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This research paper represents a numerical approximation to three interesting equations of Fisher, which are linear, non-linear and coupled linear one dimensional reaction diffusion equations from population genetics. We studied accuracy in term of L∞ error norm by random selected grids along time levels for comparison with exact results. The test example demonstrates the accuracy, efficiency and versatility of the proposed schemes. It is shown that the numerical schemes give better solutions. Moreover, the schemes can be easily applied to a wide class of higher dimension non-linear reaction diffusion equations.

Reaction diffusion equations arise as models for the densities of substances or organisms that disperse through space by Brownian motion, random walks, hydrodynamic turbulence, or similar mechanisms, and that react to each other and their surroundings in ways that affect their local densities [

Diffusion is a description of movement that arises as a result of an object or organism making many short movements in random directions. The diffusive description of random motion emerges as a continuum limit of such random walks when the length Δ x of each step and the time Δ t required for each step go to zero in such a way that the ratio ( Δ x ) 2 / Δ t remains constant. To understand how this works it is useful to consider a simple example in one space dimension. Suppose that an organism moves along a line by moving a distance Δ x to the left with probability 1/2 or a distance Δ x to the right with probability 1/2 at each time step Δ t . Suppose that ρ ( x , t ) is the probability that the organism is at location x at time t. To arrive at that point at that time it must have been either one step to the left at time t − Δ t and then moved to the right, or one step to the right and have moved to the left. Thus, we have

ρ ( x , t ) = 1 2 ρ ( x + Δ x , t − Δ t ) + 1 2 ρ ( x − Δ x , t − Δ t ) (1)

If we subtract ρ ( x , t − Δ t ) from both sides and divide by Δ t in Equation (1), we obtain,

ρ ( x , t ) − ρ ( x , t − Δ t ) Δ t = 1 2 Δ t [ ρ ( x + Δ x , t − Δ t ) − 2 ρ ( x , t − Δ t ) + ρ ( x − Δ x , t − Δ t ) ] (2)

Suppose that we now impose the diffusive scaling, ( Δ x ) 2 / Δ t = 2 D . Let us look at Equation (2),

ρ ( x , t ) − ρ ( x , t − Δ t ) Δ t = D ( Δ x ) 2 [ ρ ( x + Δ x , t − Δ t ) − 2 ρ ( x , t − Δ t ) + ρ ( x − Δ x , t − Δ t ) ] (3)

From above Equation (3), the expression on the left is a difference quotient in t and also the expression on the right is a second difference in x. Taking the limit of expression in Equation (3), as ( Δ x , Δ t ) → 0 , while in Equation (2) remains in force yields the diffusion equation,

∂ ρ ∂ t = D ∂ 2 ρ ∂ x 2 . (4)

Mathematically this is identical to the heat equation. Note that the scaling, where D is the square of the distance Δ x moved by the organism in a time unit Δ t , produces a coefficient in front of the term ∂ 2 ρ / ∂ x 2 , which is equal to 1/2 of the square of the distance moved per unit time. This interpretation of the diffusion coefficient D is valid in any number of dimensions.

In the context of ecological models, the reaction terms in reaction diffusion equations and systems are typically the same as those that are used in non- spatial population models based on ordinary differential equations. Thus, for a single population, the reaction terms would be those that might occur in a model for a population density ρ ( t ) of the form

∂ ρ ∂ t = f ( ρ ) , (5)

where f ( ρ ) often has the form f ( ρ ) = g ( ρ ) ρ . Common choices for f ( ρ ) are f ( ρ ) = R ρ (linear growth), f ( ρ ) = R ρ ( 1 − ρ / K ) (logistic growth), or f ( ρ ) = R ρ ( ρ − a ) ( 1 − ρ / K ) with a ∈ ( 0, K ) (growth with Allee effect). For systems, typical reaction terms are those that occur in non-spatial models for competition, mutualism, or predator-prey interactions. Those include Lotka-Volterra models, but also more general models such as predator-prey models with a functional response. In the case of systems the stability analysis often involves the eigenvalues of matrices obtained by linearising about the equilibria. Equilibria and eigenvalues play a similar role in the analysis of reaction diffusion models, but the eigenvalues generally are associated with differential operators rather than matrices.

Here is the outline of the article. In Section 2, we mentioned literature review according to scope of the equation and numerical treatment also in section 3, we derived governing equation and its three interesting types and in section 4, methodology is explained. In the section 5, we discussed results in detail.

A well known researchers have studied such model problem, for example, Abdullaev [

Recently, numerical solution to Fisher’s equation, have studied by many researchers, such as Wang [

In 1937 Fisher [

The linear form of Fisher’s equation is as follows,

u t = β u x x + α ( 1 − u ( x , t ) ) , (6)

where β is diffusive constant with value 0 ≤ β ≤ 1 and α is reactive constant with value 0 ≤ α ≤ 1 . Also analytical solution to above Equation (6) is,

u ( x , t ) = 1 − cosh x cosh 1 − 16 π 2 ∑ n = 1 ∞ ( − 1 ) n cos ( 2 n − 1 ) π x / 2 ( 2 n − 1 ) [ ( 2 n − 1 ) 2 π 2 + 4 ] e − ( 1 + ( 2 n − 1 ) 2 π 2 / 4 ) t , (7)

with boundary conditions are

u ( − 1 , t ) = u ( 1 , t ) = 0 , (8)

and initial condition also,

u ( x , 0 ) = 0. (9)

The coupled linear system is as follows,

u t = u x x + ( u ( x , t ) − v ( x , t ) ) + F ( x , t ) v t = v x x + ( u ( x , t ) + v ( x , t ) ) + G ( x , t ) } (10)

with analytical solution,

u ( x , t ) = e t sin x v ( x , t ) = e t cos x } (11)

The generalized form of nonlinear Fisher’s equation is as follows,

u t = u x x + u ( 1 − u ) ( u − α 1 ) , 0 < α 1 < 1 (12)

with analytical solution,

u ( x , t ) = 1 2 ( 1 + α 1 ) + ( 1 2 − 1 2 α 1 ) tanh [ 2 ( 1 − α 1 ) x 4 + ( 1 − α 1 2 ) 4 t ] , (13)

the initial and boundary conditions are taken from the exact solution (13).

Let us apply numerical methods like Finite Difference Schemes (Forward in time and central in space (FTCS), Crank Nicolson (CN) and Douglas), to solve such Equations ((6), (10), (12)) in finite domain Ω = [ 0 , 1 ] . We partitioned the interval [ a , b ] into n equal parts of width h. Place a grid on the rectangle region R by drawing vertical and horizontal lines through the points with coordinates x i , where x i = a + i h for each i = 0 , 1 , 2 , ⋯ , n also the lines x = x i represent grid lines, we assume t n = n t , n = 0 , 1 , ⋯ where n is the time grid step size. We denote the exact and numerical solutions at the grid point ( x m , t n ) by U m n and u m n respectively.

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 Equations ((6), (10), (12)) respectively, we get the following

u i n + 1 = u i n + R ( u i + 1 n − 2 u i n + u i − 1 n ) + Q ( 1 − u i n ) , (14)

where R = k β h 2 , and Q = k α . Above equation (14) represents descritezation to linear form of Fisher’s equation.

u i n + 1 = u i n + R ( u i + 1 n − 2 u i n + u i − 1 n ) + k u i n − k v i n + k F ( u i n , v i n , x , t ) v i n + 1 = v i n + R ( v i + 1 n − 2 v i n + v i − 1 n ) + k u i n + k v i n + k G ( u i n , v i n , x , t ) } (15)

above Equation (15) represents descritezation to coupled linear system.

u i n + 1 = u i n + R ( u i + 1 n − 2 u i n + u i − 1 n ) + k u i n ( 1 − u i n ) ( u i n − α 1 ) , (16)

above Equation (16) represents descritezation to nonlinear generalized form of Fisher’s equation.

Since the one dimensional Fisher’s equation or system is well posed, make sure the spacing h for spatial and k for time of the finite difference grid are made sufficiently small [

analysis, the FTCS scheme is always conditionally stable, which is 0 < R ≤ 1 2 .

Let us apply implicit finite difference scheme, which is Crank Nicolson. This method uses central finite difference approximation for both time and space derivatives at the point ( x m , t n ) [

u i n + 1 − u i n k = k β 2 h 2 δ x 2 [ u i n + 1 + u i n ] + α ( 1 − 1 2 ( u i n + 1 + u i n ) ) − R 1 u i + 1 n + 1 + ( 1 + 2 R 1 + 0.5 Q 1 ) u i n + 1 − R 1 u i − 1 n + 1 = R 1 u i + 1 n + ( 1 − 2 R 1 − 0.5 Q 1 ) u i n + R 1 u i − 1 n + Q 1 (17)

where R 1 = k β 2 h 2 and Q 1 = k α . Above Equation (17) represents descritezation

using Crank Nicolson to linear form of Fisher’s equation. Now let us look at coupled linear system, in the following way,

− R 1 u i + 1 n + 1 + ( 1 + 2 R 1 − 0.5 k ) u i n + 1 + 0.5 k v i n + 1 − R 1 u i − 1 n + 1 = R 1 u i + 1 n + ( 1 − 2 R 1 + 0.5 k ) u i n − 0.5 k v i n + k F ( 1 2 ( u i n + 1 + u i n ) , 1 2 ( v i n + 1 + v i n ) , x , t ) − R 1 v i + 1 n + 1 + ( 1 + 2 R 1 − 0.5 k ) v i n + 1 − 0.5 k u i n + 1 − R 1 v i − 1 n + 1 = R 1 v i + 1 n + ( 1 − 2 R 1 + 0.5 k ) v i n + 0.5 k u i n + k G ( 1 2 ( u i n + 1 + u i n ) , 1 2 ( v i n + 1 + v i n ) , x , t ) } (18)

Above equation (18) represents descritezation using Crank Nicolson to coupled linear system. Now let us look at generalized nonlinear Fisher’s equation using Crank Nicolson,

− R 1 u i + 1 n + 1 + ( 1 + 2 R 1 ) u i n + 1 − R 1 u i − 1 n + 1 = R 1 u i + 1 n + ( 1 − 2 R 1 ) u i n + k 2 ( u i n + 1 + u i n ) ( 1 − 0.5 ( u i n + 1 + u i n ) ) ( 0.5 ( u i n + 1 + u i n ) − α 1 ) (19)

Let us apply another implicit scheme to Equations ((6), (12)) in an order respectively.

u i n + 1 − u i n k = β 2 h 2 [ 1 + δ x 2 ] − 1 ( u i n + 1 + u i n ) + α ( 1 − 1 2 ( u i n + 1 + u i n ) ) [ 1 + δ x 2 ] ( u i n + 1 − u i n k ) = β 2 h 2 ( u i n + 1 + u i n ) + [ 1 + δ x 2 ] α ( 1 − 1 2 ( u i n + 1 + u i n ) ) ( u i n + 1 − u i n ) + δ x 2 ( u i n + 1 − u i n ) = R 1 ( u i n + 1 + u i n ) + [ 1 + δ x 2 ] α ( 1 − 1 2 ( u i n + 1 + u i n ) ) } (20)

Now Douglas scheme to nonlinear generalized Fisher’s equation,

u i n + 1 − u i n k = β 2 h 2 [ 1 + δ x 2 ] − 1 ( u i n + 1 + u i n ) + 1 2 ( u i n + 1 + u i n ) ( 1 − 1 2 ( u i n + 1 + u i n ) ) ( 1 2 ( u i n + 1 + u i n ) − α 1 ) [ 1 + δ x 2 ] ( u i n + 1 − u i n k ) = β 2 h 2 ( u i n + 1 + u i n ) + [ 1 + δ x 2 ] 1 2 ( u i n + 1 + u i n ) ( 1 − 1 2 ( u i n + 1 + u i n ) ) ( 1 2 ( u i n + 1 + u i n ) − α 1 ) [ 1 + δ x 2 ] ( u i n + 1 − u i n ) = R 1 ( u i n + 1 + u i n ) + [ 1 + δ x 2 ] 1 2 ( u i n + 1 + u i n ) ( 1 − 1 2 ( u i n + 1 + u i n ) ) ( 1 2 ( u i n + 1 + u i n ) − α 1 ) } (21)

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 L ∞ , which is outlined below:

L ∞ = max i | u i Exact − u i Approximation | (22)

Numerical computations have been performed using the uniform grid. We used FTCS, Crank Nicolson and Douglas finite difference schemes to analyse numerical behaviour of simple linear Fisher;s equation, one dimensional linear coupled system and non-linear Fisher’s equation respectively. First we look at the linear Fisher’s equation by finite difference schemes as in

Secondly, we look at the coupled linear system by finite difference schemes as in

Lastly, we look at the generalized Fisher’s equation by finite difference schemes as in

Grids | k = Time Step | h = Space Step | |
---|---|---|---|

51 × 51 | 0.0001 | 0.115 | 0.0400 |

101 × 101 | 0.0001 | 0.0115 | 0.0200 |

225 × 225 | 0.0001 | Inf | 0.0089 |

1011 × 1011 | 0.0001 | Inf | 0.0020 |

k = Time | t | |
---|---|---|

0.01 | 1 | |

0.001 | Inf | 1 |

0.0001 | 0.0115 | 1 |

Grids | k = Time Step | h = Space Step | |
---|---|---|---|

51 × 51 | 0.0001 | 0.0400 | |

101 × 101 | 0.0001 | 0.0200 | |

225 × 225 | 0.0001 | 0.0089 | |

1011 × 1011 | 0.0001 | 0.0020 |

k = Time | t | |
---|---|---|

0.01 | 1 | |

0.001 | 1 | |

0.0001 | 1 |

Grids | k = Time Step | h = Space Step | |
---|---|---|---|

51 × 51 | 0.0001 | 0.0400 | |

101 × 101 | 0.0001 | 0.0200 | |

225 × 225 | 0.0001 | 0.0089 | |

1011 × 1011 | 0.0001 | 0.0020 |

k = Time | t | |
---|---|---|

0.01 | 1 | |

0.001 | 1 | |

0.0001 | 1 |

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

Grid | h = Space Step | Grid | h = Space Step | ||

31 × 31 | 0.0667 | 31 × 31 | 0.0667 | ||

45 × 45 | 0.0455 | 45 × 45 | 0.0455 | ||

77 × 77 | 0.0263 | 77 × 77 | 0.0263 |

Grid | ||||||||
---|---|---|---|---|---|---|---|---|

51 × 51 | −0.41156459 | −0.375022428 | 0.0365 | 0.0981 | −1.02567897 | −0.93461058 | 0.0911 | 0.0981 |

71 × 71 | 0.77048108 | 0.706718997 | 0.0638 | 0.0914 | 0.79231412 | 0.72674521 | 0.0656 | 0.0915 |

151 × 151 | 0.98573361 | 0.947763396 | 0.0380 | 0.0426 | −0.49973194 | −0.48048240 | 0.0192 | 0.0426 |

201 × 201 | 0.94172500 | 0.946871908 | 0.0051 | 0.0060 | 0.57840883 | 0.58157006 | 0.0032 | 0.0060 |

271 × 271 | 0.37021916 | 0.402856751 | 0.0326 | 0.0974 | −1.04131673 | −1.13311659 | 0.0918 | 0.0974 |

Grid | ||||||||
---|---|---|---|---|---|---|---|---|

51 × 51 | −0.99034809 | −0.9903612342 | −0.14564560 | −0.1456475382 | ||||

71 × 71 | 0.910207179 | 0.91021335413 | −0.41656319 | −0.4165660175 | ||||

151 × 151 | −0.79446192 | −0.7944631049 | 0.608959970 | 0.60896087175 | ||||

201 × 201 | −0.84231287 | −0.8423135783 | 0.540842878 | 0.54084332896 | ||||

271 × 271 | −0.09140859 | −0.0914086402 | −0.99681817 | −0.9968186284 |

Grids | k = Time Step | h = Space Step | |
---|---|---|---|

21 × 21 | 0.0001 | 0.0047 | 1 |

61 × 61 | 0.0001 | 0.0047 | 0.3333 |

121 × 121 | 0.0001 | 0.0047 | 0.1667 |

301 × 301 | 0.0001 | 0.0047 | 0.0667 |

Grids | k = Time Step | h = Space Step | |
---|---|---|---|

21 × 21 | 0.0001 | 1 | |

61 × 61 | 0.0001 | 0.3333 | |

121 × 121 | 0.0001 | 0.1667 | |

301 × 301 | 0.0001 | 0.0667 |

A | Grid | k = Time | h = Space Step | |
---|---|---|---|---|

0.1 | 41 × 41 | 0.0029 | 0.0001 | 0.5000 |

0.4 | 60 × 60 | 0.0103 | 0.0001 | 0.3390 |

0.6 | 131 × 131 | 0.0079 | 0.0001 | 0.1538 |

To analyse the graphic representation to linear Fisher’s equation, we have

In this paper, the solution to linear form of the Fishers equation, coupled linear system and generalized Fisher’s equation is successfully approximated by a various numerical finite difference schemes. Two of them are implicit in nature such as Crank Nicolson and Douglas and one is explicit FTCS schemes. We have to pay attention to parameter R , which can stabilize the results as we can see from figures and tables. For instant, Von-Neumann’s method of stability analysis can not be used other than locally, since it only applies to linear finite

difference schemes. In many cases, numerical experimentation, such as solving the finite difference schemes using progressively smaller grid spacing and examining the behaviour of the sequence of the values of u ( x , t ) obtained at given points, is the suitable method available with which to assess the numerical model. The various methods of obtaining a finite difference numerical model corresponding to a particular mathematical model may result in either explicit or implicit finite difference schemes. Explicit schemes are conditionally stable and implicit schemes are unconditionally stable. Two implicit schemes are also applied to improve accuracy, stability restrictions and consistency in solution. It can be observed that the computed results show excellent agreement with the analytical solution. Our main purpose of this research is to improve accuracy in result. Accuracy in results is glanced from figures and tables.

Bader Saad Alshammari and Prof. Daoud Mashat are very 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.

Alshammari, B.S. and Mashat, D.S. (2017) Numerical Study of Fisher’s Equation by Finite Difference Schemes. Applied Mathematics, 8, 1100- 1116. https://doi.org/10.4236/am.2017.88083