RBFs are a type of scalar function based on distance measurement, whose core characteristic is that the function value only depends on the distance from the two points. The advantages of RBFs include local response characteristics, efficient processing of sparse or high-dimensional data, simple mathematical form, easy-to-implement, and parallel computing.

For 2D steady-state problems, the commonly-used RBFs include three types, the detailed formula is shown in Equation 3 φ r = 1 + ε r 2 , Multiquadric , e − ε r 2 , Gaussian , r 2 ⁡ log ⁡ r , Thin Plate Spline . (3)

Here, r = X i − X j is the Euclidean distance between two points X i = x i , y i and X j = x j , y j , ε is the RBF shape parameter.

Since there is only one space variable in Fisher’s Equation 1, we consider the time variable “equally” as a new space variable. More specifically, the Fisher’s equation is considered as a “equally” steady-state equation. The corresponding space-time RBFs has the form ϕ r ¯ = 1 + ε r ¯ 2 , Multiquadric , e − ε r ¯ 2 , Gaussian , r ¯ 2 ⁡ log r ¯ , Thin Plate Spline . (4)

Here, r ¯ = x i − x j 2 + t i − t j 2 is the Euclidean distance between two space-time points X ¯ i = x i , t i and X ¯ j = x j , t j .

Before implementation of the one-level meshless method, collocation points should be provided. More specifically, the space variable interval a , b is divided into small segments a = x 0 < x 1 < . . . < x n = b and the time variable interval 0 , T is divided into segments 0 = t 0 < t 1 < . . . < t n = T . The interval division is usually under uniform scheme, but it is also workable for un-uniform scheme.

According to the basic theory of collocation methods, the approximate solution of the function u x , t at an arbitrary point X ¯ = x , t in Fisher’s equation has the form u ¯ · ≈ ∑ j = 1 N λ j ϕ j · (5) with λ j j = 1 N the unknown coefficients and ϕ j · = 1 + ε r ¯ j 2 = 1 + ε 2 x − x j 2 + ε 2 t − t j 2 , where j is the index of collocation points and N is the total collocation point number.

To illustrate the one-level meshless method, we substitute Equation 6 into Equations 1, 2 at space-time points X ¯ k = x k , t k k = 1 n × n . Then, one can obtain the following equations ∑ j = 1 N λ j L ϕ j X ¯ k , X ¯ j = 0 , k = 1 , . . . , n − 2 2 , (6) ∑ j = 1 N λ j ϕ j X ¯ k , X ¯ j = u 0 X ¯ k , ∑ j = 1 N λ j ϕ j X ¯ k , X ¯ j = u ¯ X ¯ k , k = n − 2 2 + 1 , . . . , n 2 . (7)

Here, the operator is shown in the following Equation 8 L ϕ j = α ∂ 2 ϕ j ∂ x 2 − ∂ ϕ j ∂ t + γ ϕ j 1 − ϕ j . (8)

For multiquadric RBF ϕ j r = 1 + ε r ¯ j 2 = 1 + ε 2 x − x j 2 + ε 2 t − t j 2 , we have the corresponding derivatives in Equations 9–11 ∂ ϕ j ∂ x = ε 2 x − x j 1 + ε 2 r ¯ j 2 − 1 2 , (9) ∂ 2 ϕ j ∂ x 2 = ε 2 1 + ε 2 r ¯ j 2 − 1 2 1 − ε 2 x − x j 2 / 1 + ε 2 r ¯ j 2 , (10) ∂ ϕ j ∂ t = ε 2 t − t j 1 + ε 2 r ¯ j 2 − 1 2 . (11)

In order to obtain a square interpolation matrix, we consider N = n × n . Equations 6, 7 has the matrix form as shown in Equation 12 A λ = f , (12) where A = ϕ k j is N × N interpolation matrix, λ and f are N × 1 vectors.

Equation 9 can be directly solved to get the unknowns λ . Then the approximation solution of the unknown function in the Fisher’s equation can be solved by using Equation 5.

For the constant diffusion coefficient α = 1 and the reaction factor γ = 6 , the fisher’s equation has the form in Equation 13 u t = u x x + 6 u 1 − u , − 1 < x < 1 , 0 < t < T . (13)

The corresponding exact solution is u x , t = 1 1 + e x − 5 t 2 . (14)

The corresponding initial condition and boundary condition can be deduced from Equation 14.

At time t = 1 , the variation of shape parameter versus the L 2 − error is presented in Figure 1 for fixed collocation point number N = 117 . It can be seen that the quasi-optimal L 2 − error is 6.8893 × 10 − 8 for shape parameter ε = 1.45 . This is more accurate than the most accurate numerical result 4.33 × 10 − 5 in [14]. For shape parameter ε = 1.45 , Figure 2 is plotted to show that the numerical solution is highly consistent with the analytical solution.

Here, we consider the following Fisher’s equation as shown in Equation 15 u t = u x x + u 1 − u 6 , − 1 < x < 1 , 0 < t < T (15)

The corresponding exact solution is shown in Equation 16 u x , t = 3 1 2 tanh − 3 x 4 + 15 t 8 + 1 2 . (16)

The quasi-optimal choice of shape parameter is the same as Example 4.1. For shape parameter ε = 0.44 , the L 2 − error is 2.8366 × 10 − 10 at time t = 1 , which is also far more accurate than the most accurate result 1.27 × 10 − 5 in [14]. Figure 3 is plotted to show that the numerical solution is highly consistent with the analytical solution at three different times t = 0.1 , t = 0.5 , t = 1 .

In this example, we consider the following Fisher’s equation as shown in Equation 17 u t = u x x + u 1 − u 2 , − 1 < x < 1 , 0 < t < T . (17)

The corresponding analytical solution is shown in Equation 18 u x , t = − 1 2 tanh 2 4 x − 18 t 2 + 1 2 . (18)

At time t = 1 , the quasi-optimal choice of shape parameter is ε = 0.49 with the corresponding L 2 − error 2.3819 × 10 − 8 . It is also more accurate than the most accurate result 6.00 × 10 − 5 in [14]. Figure 4 is plotted to show that the numerical solution is highly consistent with the analytical solution at three different times t = 0.1 , t = 0.5 , t = 1 .