Startseite Technik An unconditionally stable numerical scheme for solving nonlinear Fisher equation
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An unconditionally stable numerical scheme for solving nonlinear Fisher equation

  • Vikash Vimal EMAIL logo , Rajesh Kumar Sinha und Pannikkal Liju
Veröffentlicht/Copyright: 3. September 2024
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Nonlinear Engineering
Aus der Zeitschrift Nonlinear Engineering Band 13 Heft 1

Abstract

In this study, novel numerical methods are presented for solving nonlinear Fisher equations. These equations have a wide range of applications in various scientific and engineering fields, particularly in the biomedical sciences for determining the size of brain tumors. The challenges posed by the nonlinearity of the equations are effectively addressed through the development of numerical techniques. The nonlinearity is tackled using a combination of the method of lines and backward differentiation formulas of varied orders. This method is unconditionally stable, and its accuracy is evaluated using error norms. The methods are successfully validated against test problems with known solutions, demonstrating their superiority through comparative analyses with existing methodologies in the literature.

Keywords: Fisher equation; finite differences; backward differentiation formula; method of line; Newton method

1 Introduction

The Fisher equation, first developed in 1937 [1], is expressed as follows in its original form:

(1) u t = μ u x x + α u δ ( 1 − u ) , x ∈ ( − ∞ , ∞ ) , t ≥ 0 ,

where u ( x , t ) represents the population density.

This equation is commonly referred to as the Kolmogorov–Petrovsky–Piscunov equation [2]. The first exact solution of Eq. (1) was obtained by Ablowitz and Zeppetella [3]. Various alternative formulations of Eq. (1) were subsequently explored in multiple instances [4–6]. The Fisher equation has found numerous applications in various scientific and engineering disciplines, as evidenced in previous literature [7–11]. Researchers have explored it in various forms and extended its scope, as indicated in previous studies [12–16].

Various techniques have been employed to solve modified versions of the nonlinear Fisher’s reaction–diffusion equation. These approaches include the use of radial basis functions based on differential quadrature methods (RBFs with DQMs) [17], the Faedo–Galerkin method with homogeneous Dirichlet conditions [18], and the extended homogeneous balance method [19]. Furthermore, researchers have explored the application of Lie symmetries [20] and fractional extensions [21] to this equation. In recent years, there has been a concerted effort to develop efficient numerical techniques for computing solutions to the Fisher equation.

In this study, we utilize the method of lines (MOL) to semi-discretize the nonlinear reaction–diffusion equation, which effectively transforms it into a first-order ordinary differential equation (ODE). This approach, initially introduced by Schiesser in 1991 and documented in his research [22], has proven to be a powerful and effective tool for tackling time-dependent partial differential equations (PDEs). Within the MOL, we approximate spatial derivatives using finite differences. This method allows us to convert the PDEs into a set of ODEs that can be integrated over the time domain. To tackle the nonlinearity head-on, the study utilizes Newton’s method, providing implicit solutions that obviate the need for restrictive time step constraints.

We apply the finite difference method to introduce three distinct numerical approaches for addressing the generalized Fisher equation. These methods, referred to as backward differential formula (1) (BDF1), BDF2, and BDF3, involve replacing the differential equation with a discrete difference equation at each computational node. This substitution allows us to effectively employ backward differentiation techniques. After including boundary conditions in these equations, we proceed to solve the resulting system of algebraic equations. Various numerical solutions for Eq. (1) under different initial and boundary conditions have been investigated in previous studies [23–25]. Within the domain of numerical error analysis, the fully discretized system, which includes BDF1, demonstrates a time accuracy of the first order and spatial accuracy of the second order. On the other hand, BDF2 achieves quadratic accuracy not only in time but also in spatial dimensions. In comparison, BDF3 attains a significant third-order accuracy in the time dimension while maintaining a second-order accuracy in spatial computations.

2 Test example

Consider the one-dimensional Fisher equation (1) for μ = 1 ,

(2) ∂ u ∂ t = ∂ 2 u ∂ x 2 + α u δ ( 1 − u ) , 0 ≤ x ≤ 1 , t > 0 ,

with initial condition

u ( x , 0 ) = u 0 ( x ) , 0 ≤ x ≤ 1 ,

and the boundary conditions

u ( 0 , t ) = f 1 ( t ) , 0 ≤ t ≤ T , u ( 1 , t ) = f 2 ( t ) , 0 ≤ t ≤ T ,

where α represents the reactive factor, and T stands for the final time. Additionally, the functions u 0 ( x ) , f 1 , and f 2 denote the initial and boundary conditions.

3 Numerical scheme

In this section, we transform the PDE into an ODE with initial and boundary conditions using the MOL. The nonlinear reaction–diffusion equation, represented as Eq. (2), undergoes discretization exclusively along the spatial dimension ‘ x ’ through the utilization of the MOL [26].

3.1 Semi-discretization

The interval [ a , b ] is divided into N equal subintervals, where each subinterval maintains a uniform spacing of h = b − a N . We can represent the discrete spatial points as x i = a + i h for i = 1 , 2 , … , N , with the sequence: a = x 0 ≤ x 1 ≤ … ≤ x N = b . When we discretize the variable ‘ x ’ using the MOLs, the second-order spatial derivative u x x is obtained.

(3) ∂ 2 u ∂ x 2 ( x i , t ) = 1 h 2 ( u i + 1 ( t ) − 2 u i ( t ) + u i − 1 ( t ) ) , i = 0 , 1 , 2 , … N .

Substituting this into Eq. (2) yields a system of ODEs with the required initial condition.

(4) ∂ u i ∂ t = 1 h 2 ( u i + 1 ( t ) − 2 u i ( t ) + u i − 1 ( t ) ) + α u i ( 1 − u i ) ,

(5) u ′ ( t ) = 1 h 2 A u ( t ) , u ( 0 ) = u 0 .

The term u ( t ) = [ u 0 ( t ) , … , u N ( t ) ] T stands for the vector, while u 0 = [ u 0 ( 0 ) , … , u N ( 0 ) ] T represents the initial condition. Additionally, the matrix in question is a tridiagonal matrix of dimensions ( N + 1 ) * ( N + 1 ) .

A = − 2 2 1 − 2 1 ⋱ ⋱ ⋱ 1 − 2 1 2 − 2 .

3.2 Full discretization

We divide the time interval [ 0 , T ] into M identical subintervals as follows: 0 = t 0 ≤ t 1 ≤ … ≤ t M = T , where each subinterval has a length of Δ t = T ∕ M . In this scheme, we can express t n as t n = n Δ t , with n ranging from 1 to M . We numerically solve the system of first-order ODEs as described in Eq. (5) using various backward differentiation formulas (BDFs). The typical expression for a BDF of order p used to solve the subsequent initial value problem can be illustrated as follows:

u ′ ( t ) = f ( t , u ) , u ( 0 ) = u 0

is given by

(6) u n + 1 = ∑ k = o p − 1 α k u n − k + Δ t β f ( t n + 1 , u n + 1 ) .

In this context, where u n + 1 = u ( x i , t n + 1 ) = [ u 0 , n + 1 , … u N , n + 1 ] T , and Δ t signifies the time increment, the precise values of the coefficients α k for j = 0 , 1 , … , p − 1 , as well as the coefficient β , are detailed in the study by Atkinson et al. [27].

3.3 BDF of order p = 1 (BDF1)

In Eq. (6), when we set α 0 = 1 , β = 1 , and the function f ( t , u ) = 1 h 2 A u , we obtain the following expression:

(7) u n + 1 = u n + Δ t h 2 A u n + 1 , n = 0 , 1 , … , M − 1 ,

where u 0 represents the initial condition and is commonly referred to as the backward Euler method.

3.4 BDF of order p = 2 (BDF2)

With α 0 = 4 3 , α 1 = − 1 3 , β = 2 3 , and f ( t , u ) = 1 h 2 A u , Eq. (6) becomes

(8) u n + 1 = 4 3 u n − 1 3 u n − 1 + 2 3 Δ t h 2 A u n + 1 , n = 1 , … , M − 1 .

This is the solution at the initial time level, denoted as u 1 and is computed using the BDF1 method.

3.5 BDF of order p = 3 (BDF3)

By substituting α 0 = 18 11 , α 1 = − 9 11 , α 2 = 2 11 , β = 6 11 , and f ( t , u ) = 1 h 2 A u in Eq. (6), we obtain

(9) u n + 1 = 18 11 u n − 9 11 u n − 1 + 2 11 u n − 2 + 6 11 Δ t h 2 A u n + 1 , n = 2 , … , M − 1 .

The solutions at the first time level, denoted as u 1 , and at the second time level, denoted as u 2 , are both obtained using the BDF1 method.

4 BDF (BDF1) scheme

(10) ∂ u ∂ t = u i − 1 − 2 u i + u i + 1 ( Δ x ) 2 + α u i δ ( 1 − u i ) , = λ u i − 1 + u i ( α u i δ − 1 − α u i δ − 2 λ ) + λ u i + 1 .

Substituting Eq. (10) in Eq. (7), we obtain

u n + 1 = u n + Δ t ( λ u i − 1 + u i ( α u i δ − 1 − α u i δ − 2 λ ) + λ u i + 1 )

λ Δ t u i − 1 n + 1 + ( Δ t ( α ( u i n + 1 ) δ − 1 − α ( u i n + 1 ) δ − 2 λ ) − 1 ) u i n + 1 + λ Δ t u i + 1 n + 1 + u i n = 0 ,

where λ = 1 ∕ ( Δ x ) 2 , R i = Δ t ( α ( u i n + 1 ) δ − 1 − α ( u i n + 1 ) δ − 2 λ ) − 1 ,

I i = Δ t ( α − α u i n + 1 − 2 λ ) − 1 ,  where i = 1 , 2 , … , N − 1 .

The Jacobian matrix ( J ) is given by

J = I 1 λ Δ t λ Δ t I 2 λ Δ t ⋱ ⋱ ⋱ λ Δ t I N − 1 λ Δ t λ Δ t I N ,

B = R 1 λ Δ t λ Δ t R 2 λ Δ t ⋱ ⋱ ⋱ λ Δ t R N − 2 λ Δ t λ Δ t R N − 1 u 1 1 u 2 1 ⋮ u N − 1 1 u N 1 + λ Δ t u 0 1 0 ⋮ 0 λ Δ t u N 1 + u 1 0 u 2 0 ⋮ u N − 1 0 u N 0 = 0 .

5 BDF (BDF2) scheme

Substituting Eq. (10) in Eq. (8), we obtain

u i n + 1 = 4 3 u i n − 1 3 u i ( n − 1 ) + 2 3 Δ t ( λ u i − 1 n + 1 + u i n + 1 ( α ( u i n + 1 ) δ − 1 − α ( u i n + 1 ) δ − 2 λ ) + λ u i + 1 n + 1 ) 2 3 Δ t λ u i − 1 n + 1 + u i n + 1 × 2 3 Δ t ( α ( u i n + 1 ) δ − 1 − α ( u i n + 1 ) δ − 2 λ ) − 1 + 2 3 Δ t λ u i + 1 n + 1 + 4 3 u i n − 1 3 u i n − 1 = 0 ,

where A i = 2 3 Δ t ( α ( u i n + 1 ) δ − 1 − α ( u i n + 1 ) δ − 2 λ ) − 1 , H = 2 3 Δ t λ , and C i = 2 3 Δ t ( α − 2 α u i n + 1 − 2 λ ) − 1 .

J 1 is the Jacobian matrix given by

J 1 = C 1 H H C 2 H ⋱ ⋱ ⋱ H C N − 2 H H C N − 1

M = A 1 H H A 2 H ⋱ ⋱ ⋱ H A N − 2 H H A N − 1 u 1 2 u 2 2 ⋮ u N − 2 2 u N − 1 2 + H u 0 1 0 ⋮ 0 H u N − 1 1 + 4 3 u 1 1 u 2 1 ⋮ u N − 1 1 u N 1 − 1 3 u 1 0 u 2 0 ⋮ u N − 1 0 u N 0 = 0 .

6 BDF (BDF3) scheme

Substituting Eq. (10) in Eq. (9), we obtain

u i n + 1 = 8 11 u i n − 9 11 u i ( n − 1 ) + 2 11 u i ( n − 2 ) + 6 11 Δ t ( λ u i − 1 n + 1 + u i n + 1 ( α ( u i n + 1 ) δ − 1 − α ( u i n + 1 ) δ − 2 λ ) + λ u i + 1 n + 1 )

6 11 Δ t λ u i − 1 n + 1 + u i n + 1 × 6 11 Δ t ( α ( u i n + 1 ) δ − 1 − α ( u i n + 1 ) δ − 2 λ ) − 1 ) + 6 11 Δ t λ u i + 1 n + 1 + 2 11 u i n − 2 − 9 11 u i n − 1 + 18 11 u i n = 0 ,

where L i = 6 11 Δ t ( α ( u i n + 1 ) δ − 1 − α ( u i n + 1 ) δ − 2 λ ) − 1 , K = 6 11 Δ t λ ,

Q i = 6 11 Δ t ( α − 2 α u i n + 1 − 2 λ ) − 1 .

J 2 is the Jacobian matrix given by

J 2 = Q 1 K K Q 2 K ⋱ ⋱ ⋱ K Q N − 2 K K Q N − 1 ,

N = L 1 K K L 2 K ⋱ ⋱ ⋱ K L N − 2 K K L N − 1 u 1 3 u 2 3 ⋮ u N − 2 3 u N − 1 3 + K u 0 3 0 ⋮ 0 K u N − 1 3 + 2 11 u 1 0 u 2 0 ⋮ u N − 1 0 u N 0 − 9 11 u 1 1 u 2 1 ⋮ u N − 1 1 u N 1 + 18 11 u 1 2 u 2 2 ⋮ u N − 1 2 u N 2 = 0 .

7 Stability analysis

In this section, we assess the stability of the proposed numerical scheme using the von Neumann method. We have examined the linear version of the Fisher equation, assuming the constant b for the nonlinear term. We obtain

f i ( u 1 , u 2 , … , u N − 1 , t ) = λ ( u i + 1 ( t ) − 2 u i ( t ) + u i − 1 ( t ) ) + b ,

where λ = 1 ( Δ x ) 2 .

Applying the time integration to Eq. (4) results in

u i ( n + 1 ) = u i n + λ Δ t 2 ( u i + 1 ( n + 1 ) − 2 u i ( n + 1 ) + u i − 1 ( n + 1 ) ) + λ Δ t 2 ( u i + 1 ( n ) − 2 u i ( n ) + u i − 1 ( n ) ) + b .

By substituting u i n = ξ n e j k α i Δ x = ξ n e j i ϕ , where j = − 1 , ϕ = k α Δ x , Δ x represents the space step size, and, k α = π α where α = 1 , 2 , … , N , in the given equation, we derive the amplification factor, G , as

G = ξ ( n + 1 ) ξ n = X − j Y X 1 + j Y ,

where

X = 1 − Δ t λ ( 1 − cos ϕ ) , X 1 = 1 + Δ t λ ( 1 − cos ϕ ) , Y = 0 .

As both Δ t and Δ x are positive, therefore

X 1 − X = 2 Δ t λ ( 1 − c o s ϕ ) = 2 Δ t ( Δ x ) 2 ( 1 − cos ϕ ) ≥ 0 , and

X ≤ X 1 . So

∣ G ∣ = G G ¯ = X 2 + Y 2 X 1 2 + Y 2 ≤ 1 .

Hence the proposed scheme is unconditionally stable.

8 Error analysis

We examine the error analysis of three different numerical techniques using the backward differentiation schemes of the first, second, and third order, based on the Taylor series expansion of the completely discretized numerical method.

8.1 BDF of order one (BDF1)

This scheme, fully discretized, can be expressed in the following manner using the BDF of the order one as specified in Eq. (7),

(11) u ( x i , t n + 1 ) = u ( x i , t n ) + γ h 2 [ u ( x i − 1 , t n + 1 ) − 2 u ( x i , t n + 1 ) + u ( x i + 1 , t n + 1 ) ] + γ α u ( x i , t n + 1 ) ( 1 − u ( x i , t n + 1 ) ) .

In this context, with γ = Δ t and h = Δ x , where i = 1 , 2 , … , N − 1 and n = 0 , 1 , … , M − 1 , the local truncation error (LTE) can be described as follows:

(12) LTE = u ( x i , t n + 1 ) − u ( x i , t n ) − γ h 2 × [ u ( x i − 1 , t n + 1 ) − 2 u ( x i , t n + 1 ) + u ( x i + 1 , t n + 1 ) ] − γ α u ( x i , t n + 1 ) ( 1 − u ( x i , t n + 1 ) ) .

By applying the Taylor series expansion and subsequently simplifying, we deduce the following expression:

LTE = γ ∂ u ∂ t + γ 2 2 ∂ 2 u ∂ t 2 + γ 3 6 ∂ 3 u ∂ t 3 − γ ∂ 2 u ∂ x 2 − γ h 2 12 ∂ 4 u ∂ x 4 + … = γ 2 2 ∂ 2 u ∂ t 2 + γ 3 6 ∂ 3 u ∂ t 3 − γ h 2 12 ∂ 4 u ∂ x 4 + … = O ( γ 2 + h 2 γ )

The expression for the truncation error (TE) is as follows:

T E = γ − 1 ( LTE ) = γ 2 ∂ 2 u ∂ t 2 + γ 3 ∂ 3 u ∂ t 3 − γ h 2 12 ∂ 4 u ∂ x 4 + … = O ( γ + h 2 ) = O ( Δ t + ( Δ x ) 2 ) .

Numerical errors grow in proportion to both the time step size and the square of the spatial step size. As a result, the recommended completely discretized system, when combined with the first-order BDF, exhibits second-order accuracy in spatial aspects and first-order accuracy in temporal aspects.

8.2 BDF of order two (BDF2)

The LTE associated with the scheme, fully discretized, can be expressed in the following manner using the BDF of the order two as specified in Eq. (8):

LTE = u ( x i , t n + 1 ) − 4 3 u ( x i , t n ) − 1 3 u ( x i , t n − 1 ) − 2 γ 3 h 2 × [ u ( x i − 1 , t n + 1 ) − 2 u ( x i , t n + 1 ) + u ( x i + 1 , t n + 1 ) ] − 2 3 γ α u ( x i , t n + 1 ) ( 1 − u ( x i , t n + 1 ) ) .

By applying the Taylor series expansion and subsequently simplifying, we deduce the following expression:

LTE = 2 γ 3 ∂ u ∂ t + 2 γ 2 3 ∂ 2 u ∂ t 2 + γ 3 9 ∂ 3 u ∂ t 3 − 2 γ 3 ∂ 2 u ∂ x 2 − γ h 2 18 ∂ 4 u ∂ x 4 + … = 2 γ 3 ∂ u ∂ t − ∂ 2 u ∂ x 2 + 2 γ 2 3 ∂ 2 u ∂ t 2 + γ 3 9 ∂ 3 u ∂ t 3 − γ h 2 18 ∂ 4 u ∂ x 4 + … = O ( γ 3 + γ h 2 )

The expression for the TE is as follows:

TE = γ − 1 ( L T E ) = = 2 3 ∂ u ∂ t − ∂ 2 u ∂ x 2 + 2 γ 3 ∂ 2 u ∂ t 2 + γ 2 9 ∂ 3 u ∂ t 3 − h 2 18 ∂ 4 u ∂ x 4 + … = O ( γ 2 + h 2 ) = O ( ( Δ t ) 2 + ( Δ x ) 2 ) .

Therefore, the errors demonstrate quadratic dependence on both the time and space step sizes.

8.3 BDF of order three (BDF3)

Similarly, we calculate the LTE for the entirely discretized system by employing the third-order BDF comparably.

LTE = u ( x i , t n + 1 ) − 4 3 u ( x i , t n ) − 1 3 u ( x i , t n − 1 ) − 2 γ 3 h 2 [ u ( x i − 1 , t n + 1 ) − 2 u ( x i , t n + 1 ) + u ( x i + 1 , t n + 1 ) ] − 2 γ 3 u ( x i , t n + 1 ) ( 1 − u ( x i , t n + 1 ) ) = O ( γ 4 + γ h 2 ) .

The expression for the TE is as follows:

TE = γ − 1 ( L T E ) = O ( γ 3 + h 2 ) = O ( ( Δ t ) 3 + ( Δ x ) 2 )

The computational inaccuracies exhibit a cubic relationship with the time increment and a quadratic relationship with the spatial increment. As a result, the errors are reduced to a minimum when employing the completely discretized method with BDF3, and this finding is further substantiated by the outcomes from multiple test scenarios.

9 Numerical tests and conversations

In this section, we evaluate the accuracy of the proposed techniques [28] by computing errors in both the L 2 and L ∞ norms, as defined in Eqs (13) and (14), at designated time points.

(13) L 2 = 1 M ∑ j = 1 M ∣ ϕ j ( exact ) − U j ( num ) ∣ 2 1 ∕ 2 ,

(14) L ∞ = ∣ ∣ ϕ j ( exact ) − U j ( num ) ∣ ∣ ∞ = max j ∣ ϕ j ( exact ) − U j ( num ) ∣ .

The rate of numerical convergence for these methods has been determined using the subsequent formula:

(15) ROC ≈ log ( E ( N 2 ) ∕ E ( N 1 ) ) log ( N 1 ∕ N 2 ) ,

where the L ∞ error value obtained by employing N j number of grid points is denoted as E ( N j ) . The L 2 and L ∞ error norms are computed for Tables 14 and 16 at the final time T = 0.1 , with α = δ = 1 and a time step of Δ t = 0.00005 . The tables show that as the grid sizes increased, both L 2 and L ∞ errors decreased. The table also displays the rate of convergence (ROC) for the suggested methods across various grid sizes.

Table 14

Comparing numerical and exact solutions for Example 1 at various spatial points with Δ t = 0.00005 , T = 0.1 , and α = δ = 1

BDF1 BDF2 BDF3
N L 2 L ∞ ROC L 2 L ∞ ROC L 2 L ∞ ROC
10 1.10 × 1 0 ‒ 6 9.52 × 1 0 ‒ 7 4.94 × 1 0 ‒ 7 3.66 × 1 0 ‒ 7 3.66 × 1 0 ‒ 7
15 1.03 × 1 0 ‒ 6 7.81 × 1 0 ‒ 7 1.65 × 1 0 0 2.20 × 1 0 ‒ 7 1.63 × 1 0 ‒ 7 2.00 × 1 0 00 2.44 × 1 0 ‒ 7 1.66 × 1 0 ‒ 7 1.74 × 1 0 00
20 1.02 × 1 0 ‒ 6 7.27 × 1 0 ‒ 7 1.32 × 1 0 0 1.24 × 1 0 ‒ 7 9.19 × 1 0 ‒ 8 1.99 × 1 0 00 3.41 × 1 0 ‒ 7 2.37 × 1 0 ‒ 7 1.16 × 1 0 00
Table 16

Comparison of numerical and exact solutions (BDF1, BDF2, BDF3) for Example 2 at Δ t = 0.0005 , T = 0.1 , and α = δ = 1

Computed solution Exact solution
x BDF1 BDF2 BDF3
0.1 0.2606471000 0.260738174 0.260752490 0.260738428
0.2 0.2503452690 0.250420646 0.250447850 0.250421096
0.3 0.2402488598 0.240311103 0.240348491 0.240311688
0.4 0.2303642690 0.230417726 0.230461597 0.230418385
0.5 0.2206991200 0.220747972 0.220794013 0.220748648
0.6 0.2112596750 0.211308559 0.211352270 0.211309201
0.7 0.2020521340 0.202105453 0.202142581 0.202106010
0.8 0.1930823300 0.193143852 0.193170799 0.193144276
0.9 0.1843556660 0.184428190 0.184442349 0.184428430

9.1 Test problems

We conducted various numerical experiments to compare our proposed numerical method with the exact solution. Two specific examples were analyzed, and all calculations were carried out in MATLAB.

Example 1

Consider the Fisher Eq. (2) with initial condition,

u t = u x x + u ( 1 − u ) ,

subject to the initial condition

u ( x , 0 ) = 1 ( 1 + e α 6 x ) 2 , 0 ≤ t ≤ 1 ,

where the exact solution is presented in the study by Wazwaz and Gorguis [29] given by

u ( x , t ) = 1 ( 1 + e α 6 x − 5 6 α t ) 2 .

We have tabulated both the exact solution and numerical results obtained at various node points for different values of α .

In this study, we compare numerical results and exact solutions derived from different BDFs. Tables 1 and 2 focus on the evaluation of three BDF methods at fixed time instances but varying spatial positions, providing insights into their spatial performance. In parallel, Tables 3 and 4 explore error comparisons under consistent time points but different spatial locations, offering valuable information about precision. Figures 1, 2, 3, 4, 5, 6 illustrate the computed solutions produced by BDF methods of first, second, and third orders, together with the exact solution, across a range of alpha values. In Figures 7 and 8, the absolute errors are depicted at the same time points but in different spatial locations. In a concurrent analysis, we expanded our comparison by using a time step of Δ t = 0.0005 , and the results are documented in Tables 5 and 6. Absolute errors, which are essential for evaluating accuracy, are presented in Tables 7 and 8. Figures 9, 10, 11, 12, 13, 14 display the numerical solutions, while Figures 15 and 16 provide a detailed view of the absolute errors. Notably, under these conditions, BDF2 demonstrates superior accuracy compared to BDF1 and BDF3.

Table 1

Comparison of numerical and exact solutions (BDF1, BDF2, BDF3) for Example 1 at Δ t = 0.000005 , T = 0.1 , and α = δ = 1

x BDF1 BDF2 BDF3 Exact solution
0.2 0.250420019 0.250420775 0.250421049 0.250421096
0.3 0.240310644 0.240311272 0.240311649 0.240311688
0.4 0.230417375 0.230417912 0.230418355 0.230418385
0.5 0.220747663 0.220748154 0.220748619 0.220748648
0.6 0.211308229 0.211308721 0.211309162 0.211309201
0.7 0.202105045 0.202105581 0.202105956 0.202106010
0.8 0.193143323 0.193143940 0.193144212 0.193144276
0.9 0.184427509 0.184428235 0.184428378 0.184428430
Table 2

Comparison of numerical and exact solutions (BDF1, BDF2, BDF3) for Example 1 at Δ t = 0.000005 , T = 0.1 , and α = 6 , δ = 1

x BDF1 BDF2 BDF3 Exact solution
0.1 0.3584088969 0.3584211896 0.3584221152 0.3584269144
0.2 0.3299634211 0.3299734888 0.3299752538 0.3299842051
0.3 0.3022946501 0.3023027800 0.3023052162 0.3023174246
0.4 0.2755792279 0.2755858557 0.2755887275 0.2756031473
0.5 0.2499758181 0.2499814841 0.2499845115 0.2500000000
0.6 0.2256212575 0.2256265533 0.2256294397 0.2256447723
0.7 0.2026276009 0.2026331139 0.2026355750 0.2026494300
0.8 0.1810801962 0.1810864553 0.1810882474 0.1810991715
0.9 0.1610368521 0.1610442798 0.1610452236 0.1610515941
Table 3

Error comparison (BDF1, BDF2, BDF3) for Example 1 at Δ t = 0.000005 , T = 0.1 with α = δ = 1

x BDF1 BDF2 BDF3
0.1 1.09612 × 1 0 ‒ 6 1.84267 × 1 0 ‒ 7 3.98471 × 1 0 ‒ 8
0.2 1.07701 × 1 0 ‒ 6 3.21258 × 1 0 ‒ 7 4.67977 × 1 0 ‒ 8
0.3 1.04363 × 1 0 ‒ 6 4.16011 × 1 0 ‒ 7 3.87691 × 1 0 ‒ 8
0.4 1.01011 × 1 0 ‒ 6 4.72709 × 1 0 ‒ 7 2.99744 × 1 0 ‒ 8
0.5 9.85211 × 1 0 ‒ 7 4.93894 × 1 0 ‒ 7 2.91792 × 1 0 ‒ 8
0.6 9.71300 × 1 0 ‒ 7 4.79938 × 1 0 ‒ 7 3.86630 × 1 0 ‒ 8
0.7 9.64053 × 1 0 ‒ 7 4.28806 × 1 0 ‒ 7 5.39126 × 1 0 ‒ 8
0.8 9.52847 × 1 0 ‒ 7 3.36153 × 1 0 ‒ 7 6.40192 × 1 0 ‒ 8
0.9 9.21778 × 1 0 ‒ 7 1.95717 × 1 0 ‒ 7 5.27188 × 1 0 ‒ 8
Table 4

Error comparison (BDF1, BDF2, BDF3) for Example 1 at Δ t = 0.000005 , T = 0.1 with α = 6 , δ = 1

x BDF1 BDF2 BDF3
0.1 1.80175 × 1 0 ‒ 5 5.72477 × 1 0 ‒ 6 4.79917 × 1 0 ‒ 6
0.2 2.07840 × 1 0 ‒ 5 1.07164 × 1 0 ‒ 5 8.95132 × 1 0 ‒ 6
0.3 2.27745 × 1 0 ‒ 5 1.46446 × 1 0 ‒ 5 1.22084 × 1 0 ‒ 5
0.4 2.39194 × 1 0 ‒ 5 1.72915 × 1 0 ‒ 5 1.44198 × 1 0 ‒ 5
0.5 2.41819 × 1 0 ‒ 5 1.85159 × 1 0 ‒ 5 1.54885 × 1 0 ‒ 5
0.6 2.35148 × 1 0 ‒ 5 1.82190 × 1 0 ‒ 5 1.53326 × 1 0 ‒ 5
0.7 2.18291 × 1 0 ‒ 5 1.63161 × 1 0 ‒ 5 1.38550 × 1 0 ‒ 5
0.8 1.89753 × 1 0 ‒ 5 1.27162 × 1 0 ‒ 5 1.09241 × 1 0 ‒ 5
0.9 1.47420 × 1 0 ‒ 5 7.31436 × 1 0 ‒ 6 6.37055 × 1 0 ‒ 6
Figure 1 Error propagation of Example 1 at T = 0.1 T=0.1 , Δ t = 0.000005 \Delta t=0.000005 for α = δ = 1 \alpha =\delta =1 .
Figure 1

Error propagation of Example 1 at T = 0.1 , Δ t = 0.000005 for α = δ = 1 .

Figure 2 Error propagation of Example 1 at T = 0.1 T=0.1 , Δ t = 0.000005 \Delta t=0.000005 for α = δ = 1 \alpha =\delta =1 .
Figure 2

Error propagation of Example 1 at T = 0.1 , Δ t = 0.000005 for α = δ = 1 .

Figure 3 Error propagation of Example 1 at T = 0.1 T=0.1 , Δ t = 0.000005 \Delta t=0.000005 for α = δ = 1 \alpha =\delta =1 .
Figure 3

Error propagation of Example 1 at T = 0.1 , Δ t = 0.000005 for α = δ = 1 .

Figure 4 Error propagation of Example 1 at T = 0.1 T=0.1 , Δ t = 0.000005 \Delta t=0.000005 for α = 6 \alpha =6 and δ = 1 \delta =1 .
Figure 4

Error propagation of Example 1 at T = 0.1 , Δ t = 0.000005 for α = 6 and δ = 1 .

Figure 5 Error propagation of Example 1 at T = 0.1 T=0.1 , Δ t = 0.000005 \Delta t=0.000005 for α = 6 \alpha =6 and δ = 1 \delta =1 .
Figure 5

Error propagation of Example 1 at T = 0.1 , Δ t = 0.000005 for α = 6 and δ = 1 .

Figure 6 Error propagation of Example 1 at T = 0.1 T=0.1 , Δ t = 0.000005 \Delta t=0.000005 for α = 6 \alpha =6 and δ = 1 \delta =1 .
Figure 6

Error propagation of Example 1 at T = 0.1 , Δ t = 0.000005 for α = 6 and δ = 1 .

Figure 7 Error comparison of Example 1 at α = δ = 1 \alpha =\delta =1 , T = 0.1 T=0.1 , and Δ t = 0.000005 \Delta t=0.000005 .
Figure 7

Error comparison of Example 1 at α = δ = 1 , T = 0.1 , and Δ t = 0.000005 .

Figure 8 Error comparison of Example 1 at α = 6 \alpha =6 , δ = 1 \delta =1 , T = 0.1 T=0.1 , and Δ t = 0.000005 \Delta t=0.000005 .
Figure 8

Error comparison of Example 1 at α = 6 , δ = 1 , T = 0.1 , and Δ t = 0.000005 .

Table 5

Comparison of numerical and exact solutions (BDF1, BDF2, BDF3) for Example 1 at Δ t = 0.0005 , T = 0.1 , and α = δ = 1

Computed solution Exact solution
x BDF1 BDF2 BDF3
0.1 0.260555346 0.260738174 0.260752490 0.260738428
0.2 0.250268689 0.250420646 0.250447850 0.250421096
0.3 0.240184531 0.240311103 0.240348491 0.240311688
0.4 0.230309036 0.230417726 0.230461597 0.230418385
0.5 0.220648412 0.220747972 0.220794013 0.220748648
0.6 0.211208986 0.211308559 0.211352270 0.211309201
0.7 0.201997201 0.202105453 0.202142581 0.202106010
0.8 0.193019543 0.193143852 0.193170799 0.193144276
0.9 0.184282411 0.184428190 0.184442349 0.184428430
Table 6

Comparison of numerical and exact solutions (BDF1, BDF2, BDF3) for Example 1 at Δ t = 0.0005 , T = 0.1 , and α = 6 , δ = 1

Computed solution Exact solution
x BDF1 BDF2 BDF3
0.1 0.357193283 0.358420734 0.358512716 0.358426914
0.2 0.328968781 0.329972662 0.330148039 0.329984205
0.3 0.301492288 0.302301712 0.302543759 0.302317425
0.4 0.274925758 0.275584698 0.275870007 0.275603147
0.5 0.249417565 0.249980376 0.250281128 0.250000000
0.6 0.225099525 0.225625602 0.225912319 0.225644772
0.7 0.202084201 0.202632381 0.202876840 0.202649430
0.8 0.180462762 0.181085967 0.181263959 0.181099172
0.9 0.160303577 0.161044038 0.161137779 0.161051594
Table 7

Error comparison (BDF1, BDF2, BDF3) for Example 1 at Δ t = 0.0005 , T = 0.1 with α = δ = 1

x BDF1 BDF2 BDF3
0.1 0.000183082 2.538 × 1 0 ‒ 7 1.40619 × 1 0 ‒ 5
0.2 0.000152407 4.497 × 1 0 ‒ 7 2.67542 × 1 0 ‒ 5
0.3 0.000127158 5.848 × 1 0 ‒ 7 3.68024 × 1 0 ‒ 5
0.4 0.000109349 6.594 × 1 0 ‒ 7 4.32119 × 1 0 ‒ 5
0.5 0.000100236 6.768 × 1 0 ‒ 7 4.53644 × 1 0 ‒ 5
0.6 0.000100215 6.414 × 1 0 ‒ 7 4.30690 × 1 0 ‒ 5
0.7 0.000108809 5.567 × 1 0 ‒ 7 3.65716 × 1 0 ‒ 5
0.8 0.000124734 4.238 × 1 0 ‒ 7 2.65234 × 1 0 ‒ 5
0.9 0.000146020 2.400 × 1 0 ‒ 7 1.39185 × 1 0 ‒ 5
Table 8

Error comparison (BDF1, BDF2, BDF3) for Example 1 at Δ t = 0.0005 , T = 0.1 with α = 6 , δ = 1

x BDF1 BDF2 BDF3
0.1 0.001233632 6.18020 × 1 0 ‒ 6 8.58017 × 1 0 ‒ 5
0.2 0.001015424 1.15434 × 1 0 ‒ 5 0.000163834
0.3 0.000825137 1.57122 × 1 0 ‒ 5 0.000226335
0.4 0.000677389 1.84491 × 1 0 ‒ 5 0.000266860
0.5 0.000582435 1.96236 × 1 0 ‒ 5 0.000281128
0.6 0.000545247 1.91708 × 1 0 ‒ 5 0.000267547
0.7 0.000565229 1.70489 × 1 0 ‒ 5 0.000227410
0.8 0.000636410 1.32046 × 1 0 ‒ 5 0.000164788
0.9 0.000748017 7.55610 × 1 0 ‒ 6 8.61852 × 1 0 ‒ 5
Figure 9 Error propagation at Δ t = 0.0005 \Delta t=0.0005 , T = 0.1 T=0.1 , and α = δ = 1 \alpha =\delta =1 of Example 1.
Figure 9

Error propagation at Δ t = 0.0005 , T = 0.1 , and α = δ = 1 of Example 1.

Figure 10 Error propagation at Δ t = 0.0005 \Delta t=0.0005 , T = 0.1 T=0.1 , and α = δ = 1 \alpha =\delta =1 of Example 1.
Figure 10

Error propagation at Δ t = 0.0005 , T = 0.1 , and α = δ = 1 of Example 1.

Figure 11 Error propagation at Δ t = 0.0005 \Delta t=0.0005 , T = 0.1 T=0.1 , and α = δ = 1 \alpha =\delta =1 of Example 1.
Figure 11

Error propagation at Δ t = 0.0005 , T = 0.1 , and α = δ = 1 of Example 1.

Figure 12 Error propagation at Δ t = 0.0005 \Delta t=0.0005 , T = 0.1 T=0.1 , and α = 6 \alpha =6 , δ = 1 \delta =1 of Example 1.
Figure 12

Error propagation at Δ t = 0.0005 , T = 0.1 , and α = 6 , δ = 1 of Example 1.

Figure 13 Error propagation at Δ t = 0.0005 \Delta t=0.0005 , T = 0.1 T=0.1 , and α = 6 \alpha =6 , δ = 1 \delta =1 of Example 1.
Figure 13

Error propagation at Δ t = 0.0005 , T = 0.1 , and α = 6 , δ = 1 of Example 1.

Figure 14 Error propagation at Δ t = 0.0005 \Delta t=0.0005 , T = 0.1 T=0.1 , and α = 6 \alpha =6 , δ = 1 \delta =1 of Example 1.
Figure 14

Error propagation at Δ t = 0.0005 , T = 0.1 , and α = 6 , δ = 1 of Example 1.

Figure 15 Error comparison of Example 1 at α = δ = 1 \alpha =\delta =1 , T = 0.1 T=0.1 , and Δ t = 0.0005 \Delta t=0.0005 .
Figure 15

Error comparison of Example 1 at α = δ = 1 , T = 0.1 , and Δ t = 0.0005 .

Figure 16 Error comparison of Example 1 at α = 6 \alpha =6 , δ = 1 \delta =1 , T = 0.1 T=0.1 , and Δ t = 0.0005 \Delta t=0.0005 .
Figure 16

Error comparison of Example 1 at α = 6 , δ = 1 , T = 0.1 , and Δ t = 0.0005 .

We further examined the numerical solutions of BDF1, BDF2, and BDF3 in Tables 9, 10, and 11, comparing them with findings from previous publications [30,31] under the specific conditions of Δ t = 0.00005 , T = 0.1 , and α = 6 . Our numerical results for BDF1, BDF2, and BDF3 perform better than a few existing methods. The accuracy of these BDF methods was assessed by employing the error norms L 2 and L ∞ as detailed in Tables 12 and 13, revealing that BDF3 exhibits higher accuracy values compared to BDF1 and BDF2.

Table 9

Numerical and exact solutions for Example 1 are compared at different points of space at Δ t = 0.00005 , T = 0.1 for α = 6 , δ = 1

T x DQM [31] BDF1 Exact solution
0.5 0.25 0.81847 0.818313 0.818393
0.75 0.72592 0.725715 0.725824
1.0 0.25 0.98293 0.982909 0.982919
0.75 0.97208 0.972055 0.972071
Table 10

Numerical and exact solutions for Example 1 are compared at different points of space at Δ t = 0.00005 , T = 0.1 for α = 6 , δ = 1

T x DQM [31] BDF2 Exact solution
0.5 0.25 0.81847 0.818394 0.818393
0.75 0.72592 0.725826 0.725824
1.0 0.25 0.98293 0.982918 0.982919
0.75 0.97208 0.972070 0.972071
Table 11

Numerical and exact solutions for Example 1 are compared at different points of space at Δ t = 0.00005 , T = 0.1 for α = 6 , δ = 1

T x DQM [31] BDF3 Exact solution
0.5 0.25 0.81847 0.818395 0.818393
0.75 0.72592 0.725826 0.725824
1.0 0.25 0.98293 0.982918 0.982919
0.75 0.97208 0.972070 0.972071
Table 12

Assessing differences using the L 2 and L ∞ error norm for varying values of T = 0.1 and Δ t = 0.000005 , as illustrated in Example 1

α = δ = 1 α = 6 and δ = 1
T L 2 L ∞ L 2 L ∞
BDF1 0.002677279 0.008466298 2.02786 × 1 0 ‒ 5 2.41819 × 1 0 ‒ 5
BDF2 0.002677279 0.008466298 1.34806 × 1 0 ‒ 5 1.85159 × 1 0 ‒ 5
BDF3 0.002677279 0.008466298 1.13446 × 1 0 ‒ 5 1.54885 × 1 0 ‒ 5
Table 13

Assessing differences using the L 2 and L ∞ error norm for varying values of T = 0.1 and Δ t = 0.0005 , as illustrated in Example 1

α = δ = 1 α = 6 and δ = 1
T L 2 L ∞ L 2 L ∞
BDF1 0.000131399 0.000183082 0.000796513 0.001233631
BDF2 4.96013 × 1 0 ‒ 7 6.76 × 1 0 ‒ 7 1.42598 × 1 0 ‒ 5 1.9624 × 1 0 ‒ 5
BDF3 3.20668 × 1 0 ‒ 5 4.5365 × 1 0 ‒ 5 0.000198383 0.000281128

Example 2

Consider the following Fisher’s equation-2 in domain [0, 1] for α = δ = 1 :

(16) u t = u x x + u ( 1 − u )

with initial condition

(17) u ( x , 0 ) = 1 2 tanh − α 2 2 α + 4 x + 1 2 2 α .

The exact solution is presented in [12,30] by

(18) u ( x , t ) = 1 2 tanh − α 2 2 α + 4 x − α + 4 2 α + 4 t + 1 2 2 α .

In Tables 15 and 16, we calculate the numerical solutions for Example 2, considering different time steps, specifically Δ t = 0.000005 and Δ t = 0.0005 , while maintaining constants T = 0.1 and α = 1 . Furthermore, Tables 17 and 18 provide the absolute errors for this example. Additionally, Figures 17, 18, 19, 20, 21, 22 include both 2D and 3D plots that illustrate the numerical solution in comparison to the exact solution, while Figures 23 and 24 depict 2D and 3D plots of the absolute errors.

Table 15

Comparison of numerical and exact solutions (BDF1, BDF2, BDF3) for Example 2 at Δ t = 0.000005 , T = 0.1 , and α = δ = 1

Computed solution Exact solution
x BDF1 BDF2 BDF3
0.1 0.260736413 0.2607382437 0.260738388 0.260738428
0.2 0.250419251 0.2504207746 0.250421049 0.250421096
0.3 0.240310001 0.2403112721 0.240311649 0.240311688
0.4 0.230416819 0.2304179122 0.230418355 0.230418385
0.5 0.220747153 0.2207481544 0.220748619 0.220748648
0.6 0.211307719 0.2113087208 0.211309162 0.211309201
0.7 0.202104493 0.2021055807 0.202105956 0.202106010
0.8 0.193142693 0.1931439398 0.193144212 0.193144276
0.9 0.184426775 0.1844282347 0.184428378 0.184428430
Table 17

Error Comparison (BDF1, BDF2, BDF3) for Example 2 at Δ t = 0.000005 , T = 0.1 with α = δ = 1

x BDF1 BDF2 BDF3
0.1 2.01499 × 1 0 ‒ 6 1.84222 × 1 0 ‒ 7 3.98220 × 1 0 ‒ 8
0.2 1.84504 × 1 0 ‒ 6 3.21283 × 1 0 ‒ 7 4.67833 × 1 0 ‒ 8
0.3 1.68720 × 1 0 ‒ 6 4.15963 × 1 0 ‒ 7 3.87630 × 1 0 ‒ 8
0.4 1.56573 × 1 0 ‒ 6 4.72690 × 1 0 ‒ 7 2.99897 × 1 0 ‒ 8
0.5 1.49567 × 1 0 ‒ 6 4.93901 × 1 0 ‒ 7 2.92009 × 1 0 ‒ 8
0.6 1.48139 × 1 0 ‒ 6 4.79973 × 1 0 ‒ 7 3.86734 × 1 0 ‒ 8
0.7 1.51614 × 1 0 ‒ 6 4.28834 × 1 0 ‒ 7 5.39340 × 1 0 ‒ 8
0.8 1.58283 × 1 0 ‒ 6 3.36198 × 1 0 ‒ 7 6.39982 × 1 0 ‒ 8
0.9 1.65563 × 1 0 ‒ 6 1.95727 × 1 0 ‒ 7 5.27266 × 1 0 ‒ 8
Table 18

Error Comparison (BDF1, BDF2, BDF3) for Example 2 at Δ t = 0.0005 , T = 0.1 with α = δ = 1

x BDF1 BDF2 BDF3
0.1 9.132700 × 1 0 ‒ 5 2.538 × 1 0 ‒ 7 1.40619 × 1 0 ‒ 5
0.2 7.582690 × 1 0 ‒ 5 4.497 × 1 0 ‒ 7 2.67542 × 1 0 ‒ 5
0.3 6.308980 × 1 0 ‒ 5 5.848 × 1 0 ‒ 7 3.68024 × 1 0 ‒ 5
0.4 5.411580 × 1 0 ‒ 5 6.594 × 1 0 ‒ 7 4.32119 × 1 0 ‒ 5
0.5 4.952830 × 1 0 ‒ 5 6.768 × 1 0 ‒ 7 4.53644 × 1 0 ‒ 5
0.6 4.952580 × 1 0 ‒ 5 6.414 × 1 0 ‒ 7 4.30690 × 1 0 ‒ 5
0.7 5.387586 × 1 0 ‒ 5 5.567 × 1 0 ‒ 7 3.65716 × 1 0 ‒ 5
0.8 6.194570 × 1 0 ‒ 5 4.238 × 1 0 ‒ 7 2.65234 × 1 0 ‒ 5
0.9 7.276420 × 1 0 ‒ 5 2.400 × 1 0 ‒ 7 1.39185 × 1 0 ‒ 5
Figure 17 Error propagation at Δ t = 0.000005 \Delta t=0.000005 , T = 0.1 T=0.1 , and α = δ = 1 \alpha =\delta =1 for Example 2.
Figure 17

Error propagation at Δ t = 0.000005 , T = 0.1 , and α = δ = 1 for Example 2.

Figure 18 Error propagation at Δ t = 0.000005 \Delta t=0.000005 , T = 0.1 T=0.1 , and α = δ = 1 \alpha =\delta =1 for Example 2.
Figure 18

Error propagation at Δ t = 0.000005 , T = 0.1 , and α = δ = 1 for Example 2.

Figure 19 Error propagation at Δ t = 0.000005 \Delta t=0.000005 , T = 0.1 T=0.1 , and α = δ = 1 \alpha =\delta =1 for Example 2.
Figure 19

Error propagation at Δ t = 0.000005 , T = 0.1 , and α = δ = 1 for Example 2.

Figure 20 Error propagation at Δ t = 0.0005 \Delta t=0.0005 , T = 0.1 T=0.1 , and α = δ = 1 \alpha =\delta =1 for Example 2.
Figure 20

Error propagation at Δ t = 0.0005 , T = 0.1 , and α = δ = 1 for Example 2.

Figure 21 Error propagation at Δ t = 0.0005 \Delta t=0.0005 , T = 0.1 T=0.1 , and α = δ = 1 \alpha =\delta =1 for Example 2.
Figure 21

Error propagation at Δ t = 0.0005 , T = 0.1 , and α = δ = 1 for Example 2.

Figure 22 Error propagation at Δ t = 0.0005 \Delta t=0.0005 , T = 0.1 T=0.1 , and α = δ = 1 \alpha =\delta =1 for Example 2.
Figure 22

Error propagation at Δ t = 0.0005 , T = 0.1 , and α = δ = 1 for Example 2.

Figure 23 Error comparison at Δ t = 0.000005 \Delta t=0.000005 , T = 0.1 T=0.1 , and α = δ = 1 \alpha =\delta =1 for Example 2.
Figure 23

Error comparison at Δ t = 0.000005 , T = 0.1 , and α = δ = 1 for Example 2.

Figure 24 Error comparison at Δ t = 0.0005 \Delta t=0.0005 , T = 0.1 T=0.1 , and α = δ = 1 \alpha =\delta =1 for Example 2.
Figure 24

Error comparison at Δ t = 0.0005 , T = 0.1 , and α = δ = 1 for Example 2.

In Tables 19, 20, and 21, we evaluate the numerical solutions of BDF1, BDF2, and BDF3 compared to a few existing methods [30,31]. This assessment is performed under specific conditions, including Δ t = 0.00005 , T = 0.1 , and α = 1 . BDF1, BDF2, and BDF3 exhibit more accuracy and precision compared to a few existing methods (Table 22).

Table 19

Comparing numerical results with exact solutions for Example 2 at various spatial positions, with a time step of Δ t = 0.00005 and a total time of T = 0.1 , while considering α = δ = 1 .

T x DQM [31] BDF1 Exact solution
0.5 0.25 0.33412 0.334071 0.334094
0.75 0.27838 0.278332 0.278353
1.0 0.25 0.45576 0.455714 0.455739
0.75 0.39544 0.395387 0.395411
Table 20

Comparing numerical results with exact solutions for Example 2 at various spatial positions, with a time step of Δ t = 0.00005 and a total time of T = 0.1 , while considering α = δ = 1

T x DQM [31] CFD6 [30] BDF2 Exact solution
0.5 0.25 0.33412 0.334094 0.334094 0.334094
0.75 0.27838 0.278353 0.278353 0.278353
1.0 0.25 0.45576 0.455739 0.455739 0.455739
0.75 0.39544 0.395411 0.395411 0.395411
Table 21

Comparing numerical results with exact solutions for Example 2 at various spatial positions, with a time step of Δ t = 0.00005 and a total time of T = 0.1 , while considering α = δ = 1

T x DQM [31] CFD6 [30] BDF3 Exact solution
0.5 0.25 0.33412 0.334094 0.334094 0.334094
0.75 0.27838 0.278353 0.278353 0.278353
1.0 0.25 0.45576 0.455739 0.455739 0.455739
0.75 0.39544 0.395411 0.395411 0.395411
Table 22

Comparing numerical and exact solutions for Example 2 at various spatial points with Δ t = 0.00005 , T = 0.1 , and α = δ = 1

BDF1 BDF2 BDF3
N L 2 L ∞ ROC L 2 L ∞ ROC L 2 L ∞ ROC
10 1.10 × 1 0 ‒ 6 9.52 × 1 0 ‒ 7 4.94 × 1 0 ‒ 7 3.66 × 1 0 ‒ 7 3.66 × 1 0 ‒ 7
15 1.03 × 1 0 ‒ 6 7.81 × 1 0 ‒ 7 1.65 × 1 0 0 2.20 × 1 0 ‒ 7 1.63 × 1 0 ‒ 7 2.00 × 1 0 00 2.44 × 1 0 ‒ 7 1.66 × 1 0 ‒ 7 1.74 × 1 0 00
20 1.02 × 1 0 ‒ 6 7.27 × 1 0 ‒ 7 1.32 × 1 0 0 1.24 × 1 0 ‒ 7 9.19 × 1 0 ‒ 8 1.99 × 1 0 00 3.41 × 1 0 ‒ 7 2.37 × 1 0 ‒ 7 1.16 × 1 0 00

10 Conclusion

To transform the nonlinear Fisher equation into a set of first-order ODEs, we employ a semi-discretization method along the variable “ x ” using the MOLs, and then, by utilizing first order, second order, and third-order BDFs, we can efficiently resolve this system of first-order ODEs. The numerical results generated by BDF2 and BDF3 exhibit higher accuracy compared to those produced by BDF1. In our evaluation of errors, we observe that BDF-3 achieves third-order accuracy with O ( ( Δ t ) 3 + ( Δ x ) 2 ) , BDF2 achieves second-order precision with O ( ( Δ t ) 2 + ( Δ x ) 2 ) , and BDF1 provides first-order accuracy with O ( Δ t + ( Δ x ) 2 ) . Remarkably, for smaller values of Δ t , the error diminishes, demonstrating that the computed results remain reasonably close to the exact solution. This approach effectively addresses the nonlinear, time-dependent Fisher equation.

Acknowledgements

The authors are very thankful to the reviewers for their valuable comments and suggestions, which improved the quality of this study.

  1. Funding information: The authors state that there is no funding involved.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors declare that there is no conflict of interest.

  4. Data availability statement: All data, generated or used during the study, are available within the article.

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Received: 2023-10-27
Revised: 2024-03-22
Accepted: 2024-04-02
Published Online: 2024-09-03

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

  1. Editorial
  2. Focus on NLENG 2023 Volume 12 Issue 1
  3. Research Articles
  4. Seismic vulnerability signal analysis of low tower cable-stayed bridges method based on convolutional attention network
  5. Robust passivity-based nonlinear controller design for bilateral teleoperation system under variable time delay and variable load disturbance
  6. A physically consistent AI-based SPH emulator for computational fluid dynamics
  7. Asymmetrical novel hyperchaotic system with two exponential functions and an application to image encryption
  8. A novel framework for effective structural vulnerability assessment of tubular structures using machine learning algorithms (GA and ANN) for hybrid simulations
  9. Flow and irreversible mechanism of pure and hybridized non-Newtonian nanofluids through elastic surfaces with melting effects
  10. Stability analysis of the corruption dynamics under fractional-order interventions
  11. Solutions of certain initial-boundary value problems via a new extended Laplace transform
  12. Numerical solution of two-dimensional fractional differential equations using Laplace transform with residual power series method
  13. Fractional-order lead networks to avoid limit cycle in control loops with dead zone and plant servo system
  14. Modeling anomalous transport in fractal porous media: A study of fractional diffusion PDEs using numerical method
  15. Analysis of nonlinear dynamics of RC slabs under blast loads: A hybrid machine learning approach
  16. On theoretical and numerical analysis of fractal--fractional non-linear hybrid differential equations
  17. Traveling wave solutions, numerical solutions, and stability analysis of the (2+1) conformal time-fractional generalized q-deformed sinh-Gordon equation
  18. Influence of damage on large displacement buckling analysis of beams
  19. Approximate numerical procedures for the Navier–Stokes system through the generalized method of lines
  20. Mathematical analysis of a combustible viscoelastic material in a cylindrical channel taking into account induced electric field: A spectral approach
  21. A new operational matrix method to solve nonlinear fractional differential equations
  22. New solutions for the generalized q-deformed wave equation with q-translation symmetry
  23. Optimize the corrosion behaviour and mechanical properties of AISI 316 stainless steel under heat treatment and previous cold working
  24. Soliton dynamics of the KdV–mKdV equation using three distinct exact methods in nonlinear phenomena
  25. Investigation of the lubrication performance of a marine diesel engine crankshaft using a thermo-electrohydrodynamic model
  26. Modeling credit risk with mixed fractional Brownian motion: An application to barrier options
  27. Method of feature extraction of abnormal communication signal in network based on nonlinear technology
  28. An innovative binocular vision-based method for displacement measurement in membrane structures
  29. An analysis of exponential kernel fractional difference operator for delta positivity
  30. Novel analytic solutions of strain wave model in micro-structured solids
  31. Conditions for the existence of soliton solutions: An analysis of coefficients in the generalized Wu–Zhang system and generalized Sawada–Kotera model
  32. Scale-3 Haar wavelet-based method of fractal-fractional differential equations with power law kernel and exponential decay kernel
  33. Non-linear influences of track dynamic irregularities on vertical levelling loss of heavy-haul railway track geometry under cyclic loadings
  34. Fast analysis approach for instability problems of thin shells utilizing ANNs and a Bayesian regularization back-propagation algorithm
  35. Validity and error analysis of calculating matrix exponential function and vector product
  36. Optimizing execution time and cost while scheduling scientific workflow in edge data center with fault tolerance awareness
  37. Estimating the dynamics of the drinking epidemic model with control interventions: A sensitivity analysis
  38. Online and offline physical education quality assessment based on mobile edge computing
  39. Discovering optical solutions to a nonlinear Schrödinger equation and its bifurcation and chaos analysis
  40. New convolved Fibonacci collocation procedure for the Fitzhugh–Nagumo non-linear equation
  41. Study of weakly nonlinear double-diffusive magneto-convection with throughflow under concentration modulation
  42. Variable sampling time discrete sliding mode control for a flapping wing micro air vehicle using flapping frequency as the control input
  43. Error analysis of arbitrarily high-order stepping schemes for fractional integro-differential equations with weakly singular kernels
  44. Solitary and periodic pattern solutions for time-fractional generalized nonlinear Schrödinger equation
  45. An unconditionally stable numerical scheme for solving nonlinear Fisher equation
  46. Effect of modulated boundary on heat and mass transport of Walter-B viscoelastic fluid saturated in porous medium
  47. Analysis of heat mass transfer in a squeezed Carreau nanofluid flow due to a sensor surface with variable thermal conductivity
  48. Navigating waves: Advancing ocean dynamics through the nonlinear Schrödinger equation
  49. Experimental and numerical investigations into torsional-flexural behaviours of railway composite sleepers and bearers
  50. Novel dynamics of the fractional KFG equation through the unified and unified solver schemes with stability and multistability analysis
  51. Analysis of the magnetohydrodynamic effects on non-Newtonian fluid flow in an inclined non-uniform channel under long-wavelength, low-Reynolds number conditions
  52. Convergence analysis of non-matching finite elements for a linear monotone additive Schwarz scheme for semi-linear elliptic problems
  53. Global well-posedness and exponential decay estimates for semilinear Newell–Whitehead–Segel equation
  54. Petrov-Galerkin method for small deflections in fourth-order beam equations in civil engineering
  55. Solution of third-order nonlinear integro-differential equations with parallel computing for intelligent IoT and wireless networks using the Haar wavelet method
  56. Mathematical modeling and computational analysis of hepatitis B virus transmission using the higher-order Galerkin scheme
  57. Mathematical model based on nonlinear differential equations and its control algorithm
  58. Bifurcation and chaos: Unraveling soliton solutions in a couple fractional-order nonlinear evolution equation
  59. Space–time variable-order carbon nanotube model using modified Atangana–Baleanu–Caputo derivative
  60. Minimal universal laser network model: Synchronization, extreme events, and multistability
  61. Valuation of forward start option with mean reverting stock model for uncertain markets
  62. Geometric nonlinear analysis based on the generalized displacement control method and orthogonal iteration
  63. Fuzzy neural network with backpropagation for fuzzy quadratic programming problems and portfolio optimization problems
  64. B-spline curve theory: An overview and applications in real life
  65. Nonlinearity modeling for online estimation of industrial cooling fan speed subject to model uncertainties and state-dependent measurement noise
  66. Quantitative analysis and modeling of ride sharing behavior based on internet of vehicles
  67. Review Article
  68. Bond performance of recycled coarse aggregate concrete with rebar under freeze–thaw environment: A review
  69. Retraction
  70. Retraction of “Convolutional neural network for UAV image processing and navigation in tree plantations based on deep learning”
  71. Special Issue: Dynamic Engineering and Control Methods for the Nonlinear Systems - Part II
  72. Improved nonlinear model predictive control with inequality constraints using particle filtering for nonlinear and highly coupled dynamical systems
  73. Anti-control of Hopf bifurcation for a chaotic system
  74. Special Issue: Decision and Control in Nonlinear Systems - Part I
  75. Addressing target loss and actuator saturation in visual servoing of multirotors: A nonrecursive augmented dynamics control approach
  76. Collaborative control of multi-manipulator systems in intelligent manufacturing based on event-triggered and adaptive strategy
  77. Greenhouse monitoring system integrating NB-IOT technology and a cloud service framework
  78. Special Issue: Unleashing the Power of AI and ML in Dynamical System Research
  79. Computational analysis of the Covid-19 model using the continuous Galerkin–Petrov scheme
  80. Special Issue: Nonlinear Analysis and Design of Communication Networks for IoT Applications - Part I
  81. Research on the role of multi-sensor system information fusion in improving hardware control accuracy of intelligent system
  82. Advanced integration of IoT and AI algorithms for comprehensive smart meter data analysis in smart grids
https://doi.org/10.1515/nleng-2024-0006
Schlagwörter für diesen Artikel
Fisher equation; finite differences; backward differentiation formula; method of line; Newton method
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Editorial
  2. Focus on NLENG 2023 Volume 12 Issue 1
  3. Research Articles
  4. Seismic vulnerability signal analysis of low tower cable-stayed bridges method based on convolutional attention network
  5. Robust passivity-based nonlinear controller design for bilateral teleoperation system under variable time delay and variable load disturbance
  6. A physically consistent AI-based SPH emulator for computational fluid dynamics
  7. Asymmetrical novel hyperchaotic system with two exponential functions and an application to image encryption
  8. A novel framework for effective structural vulnerability assessment of tubular structures using machine learning algorithms (GA and ANN) for hybrid simulations
  9. Flow and irreversible mechanism of pure and hybridized non-Newtonian nanofluids through elastic surfaces with melting effects
  10. Stability analysis of the corruption dynamics under fractional-order interventions
  11. Solutions of certain initial-boundary value problems via a new extended Laplace transform
  12. Numerical solution of two-dimensional fractional differential equations using Laplace transform with residual power series method
  13. Fractional-order lead networks to avoid limit cycle in control loops with dead zone and plant servo system
  14. Modeling anomalous transport in fractal porous media: A study of fractional diffusion PDEs using numerical method
  15. Analysis of nonlinear dynamics of RC slabs under blast loads: A hybrid machine learning approach
  16. On theoretical and numerical analysis of fractal--fractional non-linear hybrid differential equations
  17. Traveling wave solutions, numerical solutions, and stability analysis of the (2+1) conformal time-fractional generalized q-deformed sinh-Gordon equation
  18. Influence of damage on large displacement buckling analysis of beams
  19. Approximate numerical procedures for the Navier–Stokes system through the generalized method of lines
  20. Mathematical analysis of a combustible viscoelastic material in a cylindrical channel taking into account induced electric field: A spectral approach
  21. A new operational matrix method to solve nonlinear fractional differential equations
  22. New solutions for the generalized q-deformed wave equation with q-translation symmetry
  23. Optimize the corrosion behaviour and mechanical properties of AISI 316 stainless steel under heat treatment and previous cold working
  24. Soliton dynamics of the KdV–mKdV equation using three distinct exact methods in nonlinear phenomena
  25. Investigation of the lubrication performance of a marine diesel engine crankshaft using a thermo-electrohydrodynamic model
  26. Modeling credit risk with mixed fractional Brownian motion: An application to barrier options
  27. Method of feature extraction of abnormal communication signal in network based on nonlinear technology
  28. An innovative binocular vision-based method for displacement measurement in membrane structures
  29. An analysis of exponential kernel fractional difference operator for delta positivity
  30. Novel analytic solutions of strain wave model in micro-structured solids
  31. Conditions for the existence of soliton solutions: An analysis of coefficients in the generalized Wu–Zhang system and generalized Sawada–Kotera model
  32. Scale-3 Haar wavelet-based method of fractal-fractional differential equations with power law kernel and exponential decay kernel
  33. Non-linear influences of track dynamic irregularities on vertical levelling loss of heavy-haul railway track geometry under cyclic loadings
  34. Fast analysis approach for instability problems of thin shells utilizing ANNs and a Bayesian regularization back-propagation algorithm
  35. Validity and error analysis of calculating matrix exponential function and vector product
  36. Optimizing execution time and cost while scheduling scientific workflow in edge data center with fault tolerance awareness
  37. Estimating the dynamics of the drinking epidemic model with control interventions: A sensitivity analysis
  38. Online and offline physical education quality assessment based on mobile edge computing
  39. Discovering optical solutions to a nonlinear Schrödinger equation and its bifurcation and chaos analysis
  40. New convolved Fibonacci collocation procedure for the Fitzhugh–Nagumo non-linear equation
  41. Study of weakly nonlinear double-diffusive magneto-convection with throughflow under concentration modulation
  42. Variable sampling time discrete sliding mode control for a flapping wing micro air vehicle using flapping frequency as the control input
  43. Error analysis of arbitrarily high-order stepping schemes for fractional integro-differential equations with weakly singular kernels
  44. Solitary and periodic pattern solutions for time-fractional generalized nonlinear Schrödinger equation
  45. An unconditionally stable numerical scheme for solving nonlinear Fisher equation
  46. Effect of modulated boundary on heat and mass transport of Walter-B viscoelastic fluid saturated in porous medium
  47. Analysis of heat mass transfer in a squeezed Carreau nanofluid flow due to a sensor surface with variable thermal conductivity
  48. Navigating waves: Advancing ocean dynamics through the nonlinear Schrödinger equation
  49. Experimental and numerical investigations into torsional-flexural behaviours of railway composite sleepers and bearers
  50. Novel dynamics of the fractional KFG equation through the unified and unified solver schemes with stability and multistability analysis
  51. Analysis of the magnetohydrodynamic effects on non-Newtonian fluid flow in an inclined non-uniform channel under long-wavelength, low-Reynolds number conditions
  52. Convergence analysis of non-matching finite elements for a linear monotone additive Schwarz scheme for semi-linear elliptic problems
  53. Global well-posedness and exponential decay estimates for semilinear Newell–Whitehead–Segel equation
  54. Petrov-Galerkin method for small deflections in fourth-order beam equations in civil engineering
  55. Solution of third-order nonlinear integro-differential equations with parallel computing for intelligent IoT and wireless networks using the Haar wavelet method
  56. Mathematical modeling and computational analysis of hepatitis B virus transmission using the higher-order Galerkin scheme
  57. Mathematical model based on nonlinear differential equations and its control algorithm
  58. Bifurcation and chaos: Unraveling soliton solutions in a couple fractional-order nonlinear evolution equation
  59. Space–time variable-order carbon nanotube model using modified Atangana–Baleanu–Caputo derivative
  60. Minimal universal laser network model: Synchronization, extreme events, and multistability
  61. Valuation of forward start option with mean reverting stock model for uncertain markets
  62. Geometric nonlinear analysis based on the generalized displacement control method and orthogonal iteration
  63. Fuzzy neural network with backpropagation for fuzzy quadratic programming problems and portfolio optimization problems
  64. B-spline curve theory: An overview and applications in real life
  65. Nonlinearity modeling for online estimation of industrial cooling fan speed subject to model uncertainties and state-dependent measurement noise
  66. Quantitative analysis and modeling of ride sharing behavior based on internet of vehicles
  67. Review Article
  68. Bond performance of recycled coarse aggregate concrete with rebar under freeze–thaw environment: A review
  69. Retraction
  70. Retraction of “Convolutional neural network for UAV image processing and navigation in tree plantations based on deep learning”
  71. Special Issue: Dynamic Engineering and Control Methods for the Nonlinear Systems - Part II
  72. Improved nonlinear model predictive control with inequality constraints using particle filtering for nonlinear and highly coupled dynamical systems
  73. Anti-control of Hopf bifurcation for a chaotic system
  74. Special Issue: Decision and Control in Nonlinear Systems - Part I
  75. Addressing target loss and actuator saturation in visual servoing of multirotors: A nonrecursive augmented dynamics control approach
  76. Collaborative control of multi-manipulator systems in intelligent manufacturing based on event-triggered and adaptive strategy
  77. Greenhouse monitoring system integrating NB-IOT technology and a cloud service framework
  78. Special Issue: Unleashing the Power of AI and ML in Dynamical System Research
  79. Computational analysis of the Covid-19 model using the continuous Galerkin–Petrov scheme
  80. Special Issue: Nonlinear Analysis and Design of Communication Networks for IoT Applications - Part I
  81. Research on the role of multi-sensor system information fusion in improving hardware control accuracy of intelligent system
  82. Advanced integration of IoT and AI algorithms for comprehensive smart meter data analysis in smart grids
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