As is known to all, the boundary and initial conditions should be considered to get the analytical/numerical solutions of Equation 1. The initial condition has a simple form u x , t = u ¯ x , for  t = 0 (3)

The boundary condition has the expression B u x , t = u ^ x , t , for  x ∈ a , b (4) where B stands for the boundary operator. More specifically, B u x , t = u x , t and B u x , t = ∂ u x , t ∂ n are for Dirichlet boundary and Neumann boundary, respectively. u ¯ x and u ^ x , t are prescribed smooth functions.

As a time-dependent initial boundary value problem, numerical solution procedure for Equations 2–4 should consider two procedures including time discretization and space discretization.

In literature, one should transform Equation 2 into a suitable form using appropriate method [24]. More specifically, the time variable should be discretized first and Equation 2 will be transformed into a time-independent new equation. In this paper, we consider using the commonly-used time discretization scheme, which is computationally efficient, especially in parallel computing.

Denote t s = s Δ t , s = 0 , 1 , . . . , L with time step Δ t = T / L for total interval 0 , T . According to the definition of first-order differentiation or forward difference on the time derivative, the first term in the left side of Equation 2 has the form [25] ∂ u x , t ∂ t = u x , t s + 1 − u x , t s Δ t + ο Δ t (5)

Substituting Equation 5 into Equation 2, and considering the initial condition Equation 3, we have α ∂ 2 u s + 1 ∂ x 2 = u s + 1 − u s Δ t − β u s + 1 1 − u s + 1 γ (6) where u s + 1 = u x , t s + 1 , u s = u x , t s .

The above-mentioned time discretization scheme will be incorporated with the local meshless scheme to deal with the resulting systems Equation 6.

Since Equation 6 is an elliptic-type PDE, the RBF-based collocation methods can be employed to obtain its numerical solutions. However, the interpolation system generated by the collocation method is usually ill-conditioned which led to increased computational cost [26]. To circumvent this issue, we consider incorporate the local approach into the G-RBF collocation method in this section. Only a few neighboring points is required by the local approach and the interpolation system generated by this local method is sparse.

For generality, we denote x i i = 1 , 2 , . . . , N as a set of collocation points in a , b . For a given central point x i , one should find the nearest local points x l i l = 1 , 2 , . . . , k around x i . The nearest neighbor local template ( k = 3 ) is commonly used in one-dimensional situations, with the aim of utilizing the sparsity of local schemes to avoid ill conditioned matrices, improve computational efficiency, and demonstrate the advantages of simple and easy implementation of the method. The “local” concept implies that a subset x l i l = 1 k ⊂ x i i = 1 N that contains these k points. These overlapping subsets satisfy ∪ x l i l = 1 k = x i i = 1 N .

For the local approximation at the central point x ∈ x l i l = 1 k , one can obtain the local approximation formulation by the following Equation 7 u k x = ∑ l = 1 k λ l i G x , x l i (7) where G x , x l i = e − ε 2 x − x l i 2 (8) is the G-RBF with shape parameter ε [27].

For all collocation points in x l i l = 1 k , the corresponding local approximation is given as the following Equation 9 u k x j i = ∑ l = 1 k λ l i G x j i , x l i , j = 1 , 2 , . . . , k (9) which leads to the local system in the following Equation 10 u k i = u k x 1 i u k x 2 i ⋮ u k x k i = G 11 i   G 12 i ⋯ G 1 k i G 21 i   G 22 i ⋯ G 2 k i ⋯ ⋯ G k 1 i   G k 2 i ⋯ G k k i λ 1 i λ 2 i ⋮ λ k i = G i λ i (10)

Here, G j l i = G x j i , x l i , j , l = 1 , 2 , . . . , k is notation of Equation 8.

For the governing Equation 6 under boundary condition Equation 4, the approximate solution of with local G-RBF collocation formulation can be obtained by collocation on all points x i i = 1 , 2 , . . . , N . More specifically, the governing Equation 6 is collocated at points x i i = 2 , . . . , N − 1 α ∂ 2 u s + 1 x i ∂ x 2 = u s + 1 x i − u s x i Δ t − β u s + 1 x i 1 − u s + 1 γ x i , (11)

The boundary condition Equation 4 is collocated at points x i i = 1 , N B u x i = u ^ x i (12)

Equations 11 and 12 lead to a global sparse system [28].

As a starting example, we consider the coefficients α = 1 , β = 6 and γ = 1 in the governing Equation 2. This will lead to the following equation u t = u x x + 6 u 1 − u (13)

The physical domain is chosen as 0 ≤ x ≤ 2 , 0 ≤ t < T . The corresponding solution for Equation 13 is in a series form [29], we use the following corresponding closed form u x , t = 1 1 + e x − 5 t 2 (14)

The initial/boundary condition can be deduced from Equation 14.

For the selection of shape parameter, we use the quasi-optimal prior-tested method in [30]. Here, we consider ε = 1.36 . Figure 1 provides the convergence curve of collocation point number versus maximum error for fixed time step Δ t = 0.001 at t = 1 . From which we can see that the maximum error decreases in a smooth phenomenon with the increasing collocation point number. For fixed collocation point number N = 30 and time step Δ t = 0.001 , Figure 2 provides the absolute errors at different times t = 0.1 , t = 0.5 and t = 1 . It can be observed that the absolute errors are all at the same level for various times.

Here, we consider the coefficients α = 1 , β = 1 and γ = 6 in Equation 2. The concrete expression is u t = u x x + u 1 − u 6 (15)

The physical domain is considered in a larger scope 0 ≤ x ≤ 10 , 0 ≤ t < T and the corresponding analytical solution for Equation 15 is [12] u x , t = 1 2 tanh − 3 x 4 + 15 t 8 + 1 2 1 / 3 (16)

The initial/boundary condition can be deduced from Equation 16.

Under the quasi-optimal prior-tested method in [30], we consider shape parameter ε = 4.45 . Figure 3 presents the convergence curve, which illustrates the relationship between the number of collocation points and the maximum error at a fixed time step of Δ t = 0.001 and time t = 1 . As observed, the maximum error exhibits a smooth decreasing trend as the number of collocation points increases for N ≤ 30 . This phenomenon is similar with the previous example. For N > 30 , the solution accuracy reaches a plateau and no longer improves. This observation indicates that the proposed method’s performance is inherently constrained by the initial parameter settings, achieving its optimal accuracy under these specific conditions. With N = 30 collocation points and a fixed time step Δ t = 0.001 , Figure 4 presents the absolute errors evaluated at t = 0.1 , t = 0.5 and t = 1 . The results indicate that the absolute errors are comparable for each time instance.

The proposed method shows high efficiency, robustness and good convergence in solving one-dimensional FE. The numerical experiments show that the maximum error decreases steadily with the increase of the allocation point under the fixed time step, which proves its convergence. When the number of collocation points is fixed, the absolute error at different time points remains stable, indicating that the method has time stability. The numerical results are highly consistent with the solutions in literature [29], which verifies its effectiveness and universality for Fisher type problems. Combined with the advantages of time discretization scheme in parallel computing, this method provides an efficient computational framework for dealing with large-scale reaction-diffusion systems.

Using the advantages of other localized methods [31, 32], the deeper theoretical aspects of the local method—such as the optimal selection of local templates and rigorous stability analysis of the time integration scheme—could serve as promising directions for future research.