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This research paper represents a numerical approximation to non-linear coupled one dimension reaction diffusion system, which includes the existence and uniqueness of the time dependent solution with upper and lower bounds of the solution. Also numerical approximation is obtained by finite difference schemes to reach at reasonable level of accuracy, which is magnified by
L
_{2},
L
_{∞} and relative error norms. The accuracy of the approximations is shown by randomly selected grid points along time level and comparison with analytical results. The test example demonstrates the accuracy, efficiency and versatility of the proposed schemes. Moreover, the schemes can be easily applied to a wide class of higher dimension non-linear reaction diffusion equations with a little modifications.

Fisher and Kolmogorov Petrovsky Piscounov founded quasilinear partial dif- ferential equation which represents reaction diffusion phenomena [

here, x declares spatial coordinate position with time t, also

where the assumption on the diffusion coefficient D, is to be constant [

related to stranded scale to growth in the population [

In the year of 1948 El’ dovich, raised the major features of the diffusion reaction semi-linear equation which make useful contribution to wave of advantageous genes which are now totally classify as in the dynamics of the gas and flame with chemical kinetics [

where in above Equation (5),

Exact solution to system in Equation (5) found by E. S. Fahmy [

It is very important to enforce some assumptions on some physical parameters or related reaction functions, to analysed the smoothness and uniqueness of a positive oriented natural solution by considering lower and upper bound of the solution of the system which mentioned in Equation (5) [

imposed the following boundary conditions on Equation (8), which is,

imposed the following initial conditions on Equation (8), which is,

combined above three equations, to get the model system.

Assumption or Hypothesis (H)

Let us suppose that

where above Equations (11), (12) represents

A smooth pair of two vector functions

In the above definitions the smoothness of

Let f holds hypothesis (H). If there exist upper and lower solutions

The quality of being of practical use about above theorem, need to construct of lower and upper solutions, with existence problem to be ensured [

Let us apply numerical methods technique, to solve such system which men- tioned in Equation (5) in finite domain

We consider forward in time and center in space (FTCS) explicit scheme by substituting the forward difference approximation for the time derivative and the central difference approximation for the space derivative in system in Equation (5),

where

Finite difference schemes, must pass certain tests of accuracy, consistency, stability and convergence [

Accuracy of the FTCS scheme for system in Equation (16), we apply Taylor’s series on each term, which is as follows:

simplifies above equations, we get the following,

dividing above system, by k and simplifies some terms:

This leads to principle part of the truncation error (PPTE), which is as follows:

which shows that this scheme has 2nd order accuracy in space and first order accuracy in time such as

According to the definition of the consistency, if the difference between finite difference equation (FDE) and related partial differential equation (PDE), i.e truncation error vanishes as the sizes of the grid spacing go to zero inde- pendently, i.e

Equation (20) is consistent, because the truncation error divided by k tends to zero as h and k tends to zero [

A finite difference method is convergent if the solution of the finite difference equation approaches to exact solution of the partial differential equation as the sizes of the grid spacing h and k tends to zero [

where

where

this completes the proof of the convergence.

Another important feature of a finite difference method of solving partial dif- ferential equation is the stability of the associated finite difference equation which must be investigated. Let us look again Equation (16),

linear form of above Equation (26), is as follows:

where

where

where

where matrix

above Inequality (29) leads to the following two special cases,

1) Special Case 1

・ The right hand side of the Inequity (29) gives,

which is condition of stability to FTCS scheme in coupled non-linear PDE system.

2) Special Case 2

・ The left hand side of the Inequity (29) gives,

which is condition of stability to FTCS scheme in coupled non-linear PDE system.

Special case (1) and special case (2) lead to the following very interesting results,

The Von-Neumann stability analysis is the most common used method of de- termining stability criterion as it is generally the easiest to apply. It can only be used to establish a necessary and sufficient condition for stability of linear initial value problems with constant coefficients [

Let us apply Crank Nicolson implicit finite difference scheme to Equation (5).

after some simplification, we get the following,

combine Equations (34) (35), we get the following,

where

Accuracy of the CN scheme to Equation (36), we apply Taylor’s series on each term. After some simplification, resultant is as follows,

we divide above equation by time step k with using in system (5), to get the accuracy, in the following form,

Now principle part of the truncation error (PPTE) is as follows:

which shows that this scheme is 2nd order accurate in both time and space, such as

From accuracy, we find principle part of the truncation error along with Equ- ation (42). Which shows that Crank Nicolson scheme is consistent because

Stability of the associated finite difference Equation (36), which is in linear form,

where

where

According to Von-Neumann stability analysis, we have

where

where matrix

Above Equation (46), satisfies the Von-Neumann stability criterion, which shows that CN scheme for Equation (5) is unconditionally stable [

Let us apply another implicit scheme with improved accuracy in space to Equation (5).

where

combine Equations (49) (50), to get the following,

where

Accuracy of the Douglas scheme to Equation (36), we apply Taylor’s series on each term. After some simplification, resultant is as follows,

Dividing above Equations (54) and (55) by k and take into account Equation (5), so resultants are as follows:

let us look above equation in new way,

Now principle part of the truncation error (PPTE) is as follows:

which shows that this scheme is 4th order accurate in space, such as

From accuracy, we find principle part of the truncation error along with Equ- ation (58). Which shows that Douglas scheme is consistent because

Stability of the associated finite difference Equation (51), which is in linear form, is

Consider Equations (44) (60), in the following few important steps.

Apply Von-Neumann stability analysis to Equation (60), we get the following

Above Equation (63), satisfies the Von-Neumann stability criterion, which shows that fourth order implicit scheme is unconditionally stable [

Richardson extrapolation method lead to considerable improvement of numerical results which solving the partial differential equation system by finite difference method. Richardson’s extrapolation formulae are [

above formula leads to get fourth order accuracy [

The aim of the accuracy is assessed by some redefined norms, associated with the consistency of the finite difference schemes, such scaled measurement to error defined in term of norms specially

where

Numerical computations have been performed using the uniform grid [

time = t | x | Error | Error | ||||
---|---|---|---|---|---|---|---|

0.5 | −7.6 | 0.0046011 | 0.0041001 | 0.000501 | 0.9953994 | 0.9952149 | 0.000184 |

−3.2 | 0.0085027 | 0.0083021 | 0.000200 | 0.9914981 | 0.9911982 | 0.000299 | |

−1.6 | 0.0096107 | 0.0092104 | 0.000401 | 0.9903902 | 0.9900162 | 0.000374 | |

−0.4 | 0.0176852 | 0.01761453 | 0.000071 | 0.9823165 | 0.9821154 | 0.000202 | |

2 | 0.0975382 | 0.0972546 | 0.00013 | 0.9024705 | 0.9020103 | 0.000611 | |

4 | 0.4845664 | 0.4841066 | 0.00046 | 0.5154585 | 0.5152934 | 0.000160 | |

5.6 | 0.7288725 | 0.7281615 | 0.00071 | 0.2711472 | 0.2710380 | 0.000109 | |

8.4 | 0.8968938 | 0.8965883 | 0.00031 | 0.1031154 | 0.1030094 | 0.000106 |

time = t | x | Error | Error | ||||
---|---|---|---|---|---|---|---|

0.5 | −7.6 | 0.0041091 | 0.0041001 | 0.000009 | 0.9952994 | 0.9952149 | 0.0000845 |

−3.2 | 0.0083067 | 0.0083021 | 0.000021 | 0.9911981 | 0.9911001 | 0.0000981 | |

−1.6 | 0.0092145 | 0.0092104 | 0.000050 | 0.9900902 | 0.9900162 | 0.0000742 | |

−0.4 | 0.0176109 | 0.0176141 | 0.000011 | 0.9821165 | 0.9821154 | 0.0000220 | |

2 | 00.0972592 | 0.0972546 | 0.000017 | 0.9020705 | 0.9020103 | 0.0000601 | |

4 | 0.4841021 | 0.4841066 | 0.000060 | 0.5152585 | 0.5152934 | 0.0000351 | |

5.6 | 0.7281665 | 0.7281615 | 0.000011 | 0.2710472 | 0.2710380 | 0.0000923 | |

8.4 | 0.8965813 | 0.8965883 | 0.000005 | 0.1030154 | 0.1030094 | 0.000006 |

Time = t | x | Error | Error | ||||
---|---|---|---|---|---|---|---|

0.5 | −7.6 | 0.0041211 | 0.0041001 | 0.0000021 | 0.9952117 | 0.9952149 | 0.00000032 |

−3.2 | 0.0083227 | 0.0083021 | 0.0000021 | 0.9911012 | 0.9911001 | 0.00000011 | |

−1.6 | 0.0092607 | 0.0092104 | 0.0000053 | 0.9900121 | 0.9900162 | 0.00000041 | |

−0.4 | 0.0176152 | 0.0176141 | 0.0000011 | 0.9821116 | 0.9821154 | 0.00000038 | |

2 | 0.0972382 | 0.0972546 | 0.0000017 | 0.9020100 | 0.9020103 | 0.00000003 | |

4 | 0.4841664 | 0.4841066 | 0.0000059 | 0.5152959 | 0.5152934 | 0.00000025 | |

5.6 | 0.7281725 | 0.7281615 | 0.0000015 | 0.2710339 | 0.2710380 | 0.0000004 | |

8.4 | 0.8965938 | 0.8965883 | 0.000005 | 0.1030091 | 0.1030094 | 0.0000003 |

t | Grid | Crank Nicolson | Douglas | ||||
---|---|---|---|---|---|---|---|

1.0 | 31 × 31 | ||||||

51 × 51 | |||||||

81 × 81 | |||||||

101 × 101 |

t | Crank Nicolson | Douglas | ||||
---|---|---|---|---|---|---|

0.1 | 0.0012 | 0.000031 | 0.0000013 | 0.000067 | 0.0000011 | 0.00000079 |

0.3 | 0.0019 | 0.000043 | 0.0000027 | 0.000071 | 0.0000019 | 0.00000008 |

0.5 | 0.00017 | 0.000051 | 0.0000039 | 0.000079 | 0.0000031 | 0.000000081 |

0.7 | 0.00021 | 0.000079 | 0.0000048 | 0.000081 | 0.0000042 | 0.000000094 |

schemes CN and Douglas at different grids and time levels. Rate of convergence is defined in

In this chapter, the solution to one dimensional coupled Fisher KPP system is successfully approximated by a various numerical ﬁnite diﬀerence schemes. Explicit FTCS is conditionally stable, and we give more attention to parameter R_{1} and R_{2}, which can be used to stabilized the results as we can see from

Grid | Crank Nicolson | Douglas | ||||
---|---|---|---|---|---|---|

Rate | Rate | Rate | ||||

31 × 31 | - | - | - | - | - | - |

51 × 51 | 2.1882 | 3.0712 | 3.1098 | 2.881 | 3.0941 | 3.8172 |

81 × 81 | 2.1761 | 3.1971 | 3.2910 | 2.5619 | 3.4218 | 3.8971 |

101 × 101 | 1.4097 | 3.3468 | 4.0118 | 2.9431 | 3.7689 | 3.9981 |

as we explained in methodology section [

Shahid Hasnain, Prof. Daoud Mashat and Muhammad Saqib is thankful to Dr Muhammad Faheem Afzaal, Department of Chemical Engineering, Imperial College London and Vineet K. Srivastava, Scientist, ISTRAC/ISRO, Bangalore, India for thoughtful remarks. This research was supported by Department of Mathematics, division of Numerical Analysis, King Abdulaziz University, Jeddah, Saudi Arabia.

There is no conflict of interest in this research paper.

Hasnain, S., Mashat, D.S., Saqib, M., Ghaleb, S.A. and Mshary, N.Y. (2017) Numerical Approximation to Nonlinear One Dimensional Coupled Reaction Diffusion System. Journal of Applied Mathematics and Physics, 5, 1551-1574. https://doi.org/10.4236/jamp.2017.58129