is suitable in most problems. For periodic
problems, Fourier series expansion can be the best approximation giving the
Fourier expansion-based differential quadrature (FDQ) method. To clearly
describe the DQ method, the readers can see that the PDQ method was first
presented in the work of Shu and Richards [

Unlike some previous techniques using various
transformations to reduce the equation into more simple equation, the current
method does not require extra effort to
deal with the nonlinear terms. Therefore the equations are solved easily and
elegantly using the present method. This method has also additional advantages
over some rival techniques, ease in use and computational costeffectiveness
in order to find solutions of the given nonlinear equations. The combination of
the PDQ method in space with the low-storage third-order total
variation diminishing Runge-Kutta (TVD-RK3) scheme in time [

The present method is useful for obtaining numerical approximations of linear or nonlinear differential equations and it is also quite straightforward to write codes in any programming languages. Also, round off errors and necessity of large computer memory are not faced in this method. The computed results obtained by this way have been compared with the exact solution to show the required accuracy of it. Furthermore, the current method is of a general nature and can therefore be used for solving the nonlinear partial differential equations arising in various areas. Therefore, this paper suggests the use of this technique for solving the generalized Burgers-Huxley equation problems.

Behaviors of
many physical systems encountered in models of reaction mechanisms, convection effects, and diffusion transports give rise to
the generalized Burgers-Huxley equation. The following generalized
Burgers-Huxley equation problem arising in
various fields of science is considered:

The current work aims to demonstrate that the proposed numerical algorithm is capable of achieving high accuracy for the problems represented by the generalized Burgers-Huxley equation. The computed results are compared with the exact solutions to verify the effectiveness of the current method.

The method uses the basis of the quadrature method in driving the derivatives of a function. It follows that the partial derivative of a function with respect to a space variable can be approximated by a weighted linear combination of function values at some intermediate points in that variable.

The selection of locations of the sampling points plays an important
role in the accuracy of the solution of the differential equations. Using
uniform grids can be considered to be a convenient and easy selection method.
Quite frequently, the DQM delivers more accurate solutions using the so-called
Chebyshev-Gauss-Lobatto points [

In order to attain the accurate numerical solution of differential
equations, proper implementation of the boundary is also very important. For
prescribing the Dirichlet boundary conditions (

In order to see numerically whether the
present methodology leads to accurate solutions, the PDQ solutions will be
evaluated for some examples of the generalized Burgers-Huxley equations given above. To verify the efficiency, measure its
accuracy and the versatility of the PDQ method for the current problem in
comparison with the exact solution, absolute error for different values of

Consider
the generalized Burgers-Huxley equation in the form (

The absolute errors for various values of

0.20 | 6.841E-09 | 3.194E-07 | 2.239E-06 | 6.058E-06 | |

0.60 | 7.733E-09 | 3.610E-07 | 2.531E-06 | 6.842E-06 | |

0.80 | 7.748E-09 | 3.617E-07 | 2.535E-06 | 6.852E-06 | |

0.20 | 3.644E-08 | 1.701E-06 | 1.193E-05 | 3.226E-05 | |

0.60 | 4.226E-08 | 1.973E-06 | 1.383E-05 | 3.739E-05 | |

0.80 | 4.236E-08 | 1.977E-06 | 1.386E-05 | 3.746E-05 | |

0.20 | 1.420E-08 | 6.630E-07 | 4.649E-06 | 1.257E-05 | |

0.60 | 1.615E-08 | 7.538E-07 | 5.284E-06 | 1.428E-05 | |

0.80 | 1.618E-08 | 7.553E-07 | 5.294E-06 | 1.431E-05 |

The absolute errors for various values of

0.10 | 4.059E-14 | 5.984E-12 | 7.554E-11 | 2.802E-10 | |

0.50 | 5.888E-14 | 8.681E-12 | 1.096E-10 | 4.065E-10 | |

1.00 | 5.924E-14 | 8.733E-12 | 1.102E-10 | 4.090E-10 | |

0.10 | 2.021E-13 | 2.979E-11 | 3.760E-10 | 1.395E-09 | |

0.50 | 3.215E-13 | 4.741E-11 | 5.984E-10 | 2.220E-09 | |

1.00 | 3.239E-13 | 4.775E-11 | 6.027E-10 | 2.236E-09 | |

0.10 | 8.301E-14 | 1.224E-11 | 1.545E-10 | 5.731E-10 | |

0.50 | 1.229E-13 | 1.812E-11 | 2.288E-10 | 8.487E-10 | |

1.00 | 1.237E-13 | 1.824E-11 | 2.302E-10 | 8.541E-10 |

The absolute errors for various values of

0.30 | 2.317E-09 | 1.013E-07 | 6.628E-07 | 1.682E-06 | |

0.50 | 2.413E-09 | 1.055E-07 | 6.902E-07 | 1.751E-06 | |

0.90 | 2.428E-09 | 1.062E-07 | 6.944E-07 | 1.762E-06 | |

0.30 | 6.580E-09 | 2.878E-07 | 1.882E-06 | 4.776E-06 | |

0.50 | 6.899E-09 | 3.017E-07 | 1.974E-06 | 5.007E-06 | |

0.90 | 6.950E-09 | 3.039E-07 | 1.988E-06 | 5.043E-06 | |

0.30 | 3.695E-09 | 1.616E-07 | 1.057E-06 | 2.681E-06 | |

0.50 | 3.855E-09 | 1.686E-07 | 1.103E-06 | 2.798E-06 | |

0.90 | 3.881E-09 | 1.697E-07 | 1.110E-06 | 2.816E-06 |

The absolute errors for various values of

0.30 | 1.545E-08 | 1.854E-07 | 9.807E-07 | 1.989E-06 | |

0.50 | 1.608E-08 | 1.930E-07 | 1.021E-06 | 2.071E-06 | |

0.90 | 1.618E-08 | 1.942E-07 | 1.027E-06 | 2.082E-06 | |

0.30 | 4.387E-08 | 5.264E-07 | 2.785E-06 | 5.649E-06 | |

0.50 | 4.600E-08 | 5.520E-07 | 2.920E-06 | 5.922E-06 | |

0.90 | 4.633E-08 | 5.560E-07 | 2.941E-06 | 5.960E-06 | |

0.30 | 2.463E-08 | 2.956E-07 | 1.564E-06 | 3.172E-06 | |

0.50 | 2.570E-08 | 3.084E-07 | 1.632E-06 | 3.309E-06 | |

0.90 | 2.587E-08 | 3.105E-07 | 1.642E-06 | 3.328E-06 |

The absolute errors for various values of

0.30 | 2.039E-04 | 6.505E-06 | 2.059E-07 | 6.512E-09 | |

0.50 | 2.116E-04 | 6.771E-06 | 2.144E-07 | 6.780E-09 | |

0.90 | 2.111E-04 | 6.808E-06 | 2.157E-07 | 6.823E-09 | |

0.30 | 5.809E-04 | 1.848E-05 | 5.848E-07 | 1.849E-08 | |

0.50 | 6.071E-04 | 1.937E-05 | 6.131E-07 | 1.939E-08 | |

0.90 | 6.064E-04 | 1.950E-05 | 6.175E-07 | 1.953E-08 | |

0.30 | 3.268E-04 | 1.038E-05 | 3.283E-07 | 1.038E-08 | |

0.50 | 3.399E-04 | 1.083E-05 | 3.426E-07 | 1.083E-08 | |

0.90 | 3.392E-04 | 1.089E-05 | 3.448E-07 | 1.091E-08 |

Comparison of PDQ-Lagrange and PDQ-Chebyshev methods: the absolute errors for various
values of

PDQ-Lagrange | PDQ-Chebyshev | ||
---|---|---|---|

0.01 | 7.6338863E-09 | 7.6338860E-09 | |

0.10 | 3.1524288E-08 | 3.1524290E-08 | |

1.00 | 4.6979115E-08 | 4.6979110E-08 | |

0.01 | 1.0877730E-08 | 1.0877730E-08 | |

0.10 | 8.2905812E-08 | 8.2905810E-08 | |

1.00 | 1.3450669E-07 | 1.3450670E-07 | |

0.01 | 9.7652503E-09 | 9.7652500E-09 | |

0.10 | 4.9131274E-08 | 4.9131270E-08 | |

1.00 | 7.5113908E-08 | 7.5113910E-08 |

In Table

In Table

For the computational work in this
example, the absolute errors have been shown for various values of

Here, the absolute errors have been
shown in Table

Table

The PDQ method based on the Lagrange
interpolation functions and the PDQ method based on the Chebyshev interpolation
functions are compared in terms of the absolute errors for various values of

The maximum absolute errors
and convergence rate (CR) of the proposed method are produced for various
values of

In the examples above, although a very few number of grids are used and even when the parameters are taken to increase the nonlinearity of the problem, the present results are still seen to be very accurate. Tables

The maximum absolute errors and convergence
rate (CR) of the DQM-TVD-RK3 with

Maximum absolute error | CR | |
---|---|---|

5 | 2.103906E-03 | |

10 | 1.019676E-07 | 14.33 |

15 | 4.738276E-11 | 18.93 |

20 | 4.665157E-13 | 16.06 |

Solutions of Example

In this paper, use of a combination of polynomial-based differential quadrature method in space and a low-storage third-order total variation diminishing Runge-Kutta method in time has been proposed for the generalized Burgers-Huxley equation, with high convergence. Comparisons of the computed results with exact solutions showed that the method has the capability of solving the generalized Burgers-Huxley equation and is also capable of producing highly accurate solutions with minimal computational effort for both time and space. It was seen that the polynomial-based differential quadrature technique approximates the exact solution very well. Since the scheme is explicit, linearization is not needed. No requiring extra effort to deal with nonlinear terms, ease in use, and computational cost-effectiveness have made the current method an efficient alternative method for modelling these nonlinear behaviors. For concrete problems where an exact solution does not exist, the present method is a very good choice to achieve a high degree of accuracy while dealing with the problems.