The TPS-RBF is one of the traditional RBFs based on distance measurement. Compared with the multiquadrics and Gaussian RBF [22, 23], it is parameter-free. It also has the advantage of high flexibility, can adapt to complex data distributions, and performs particularly well when dealing with non-linear data.

For 2D problems, it is defined as φ r = r 2 ⁡ log ⁡ r . (4)

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

There is only one space variable x in F–KPP Equation 1. To facilitate the numerical procedure, the time variable t is “equally” considered as a new space variable. Hence, one can obtain a new space–time point x , t . To avoid confusion with Equation 4, the corresponding space–time RBFs can be expressed as ϕ r ¯ = r ¯ 2 ⁡ log r ¯ , (5) where r ¯ = x i − x j 2 + t i − t j 2 is the distance between the points X ¯ i = x i , t i and X ¯ j = x j , t j .

The collocation method is a numerical technique for solving PDEs by enforcing the governing equation at a set of discrete collocation points [24, 25]. In the context of RBF collocation, the unknown solution is approximated as a linear combination of RBFs centered at the collocation points. The coefficients of the linear combination are then determined by enforcing the governing equation and boundary conditions at the collocation points.

The collocation points should be determined before implementing the collocation method. To be more specific, the interval a , b is divided into x i i = 1 n by inserting n − 2 points, with x 1 = a and x n = b . The interval 0 , T is divided into t i i = 1 n by inserting n − 2 points, with t 1 = 0 and t n = T . The traditional interval division is usually based on a uniform scheme, but it can also be applied to a non-uniform scheme. The corresponding numerical results will be compared in the numerical examples.

In this study, we consider using the Chebyshev–Gauss–Lobatto scheme to generate non-uniform collocation points in the t direction [26]. However, the uniform scheme is still considered for the x direction. Considering t ∈ 0 , 1 , we obtain Equation 6 as follows t i = 1 − cos π 2 n i , i = 1 , . . . , n . (6)

The configuration of the Chebyshev–Gauss–Lobatto scheme reveals that the points are initially dense but gradually become sparse as the index increases.

Based on the fundamental principles of collocation methods, the numerical approximation for u x , t in Equation 1 can be expressed in the following combination form of Equation 5: u ¯ x , t ≈ ∑ j = 1 N λ j ϕ j r ¯ , (7) where λ j j = 1 N represents the unknown coefficients to be determined. ϕ j r ¯ = r ¯ j 2 ⁡ log ⁡ r ¯ j with r ¯ j = x − x j 2 + t − t j 2 and X ¯ j = x j , t j , j = 1 , . . . , N are the N collocation points.

To illustrate the TPS-RBF collocation method, we substitute Equation 7 into Equations 1–3 at N collocation points, X ¯ k = x k , t k , k = 1 , . . . , N . Then, one can obtain Equations 8–11 as follows ∑ j = 1 N λ j L ϕ j r ¯ k j = 0 , k = 1 , . . . , N I , (8) ∑ j = 1 N λ j ϕ j r ¯ k j = u 1 X ¯ k , k = N I + 1 , . . . , N I + N B , (9) ∑ j = 1 N λ j ϕ j r ¯ k j = u 2 X ¯ k , k = N I + N B + 1 , . . . , N T . (10)

Here, L ϕ j = α ∂ 2 ϕ j ∂ x 2 − ∂ ϕ j ∂ t + β ϕ j 1 − ϕ j k , (11)

r ¯ k j = x k − x j 2 + t k − t j 2 , N I is the interior collocation number, N B is the boundary collocation number, and N T is the total collocation number.

The detailed first- and second-order derivatives used in the collocation procedures can be computed. More specifically, for the TPS-RBF ϕ j r ¯ = r ¯ j 2 ⁡ log ⁡ r ¯ j , we have Equations 12–14 as follows: ∂ φ j r ¯ ∂ x = 2 r ¯ · r ¯ x · ⁡ log r ¯ + r ¯ 2 · 1 r ¯ . r x = 2 x − x j log r ¯ + x − x j , (12) ∂ 2 φ j r ¯ ∂ x 2 = 2 ⁡ log r ¯ + 2 x − x j · 1 r ¯ · r x + 1 , = 2 ⁡ log r ¯ + 2 x − x j 2 r ¯ 2 + 1 (13) ∂ φ j r ¯ ∂ t = 2 r ¯ · r ¯ t · ⁡ log r ¯ + r ¯ 2 · 1 r ¯ · r t = 2 t − t j log r ¯ + t − t j . (14)

Equations 8–10 have the following matrix form: Q λ = b , (15) where Q = Q k j is the N T × N T interpolation matrix with elements Q k j = L ϕ j r ¯ k j , k = 1 , . . . , N I ;   j = 1 , . . . , N T , Q k j = ϕ j r ¯ k j , k = N I + 1 , . . . , N I + N B ;   j = 1 , . . . , N T , Q k j = ϕ j r ¯ k j , k = N I + N B + 1 , . . . , N T ;   j = 1 , . . . , N T . λ = λ 1 , λ 2 , . . . , λ N T T

is the unknown coefficient vector. b = 0 , . . . , 0 ⎵ N I , u 1 X ¯ N I + 1 , . . . u 1 X ¯ N I + N B , u 2 X ¯ N I + N B + 1 , . . . u 1 X ¯ N T

is the N T × 1 right-hand side vector.

By solving Equation 15, the unknown λ can be determined, enabling the computation of an approximate solution for the unknown function in the Fisher–KPP equation using Equation 7.

The proposed TPS-RBF collocation method is implemented as follows:

Step 1: Select a set of collocation points in the problem domain.

For the constant diffusion coefficient α = 1 , the reaction factor β = 6 , and the environmental carrying capacity k = 1 , the Fisher–KPP equation takes the following form: u t = u x x + 6 u 1 − u , − 1 < x < 1 , 0 < t < T .

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

In this example, we consider the following modified Fisher–KPP equation: u t = u x x + u 1 − u 2 , 0 < x < 1 , 0 < t < T .

The corresponding analytical solution is u x , t = − 1 2 tanh 2 4 x − 18 t 2 + 1 2 .

In this study, we consider a non-linear Fisher–KPP equation with parameters α = 1 , β = − 1 , and k = 1 . u t − u x x + u 1 − u = f x , t , a < x < b , 0 < t < T .

The right-hand term is f x , t = 2 t + 2 t 2 − t 4 ⁡ sin x sin x . The corresponding exact solution is u x , t = t 2 ⁡ sin x .

All corresponding initial and boundary conditions can be deduced from exact solutions.

For example 1, at time t = 1 , the variation in the total collocation point number versus the L 2 − error is presented in Figure 1a for both uniform and non-uniform point distributions. It can be observed that the L 2 − error curve oscillates for relatively small total collocation point numbers in both two types of point distributions. As the total collocation point number increases, the L 2 − error curve converges smoothly. From this observation, one should choose relatively larger total collocation point numbers when dealing with the Fisher–KPP equation. At the same time, the non-uniform point distribution case performs better than the uniform case for relatively larger collocation points, even with one-decimal precision.

For example 2, at time t = 1 , the variation in the total collocation point number versus the L 2 − error is presented in Figure 1b for both uniform and non-uniform point distributions. It can be observed that the L 2 − error curve converges smoothly for the non-uniform point distribution case, while there is a sharp increase for the uniform point distribution case. At the same time, the non-uniform point distribution case also performs better than the uniform case.

For example 3, at time t = 10 with interval a , b = 0 , 1 , the variation in the total collocation point number versus the L 2 − error is presented in Figure 1c for both uniform and non-uniform point distributions. It can be observed that the L 2 − error curve converges smoothly for the uniform point distribution case for relatively few points. For larger points, the non-uniform case also performs better than the uniform case. Thus, we selected three different collocation numbers in the following analysis to show the numerical results. If we choose the same collocation number, the numerical results remain at the same level.

For example 1, we fix the total collocation point number N T = 517 . In order to show the consistency between the numerical and exact solutions for different times t = 0.2 , t = 0.6 , and t = 1 , the solutions for non-uniform and uniform point distribution cases are presented in Figures 2a, b, respectively. From Figure 2a, we can observe that the numerical solutions of the non-uniform case coincide exactly with the exact solutions, while there is a relatively larger error at the two sides of the solution curve for the uniform case.

For example 2 with fixed total collocation point number N T = 429 , Figure 3 demonstrates the strong consistency between the numerical and analytical solutions at different times t = 0.1 , t = 0.5 , and t = 1 for the non-uniform point distribution case. It was found that the numerical solutions coincide exactly with the exact solutions for all the different times.

For example 3 with interval a , b = − 1 , 1 and fixed total collocation point number N T = 472 , Figure 4 illustrates the close agreement between the numerical and analytical solutions for different times t = 1 , t = 5 , and t = 10 .

Numerical experiments have demonstrated the impact of two different point distributions and the total number of collocation points on solution accuracy. For all the given examples, increasing the total number of collocation points can reduce oscillations and ensure smoother convergence of the solution curve, especially for non-uniform distributions. It is notable that in cases where the total number of distribution points is relatively large, the non-uniform point distribution cases always perform better than the uniform distribution cases with higher solution accuracy. At different times, the non-uniform distribution exhibits strong consistency with the exact solution at all testing time points, while the uniform distribution shows significant errors near the domain boundaries in some cases. This may contribute to the relatively dense collocation points along the time axis; in other words, dense information at the initial boundary leads to more accurate results than in the uniform information case.

These findings emphasize the practical advantages of using non-uniform collocation with a relatively large number of points to ensure numerical stability and accuracy when solving Fisher–KPP equations.