Home Applications of the Belousov–Zhabotinsky reaction–diffusion system: Analytical and numerical approaches
Article Open Access

Applications of the Belousov–Zhabotinsky reaction–diffusion system: Analytical and numerical approaches

  • Zaki Mrzog Alaof , Khalid K. Ali EMAIL logo , Mona S. Mehanna , Abdel-Haleem Abdel-Aty and Mohamed A. Shaalan
Published/Copyright: March 5, 2025
Download Article (PDF)
Open Physics
From the journal Open Physics Volume 23 Issue 1

Abstract

The Belousov–Zhabotinsky (BZ) reaction–diffusion system is a well-known model of chemical self-organization that exhibits complex spatiotemporal patterns. The BZ reaction–diffusion system provides a useful tool for studying the behavior of waves in random and complex media. Its applications in this field are wide-ranging and have the potential to contribute to a better understanding of the behavior of waves in natural and engineered systems. In this article, we investigate the BZ system using both analytical and numerical methods. We first apply the Bernoulli sub-ordinary differential equation (ODE) technique to the BZ system to obtain a simplified system of ODEs. Then, we use the exponential cubic B-spline method and the trigonometric cubic B-spline method to solve the simplified system numerically. The results show that both methods are effective in capturing the essential features of the BZ system. We also compare the results obtained using the two numerical methods. Our findings analytically contribute to a better understanding of the BZ system through graphs of the soliton solutions.

Keywords: Belousov–Zhabotinsky reaction–diffusion model; Bernoulli sub-ODE technique; soliton solutions; exponential cubic B-spline method; trigonometric cubic B-spline method

1 Introduction

The Belousov–Zhabotinsky (BZ) system was first discovered in the 1950s by Boris Belousov, a Russian chemist, who observed an unusual color change in a mixture of chemicals. Later, Anatol Zhabotinsky expanded on Belousov’s work and discovered that the reaction could produce oscillations and other dynamic patterns. The BZ reaction is described by a system for the first time in 1983 [1]. The BZ reaction–diffusion system is a well-known example of a chemical oscillator that exhibits periodic and chaotic behavior. The system involves the oxidation of a reducing agent by a chemical oxidant, with the product of the reaction then oxidizing the original reducing agent in a cyclic process. This process is accompanied by a change in the concentration of one or more species over time. The BZ reaction–diffusion system can also exhibit spatial patterns, which can be modeled using a reaction–diffusion system. In this system, the concentration of the reactants and products is allowed to vary both in time and space and is governed by a set of partial differential equations. The BZ reaction–diffusion system is notable for its ability to produce a wide variety of complex spatiotemporal patterns, including spirals, stripes, and spots. The system has been studied extensively both experimentally and mathematically and has applications in fields such as chemistry, physics, and biology. The BZ reaction illustrates oscillatory behavior in chemical reactions and is a famous example of non-equilibrium thermodynamics. Because of the fluctuating reactant and product concentrations, this reaction, which includes the interaction of bromate ions with organic acids, causes periodic color variations. The BZ reaction has a wide range of physical phenomena, from biological systems to materials science and environmental science, including modeling the electrical impulses that regulate heartbeats, creating self-assembling materials with complex structures and modeling the spread of pollutants in water bodies [2–4]. The study of the BZ reaction–diffusion system has led to important insights into the behavior of complex chemical systems and has inspired the development of new mathematical and computational techniques for studying these systems. The system continues to be an active area of research, with new applications and insights being discovered all the time. The principle idea that the main factor determining the wavefront’s speed is how concentrated certain chemical substances, the bromous acid ( HBrO 2 ) and the bromide ion ( Br − ) given by U and V , respectively. The sequence of simplified reaction

A + V ⇌ k 1 U + P , U + V ⇌ k 2 2 P , A + U ⇌ k 3 2 U , 2 U ⇌ k 4 P + A ,

where k 1 , k 2 , k 3 , and k 4 are known rate constants, P is the compound HBrO, and A is the concentration of BrO 3 − . By using lowercase letters for concentrations, applying the Law of Mass Action to this scheme

U x = k 1 a V − k 2 U V + k 3 a U − k 4 U 2 + D U x x , V t = − k 1 a V − k 2 U V + D V x x ,

where D is the diffusion coefficient. A suitable non-dimensional is

θ = k 4 U k 3 a , ν = k 2 V k 3 a r , x * = k 3 a D 1 2 x t * = k 3 a t , L = k 1 k 4 k 2 k 3 , M = k 1 k 3 , b = k 2 k 4 ,

where the parameter r is used to describe the concentration of bromide ions in the BZ reaction and can be experimentally manipulated. We omit the asterisks for simplicity

θ t = L r ν + θ ( 1 − θ − r ν ) + θ x x , ν t = − M ν − b θ ν + ν x x ,

since L ≈ M = O ( 1 0 − 4 ) , L ≪ 1 , and M ≪ 1 , We simplify the model for the BZ reaction by disregarding these terms. The resulting model is described by a system of equations

θ t = θ x x + θ ( 1 − θ − r ν ) , ν t = ν x x − b θ ν ,

we search for traveling wavefront solutions in the BZ reaction in which the level of bromide ion is lowered and the wave travels from a zone of high bromous acid concentration to one of low bromous acid concentration. So, we apply appropriate boundary conditions to solve the BZ system

θ ( − ∞ , t ) = 0 , ν ( − ∞ , t ) = 1 , θ ( ∞ , t ) = 1 , ν ( ∞ , t ) = 0 ,

where the functions θ and ν introduce the bromous acid and bromide ion concentrations, respectively. r and b are positive constants, for more details see [5]. We employ the variable change Θ = θ and ℧ = 1 − ν [6], which results in

(1) Θ t = Θ x x + Θ ( 1 − r − Θ + r ℧ ) , ℧ t = ℧ x x + b Θ ( 1 − ℧ ) .

The traveling wave solutions are studied under different conditions by various researchers: Murray [7,8], Tyson [9], Ye and Wang [10,11], Li [12], Li and Ye [13], Trofimchuk et al. studied the stability and the uniqueness of monotone wave fronts [14]; Basavaraja et al. investigated oscillating patterns and bifurcation phase diagrams for the BZ system [15], Manoranjan and Mitchell [16] used the finite-element method to obtain the wave solution; Wang et al. [17] presented solutions for four kinds of explicit wavefronts. Quinney presented the soliton solutions using the finite-difference method [18]. The soliton solutions can be investigated by a variety of analytical methods, like the Bernoulli sub-ordinary differential equation (ODE) method [19], Laplace transform method [20], and Hirota bilinear method [21,22]. The authors [23–26] have mentioned several numerical methods, including parametric continuation, shooting, finite difference, and basis spline functions collocation methods. These analytical and numerical methods are commonly used in various fields and can be applied to solve a wide range of problems.

This work is organized into several sections. In Section 2, we present a summary of the analytical method based on the Bernoulli sub-ODE technique. In Section 3, we illustrate the application of the analytical method and present the solutions obtained. In Section 4, we present the majority of our results in the form of two- and three-dimensional graphs. In Section 5, we investigate the numerical illustrations using two different cubic B-spline function schemes. In Section 6, we compare between analytical and numerical results. Finally, in Section 7, we provide a synopsis of our findings and draw a conclusion based on our analysis of the BZ reaction.

2 Bernoulli sub-ODE technique

The strategy of the Bernoulli sub-ODE method [19] is explained as follows: Exhibiting the main steps in the current section to illustrate the new form of the Bernoulli sub-ODE method. Presumption the model equation is given by

(2) G ( ζ , ζ x , ζ t , ζ x x , ζ t t , ζ x t ) = 0 ,

where G is a polynomial in ζ = ζ ( x , t ) and its partial derivatives. Inserting the wave transformations

(3) η = x + c t , ζ ( x , t ) = ζ ( η ) ,

where c > 0 is the velocity of the wave (2) switched to the next ODE

(4) Q ( ζ , ζ ′ , ζ ″ , ζ ‴ , ζ ″ ″ ) = 0 .

  1. Postulate the following finite series is the solution of (4):

    (5) ζ ( η ) = ∑ i = 0 N g i ( G ( η ) ) i ,

    where g i ≠ 0 , ( 0 ≤ i ≤ N ) are calculated from the principle of homogeneous balance to obtain the value of N and g N ≠ 0 , then

    (6) G ′ + λ G = μ G 2 , λ , μ ≠ 0 ,

    is an ODE that has a solution G ( η ) given by

    (7) G ( η ) = 1 μ λ + d e λ η .

  2. Inserting (5) and (6) into (4), we gain the solution of the acquired system when equating all coefficients of G ( η ) to zero. The previous set will be solved using the Mathematica program.

3 Applications

Employing the wave transformation (3) on (1), we obtain the next ODE system of equations:

(8) Θ ( η ) ( − 1 + r + Θ ( η ) − r ℧ ( η ) ) + c Θ ′ ( η ) − Θ ″ ( η ) = 0 , b Θ ( η ) ( − 1 + ℧ ( η ) ) + c ℧ ′ ( η ) − ℧ ″ ( η ) = 0 .

Determine the values of M and N from (8)

  1. Θ ( η ) ℧ ( η ) with Θ ″ ( η ) ⇒ M = 2 ,

  2. Θ ( η ) ℧ ( η ) with ℧ ″ ( η ) ⇒ N = 2 .

3.1 Results employing the Bernoulli sub-ODE technique

Based upon (5), the solution of (8) is stated next:

(9) Θ ( η ) = f 0 + f 1 G ( η ) + f 2 G ( η ) 2 , ℧ ( η ) = m 0 + m 1 G ( η ) + m 2 G ( η ) 2 .

By inserting (9) into (8) and setting the coefficients with equal powers of G ( η ) equal to zero, we arrive at the following system:

(10) − f 0 + r f 0 + f 0 2 − r f 0 m 0 = 0 , − f 1 + r f 1 − c λ f 1 − λ 2 f 1 + 2 f 0 f 1 − r f 1 m 0 − r f 0 m 1 = 0 , c μ f 1 + 3 λ μ f 1 + f 1 2 − f 2 + r f 2 − 2 c λ f 2 − 4 λ 2 f 2 + 2 f 0 f 2 − r f 2 m 0 − r f 1 m 1 − r f 0 m 2 = 0 , − 2 μ 2 f 1 + 2 c μ f 2 + 10 λ μ f 2 + 2 f 1 f 2 − r f 2 m 1 − r f 1 m 2 = 0 , − 6 μ 2 f 2 + f 2 2 − r f 2 m 2 = 0 , − b f 0 + b f 0 m 0 = 0 , − b f 1 + b f 1 m 0 − c λ m 1 − λ 2 m 1 + b f 0 m 1 = 0 , − b f 2 + b f 2 m 0 + c μ m 1 + 3 λ μ m 1 + b f 1 m 1 − 2 c m 2 λ − 4 λ 2 m 2 + b f 0 m 2 = 0 , − 2 m 1 μ 2 + b f 2 m 1 + 2 c μ m 2 + 10 λ μ m 2 + b f 1 m 2 = 0 , − 6 μ 2 m 2 + b f 2 m 2 = 0 .

Four sets of solutions are obtained when the former system of equations is solved as follows:

Set 1

(11) f 0 = 0 , f 1 = 0 , f 2 = μ 2 λ 2 , m 0 = − 1 + r + 6 λ 2 r , m 1 = 0 , m 2 = μ 2 − 6 λ 2 μ 2 r λ 2 , c = − 5 λ , b = 6 λ 2 .

Set 2

(12) f 0 = 1 , f 1 = − 2 μ λ , f 2 = μ 2 λ 2 , m 0 = 1 , m 1 = 2 ( − μ + 6 λ 2 μ ) r λ , m 2 = μ 2 − 6 λ 2 μ 2 r λ 2 , c = 5 λ , b = 6 λ 2 .

Set 3

(13) f 0 = 0 , f 1 = − 2 λ μ , f 2 = 2 μ 2 , m 0 = r 2 μ − r μ ( 1 + λ 2 ) + λ r 2 μ 2 ( 3 + λ 2 ) r 2 μ , m 1 = 2 r 2 ( 2 r λ μ − 3 r 2 μ 2 + r 2 λ 2 μ 2 ) , m 2 = − 4 μ 2 r , c = − r 2 μ 2 ( 3 + λ 2 ) r μ , b = 3 .

Set 4

(14) f 0 = 0 , f 1 = − 2 λ μ , f 2 = 2 μ 2 , m 0 = r 2 μ − r μ ( 1 + λ 2 ) − λ r 2 μ 2 ( 3 + λ 2 ) r 2 μ , m 1 = 2 r 2 ( 2 r λ μ + 3 r 2 μ 2 + r 2 λ 2 μ 2 ) , m 2 = − 4 μ 2 r , c = r 2 μ 2 ( 3 + λ 2 ) r μ , b = 3 .

The traveling wave solution is provided as follows:

(15) Θ ( x , t ) = f 0 + f 1 1 μ λ + d e λ ( x + c t ) + f 2 1 μ λ + d e λ ( x + c t ) 2 , ℧ ( x , t ) = m 0 + m 1 1 μ λ + d e λ ( x + c t ) + m 2 1 μ λ + d e λ ( x + c t ) 2 .

4 Discussion of graphical representations

In this section, we present some graphs in two and three dimensions to show our solutions. In Figure 1, we plot graphs of (15) with set 4 (14) employing the Bernoulli sub-ODE method at r = 1.3 , d = 0.2 , μ = 0.5 and λ = 0.5 . The graphs of Θ and ℧ of (15) with set 2 (12) at b = 1.56 , b = 1.82 , and b = 2.09 are shown in Figure 2.

Figure 1 Graph of Θ \Theta and ℧ \mho of (15) with set 4 (14) at r = 1.3 , d = 0.2 , μ = 0.5 r=1.3,d=0.2,\mu =0.5 , and λ = 0.5 \lambda =0.5 .
Figure 1

Graph of Θ and ℧ of (15) with set 4 (14) at r = 1.3 , d = 0.2 , μ = 0.5 , and λ = 0.5 .

Figure 2 Graph of Θ \Theta and ℧ \mho of (15) with set 2 (12) at b = 1.56 , b = 1.82 b=1.56,b=1.82 , and b = 2.09 b=2.09 .
Figure 2

Graph of Θ and ℧ of (15) with set 2 (12) at b = 1.56 , b = 1.82 , and b = 2.09 .

The graphs can be a very effective tool for describing and visualizing solutions to various problems, including mathematical models and systems. In the context of Figure 1, it appears that the wave being referred to is a bell-shaped soliton (dark soliton) for variable Θ and that the wave has turned down and shifted to the left (kink soliton) for variable ℧ . This could be due to changes in the parameters of the system. Figure 2 shows the variation in variables Θ and ℧ as the parameter b is changed. The parameter b determines the strength of the feedback loop between the two variables Θ and ℧ . When b is small, the feedback loop is weak and the concentration ℧ has little effect on the dynamics of the system. As b increases, the feedback loop becomes stronger, and the oscillations in the system become more pronounced. When b becomes very large, the system may exhibit chaotic behavior, which can be seen as irregular and unpredictable variations in the values of Θ and ℧ .

5 Numerical method outline

We discretize the domain [ a , b ] of the function into small intervals or knots as Ω : a = x 0 < x 1 < … < x ℒ = b . Let the interval width be denoted by h = b − a ℒ = x l + 1 − x l for l = 0 , 1 , … , ℒ and let the time levels with Δ t > 0 be t m = m Δ t for m = 0 , 1 , … , ℳ , where ℳ is +ve integer. We use two collocation methods like exponential cubic B-spline method and trigonometric cubic B-spline method to solve the BZ equation (1) numerically.

Let the initial and boundary conditions of the BZ system (1)

(16) Θ ( x , 0 ) = f 1 ( x ) , ℧ ( x , 0 ) = f 2 ( x ) , a ≤ x ≤ b , Θ ( a , t ) = f 3 ( t ) , Θ ( b , t ) = f 4 ( t ) , ℧ ( a , t ) = f 5 ( t ) , ℧ ( b , t ) = f 6 ( t ) .

Here are their outlines of two methods as follows.

5.1 Exponential cubic B-spline method

We construct the exponential cubic B-spline functions [23], E l ( x ) , at the knots x l , ( l = − 1 ( 1 ) ℒ + 1 ) can be expressed as

(17) E l ( x ) = B 2 ( x l − 2 − x ) − 1 P ( sinh ( P ( x l − 2 − x ) ) ) x l − 2 ≤ x ≤ x l − 1 , A 1 + B 1 ( x l − x ) + C 1 exp ( P ( x l − x ) ) + D 1 exp ( − P ( x l − x ) ) x l − 1 ≤ x ≤ x l , A 1 + B 1 ( x − x l ) + C 1 exp ( P ( x − x l ) ) + D 1 exp ( − P ( x − x l ) ) x l ≤ x ≤ x l + 1 , B 2 ( x − x l + 2 ) − 1 P ( sinh ( P ( x l + 2 − x ) ) ) x l + 1 ≤ x ≤ x l + 2 , 0 otherwise ,

where A 1 = P h C P h C − S , B 1 = P 2 [ C ( C − 1 ) + S 2 ( P h C − S ) ( 1 − C ) ] , B 2 = P 2 ( P h C − S ) , C 1 = exp ( − P h ) ( 1 − C ) + S ( exp ( − P h ) − 1 ) 4 ( P h C − S ) ( 1 − C ) , D 1 = exp ( P h ) ( C − 1 ) + S ( exp ( P h ) − 1 ) 4 ( P h C − S ) ( 1 − C ) , C = cosh ( P h ) , S = sinh ( P h ) , and P is a free parameter.

We assume that the numerical solutions of the BZ system (1) are Θ ˜ ( x , t ) and ℧ ˜ ( x , t ) which are approximations to the exact solutions Θ ( x , t ) and ℧ ( x , t ) as follows:

(18) Θ ˜ ( x , t ) = ∑ l = − 1 ℒ + 1 B l ( x ) δ l ( t ) , ℧ ˜ ( x , t ) = ∑ l = − 1 ℒ + 1 B l ( x ) γ l ( t ) ,

where δ l ( t ) and γ l ( t ) are the time-dependent unknown quantities, B l ( x ) the exponential cubic B-spline function and its two derivatives B l ′ ( x ) , B l ″ ( x ) at the knots x l are established from (17).

Using (17) and (18), we compute the numerical solutions of the BZ equation (1) and its derivatives up to second order as follows:

(19) Θ ˜ ( x l , t ) = ( b 1 ) δ l − 1 ( t ) + δ l ( t ) + ( b 1 ) δ l + 1 ( t ) , Θ ˜ l ′ = ( − b 2 ) ( δ l − 1 ( t ) − δ l + 1 ( t ) ) , Θ ˜ l ″ = ( b 3 ) ( δ l − 1 ( t ) − 2 δ l ( t ) + δ l + 1 ( t ) ) ,

(20) ℧ ˜ ( x l , t ) = ( b 1 ) γ l − 1 ( t ) + γ l ( t ) + ( b 1 ) γ l + 1 ( t ) , ℧ ˜ l ′ = ( − b 2 ) ( γ l − 1 ( t ) − γ l + 1 ( t ) ) , ℧ ˜ l ″ = ( b 3 ) ( γ l − 1 ( t ) − 2 γ l ( t ) + γ l + 1 ( t ) ) .

where b 1 = S − P h 2 ( P h C − S ) , b 2 = P ( C − 1 ) 2 ( P h C − S ) , b 3 = P 2 S 2 ( P h C − S ) .

Substituting from (19) and (20) into (1), a general form equation is reached for the linearization technique

(21) ( b 1 ) δ ˙ l − 1 ( t ) + δ ˙ l ( t ) + ( b 1 ) δ ˙ l + 1 ( t ) − ( b 3 ) ( δ l − 1 ( t ) − 2 δ l ( t ) + δ l + 1 ( t ) ) − ( ( b 1 ) δ l − 1 ( t ) + δ l ( t ) + ( b 1 ) δ l + 1 ( t ) ) + r ( ( b 1 ) δ l − 1 ( t ) + δ l ( t ) + ( b 1 ) δ l + 1 ( t ) ) + Z l 1 − r Z l 2 = 0 , ( b 1 ) γ ˙ l − 1 l ( t ) + γ ˙ l ( t ) + ( b 1 ) γ ˙ l + 1 ( t ) − ( b 3 ) ( γ l − 1 ( t ) − 2 γ l ( t ) + γ l + 1 ( t ) ) − b ( ( b 1 ) δ l − 1 ( t ) + δ l ( t ) + ( b 1 ) δ l + 1 ( t ) ) + b Z l 2 = 0 ,

where δ ˙ l ( t ) = d δ l d t , γ ˙ l ( t ) = d γ l d t ,

Z l 1 = Θ 2 = ( ( b 1 ) δ l − 1 ( t ) + δ l ( t ) + ( b 1 ) δ l + 1 ( t ) ) 2 , Z l 2 = Θ ℧ = ( ( b 1 ) δ l − 1 ( t ) + δ l ( t ) + ( b 1 ) δ l + 1 ( t ) ) ( ( b 1 ) γ l − 1 ( t ) + γ l ( t ) + ( b 1 ) γ l + 1 ( t ) ) .

We describe δ l ( t ) , γ l ( t ) , and their derivatives with respect to time δ ˙ l ( t ) , γ ˙ l ( t ) with the finite difference method as follows:

(22) δ l ( t ) = δ l m , γ l ( t ) = γ l m , δ ˙ l ( t ) = δ l m + 1 − δ l m − 1 2 Δ t , γ ˙ l ( t ) = γ l m + 1 − γ l m − 1 2 Δ t .

Replacing δ l ( t ) , γ l ( t ) , and their first time derivatives in (21) with (22), and eliminating four unknowns δ − 1 , δ ℒ + 1 , γ − 1 , γ ℒ + 1 via boundary conditions (16), we obtain a system ( 2 n + 2 ) × ( 2 n + 2 ) of nonlinear algebraic equations can be evaluated by the Mathematica program.

5.2 Trigonometric cubic B-spline method

We construct the trigonometric cubic B-spline functions [26], T l ( x ) , at the knots x l , ( l = − 1 ( 1 ) ℒ + 1 ) can be defined as

(23) T l ( x ) = 1 w W 3 ( x l − 2 ) x l − 2 ≤ x ≤ x l − 1 , W ( x l − 2 ) ( W ( x l − 2 ) Y ( x l ) + W ( x l − 1 ) Y ( x l + 1 ) ) + W 2 ( x l − 1 ) Y ( x l + 2 ) x l − 1 ≤ x ≤ x l , Y ( x l + 2 ) ( Y ( x l + 2 ) W ( x l ) + W ( x l − 1 ) Y ( x l + 1 ) ) + Y 2 ( x l + 1 ) W ( x l − 2 ) x l ≤ x ≤ x l + 1 , Y 3 ( x l + 2 ) x l + 1 ≤ x ≤ x l + 2 , 0 otherwise ,

where w = sin h 2 sin ( h ) sin 3 h 2 , W ( x l ) = sin x − x l 2 , Y ( x l ) = sin x l − x 2 .

We assume that the numerical solutions of the BZ system (1) are denoted as Θ ˜ ( x , t ) and ℧ ˜ ( x , t ) , which are approximations equivalent to the exact solutions Θ ( x , t ) and ℧ ( x , t ) , respectively, as follows:

(24) Θ ˜ ( x , t ) = ∑ l = − 1 ℒ + 1 T l ( x ) α l ( t ) , ℧ ˜ ( x , t ) = ∑ l = − 1 ℒ + 1 T l ( x ) β l ( t ) ,

where α l ( t ) , β l ( t ) are time-dependent unknowns, and the trigonometric cubic B-spline function T l ( x ) and its two derivatives T l ′ ( x ) , T l ″ ( x ) at the nodal points x l are established from (23).

Using (23) and (24), we obtain the numerical solutions Θ ˜ ( x , t ) and ℧ ˜ ( x , t ) of the BZ equation (1) and its derivatives up to second order in terms of α l ( t ) and β l ( t ) , respectively, as follows:

(25) Θ ˜ ( x l , t ) = ( d 1 ) α l − 1 ( t ) + ( d 2 ) α l ( t ) + ( d 1 ) α l + 1 ( t ) , Θ ˜ l ′ = ( − d 3 ) ( α l − 1 ( t ) − α l + 1 ( t ) ) , Θ ˜ l ″ = ( d 4 ) ( α l − 1 ( t ) + ( d 5 ) α l ( t ) + ( d 4 ) α l + 1 ( t ) ) ,

(26) ℧ ˜ ( x l , t ) = ( d 1 ) β l − 1 ( t ) + ( d 2 ) β l ( t ) + ( d 1 ) β l + 1 ( t ) , ℧ ˜ l ′ = ( − d 3 ) ( β l − 1 ( t ) − β l + 1 ( t ) ) , ℧ ˜ l ″ = ( d 4 ) ( β l − 1 ( t ) + ( d 5 ) β l ( t ) + ( d 4 ) β l + 1 ( t ) ) ,

where d 1 = sin 2 h 2 sin ( h ) sin 3 h 2 , d 2 = 2 1 + 2 cos ( h ) , d 3 = 3 4 sin 3 h 2 , d 4 = 3 + 9 cos ( h ) 16 sin 2 h 2 2 cos h 2 + cos 3 h 2 , d 5 = 3 cos 2 h 2 sin 2 h 2 ( 2 + 4 cos ( h ) ) .

From (25) and (26) into (1), a general form equation is reached for the linearization technique

(27) ( d 1 ) α ˙ l − 1 ( t ) + ( d 2 ) α ˙ l ( t ) + ( d 1 ) α ˙ l + 1 ( t ) − ( ( d 4 ) α l − 1 ( t ) + ( d 5 ) α l ( t ) + ( d 4 ) α l + 1 ( t ) ) − ( ( d 1 ) α l − 1 ( t ) + ( d 2 ) α l ( t ) + ( d 1 ) α l + 1 ( t ) ) + r ( ( d 1 ) α l − 1 ( t ) + ( d 2 ) α l ( t ) + ( d 1 ) α l + 1 ( t ) ) + H l 1 − r H l 2 = 0 , ( d 1 ) β ˙ l − 1 ( t ) + ( d 2 ) β ˙ l ( t ) + ( d 1 ) β ˙ l + 1 ( t ) − ( ( d 4 ) β l − 1 ( t ) + ( d 5 ) β l ( t ) + ( d 4 ) β l + 1 ( t ) ) − b ( ( d 1 ) α l − 1 ( t ) + ( d 2 ) α l ( t ) + ( d 1 ) α l + 1 ( t ) ) + b H l 2 = 0 ,

where α ˙ l ( t ) = d α l d t , β ˙ l ( t ) = d β l d t , H l 1 = Θ 2 = ( ( d 1 ) α l − 1 ( t ) + ( d 2 ) α l ( t ) + ( d 1 ) α l + 1 ( t ) ) 2 , H l 2 = Θ ℧ = ( ( d 1 ) α l − 1 ( t ) + ( d 2 ) α l ( t ) + ( d 1 ) α l + 1 ( t ) ) ( ( d 1 ) β l − 1 ( t ) + ( d 2 ) β l ( t ) + ( d 1 ) β l + 1 ( t ) ) .

We apply the finite difference method of α l ( t ) , β l ( t ) , and their derivatives with respect to time α ˙ l ( t ) , β ˙ l ( t ) as follows:

(28) α l ( t ) = α l m , β l ( t ) = β l m , α ˙ l ( t ) = α l m + 1 − α l m − 1 2 Δ t , β ˙ l ( t ) = β l m + 1 − β l m − 1 2 Δ t .

Substituting α l ( t ) , β l ( t ) , and thier first time derivatives from (28) into (27) and eliminating unknowns α − 1 , α ℒ + 1 , β − 1 , β ℒ + 1 by boundary conditions (16), we obtain a system ( 2 n + 2 ) × ( 2 n + 2 ) of nonlinear algebraic equations can be calculated by the Mathematica program.

As above, we demonstrate the efficiency and effectiveness of the numerical methods presented and compute the L 2 and L ∞ norms of the error between the numerical and exact solutions as follows:

(29) L 2 = h ∑ l = 0 n ( Θ l m − Θ ˜ l m ) 2 = h ∑ l = 0 n ( ℧ l m − ℧ ˜ l m ) 2 , L ∞ = max l = 0 n ∣ Θ l m − Θ ˜ l m ∣ = max l = 0 n ∣ ℧ l m − ℧ ˜ l m ∣ .

6 Comparison of analytical and numerical results

Based on the information provided, we obtain numerical results for the BZ reaction–diffusion model using the proposed numerical methods. In particular, we compute the errors L 2 and L ∞ at r = 1.3 , d = 0.2 , μ = 0.5 , λ = 0.5 for the numerical solutions of the bromous acid concentration Θ and bromide ion concentration ℧ (15) at different step lengths and times. In Table 1, we investigate the error norms for soliton solutions of the BZ system (15) at different times with time level Δ t = 0.1 and step size h = 0.2 via the trigonometric cubic B-spline method with parameter set (12) and the exponential cubic B-spline method with parameter set (14). The obtained results in Table 2 show the errors at different time levels by the exponential cubic B-spline method of the BZ system (15) with parameters (12), whereas the trigonometric cubic B-spline method can be applied using (15) and parameters (14) at different time levels, as shown in Table 3. Also, the graphical plots in Figures 3 and 4 show the congruence between the analytical solutions and the numerical solutions of the BZ reaction–diffusion system (15) at time t = 0.5 and step size h = 0.5 with parameters of two sets (12) and (14), respectively. Thus, we can observe the proposed numerical methods are satisfactorily accurate in solving the BZ reaction–diffusion model.

Table 1

L 2 and L ∞ norms with time level Δ t = 0.1 and step length h = 0.2 and values for the BZ equation using (15) and parameters of (12) and (14), respectively

t Trigonometric cubic B-pline using set (12) Exponential cubic B-pline using set (14)
Θ ( x , t ) ℧ ( x , t ) Θ ( x , t ) ℧ ( x , t )
L 2 L ∞ L 2 L ∞ L 2 L ∞ L 2 L ∞
0.2 6.7152 × 1 0 ‒ 4 3.5820 × 1 0 ‒ 4 2.1793 × 1 0 ‒ 3 5.2009 × 1 0 ‒ 4 4.0002 × 1 0 ‒ 5 2.3615 × 1 0 ‒ 5 1.1801 × 1 0 ‒ 4 6.6231 × 1 0 ‒ 5
0.3 8.3227 × 1 0 ‒ 4 3.9462 × 1 0 ‒ 4 2.0219 × 1 0 ‒ 3 5.1989 × 1 0 ‒ 4 3.8578 × 1 0 ‒ 5 2.1984 × 1 0 ‒ 5 1.0836 × 1 0 ‒ 4 5.9843 × 1 0 ‒ 5
0.4 1.4714 × 1 0 ‒ 3 8.8277 × 1 0 ‒ 4 4.2047 × 1 0 ‒ 3 1.0405 × 1 0 ‒ 3 8.7150 × 1 0 ‒ 5 6.9560 × 1 0 ‒ 5 3.6256 × 1 0 ‒ 4 4.9303 × 1 0 ‒ 4
0.5 9.3346 × 1 0 ‒ 3 1.1827 × 1 0 ‒ 2 5.2762 × 1 0 ‒ 3 4.3328 × 1 0 ‒ 3 1.9906 × 1 0 ‒ 3 3.2484 × 1 0 ‒ 3 1.4241 × 1 0 ‒ 2 2.2937 × 1 0 ‒ 2
Table 2

L 2 and L ∞ norms via an exponential B-spline method with step length h = 1.0 , different time level Δ t and values for the BZ equation using (15) and parameters of (12)

t Δ t = 0.1 Δ t = 0.01
Θ ( x , t ) ℧ ( x , t ) Θ ( x , t ) ℧ ( x , t )
L 2 L ∞ L 2 L ∞ L 2 L ∞ L 2 L ∞
0.2 1.7875 × 1 0 ‒ 3 8.8980 × 1 0 ‒ 4 6.8753 × 1 0 ‒ 4 3.4223 × 1 0 ‒ 4 1.7649 × 1 0 ‒ 3 9.2811 × 1 0 ‒ 4 6.7881 × 1 0 ‒ 4 3.5696 × 1 0 ‒ 4
0.3 1.7655 × 1 0 ‒ 3 1.0145 × 1 0 ‒ 3 6.7904 × 1 0 ‒ 4 3.9019 × 1 0 ‒ 4 2.6436 × 1 0 ‒ 3 1.4227 × 1 0 ‒ 3 1.0168 × 1 0 ‒ 3 5.4720 × 1 0 ‒ 4
0.4 3.5406 × 1 0 ‒ 3 1.8802 × 1 0 ‒ 3 1.3618 × 1 0 ‒ 3 7.2315 × 1 0 ‒ 4 3.5250 × 1 0 ‒ 3 1.9030 × 1 0 ‒ 3 1.3558 × 1 0 ‒ 3 7.3193 × 1 0 ‒ 4
0.5 3.5413 × 1 0 ‒ 3 2.0287 × 1 0 ‒ 3 1.3620 × 1 0 ‒ 3 7.8030 × 1 0 ‒ 4 4.4131 × 1 0 ‒ 3 2.3433 × 1 0 ‒ 3 1.6973 × 1 0 ‒ 3 9.0127 × 1 0 ‒ 4
Table 3

L 2 and L ∞ norms via trigonometric B-spline method with step length h = 1.0 , different time level Δ t and values for the BZ equation using (15) and parameters of (14)

t Δ t = 0.1 Δ t = 0.01
Θ ( x , t ) ℧ ( x , t ) Θ ( x , t ) ℧ ( x , t )
L 2 L ∞ L 2 L ∞ L 2 L ∞ L 2 L ∞
0.2 2.3897 × 1 0 ‒ 3 7.6357 × 1 0 ‒ 4 2.3713 × 1 0 ‒ 2 7.1354 × 1 0 ‒ 3 2.4482 × 1 0 ‒ 3 8.2541 × 1 0 ‒ 4 2.3721 × 1 0 ‒ 2 7.1029 × 1 0 ‒ 3
0.3 2.5338 × 1 0 ‒ 3 9.0546 × 1 0 ‒ 4 2.3726 × 1 0 ‒ 2 7.0598 × 1 0 ‒ 3 3.7364 × 1 0 ‒ 3 1.3069 × 1 0 ‒ 3 3.5521 × 1 0 ‒ 2 1.0625 × 1 0 ‒ 2
0.4 4.9227 × 1 0 ‒ 3 1.6647 × 1 0 ‒ 3 4.7213 × 1 0 ‒ 2 1.4195 × 1 0 ‒ 2 5.0800 × 1 0 ‒ 3 1.8356 × 1 0 ‒ 3 4.7279 × 1 0 ‒ 2 1.4129 × 1 0 ‒ 2
0.5 5.2898 × 1 0 ‒ 3 1.9896 × 1 0 ‒ 3 4.7356 × 1 0 ‒ 2 1.4039 × 1 0 ‒ 2 6.4874 × 1 0 ‒ 3 2.4116 × 1 0 ‒ 3 5.8994 × 1 0 ‒ 2 1.7614 × 1 0 ‒ 2
Figure 3 Numerical (exponential B-spline method) and analytical results of the BZ equation using (15) with parameters (12) at r = 1.3 r=1.3 , d = 0.2 d=0.2 , μ = 0.5 \mu =0.5 , λ = 0.5 \lambda =0.5 , time t = 0.5 t=0.5 with Δ t = 0.1 \Delta t=0.1 and h = 0.6 h=0.6 .
Figure 3

Numerical (exponential B-spline method) and analytical results of the BZ equation using (15) with parameters (12) at r = 1.3 , d = 0.2 , μ = 0.5 , λ = 0.5 , time t = 0.5 with Δ t = 0.1 and h = 0.6 .

Figure 4 Numerical (trigonometric B-spline method) and analytical results of the BZ equation using (15) with parameters (14) at r = 1.3 , d = 0.2 , μ = 0.5 , λ = 0.5 r=1.3,d=0.2,\mu =0.5,\lambda =0.5 , time t = 0.5 t=0.5 with Δ t = 0.1 \Delta t=0.1 and h = 0.6 h=0.6 .
Figure 4

Numerical (trigonometric B-spline method) and analytical results of the BZ equation using (15) with parameters (14) at r = 1.3 , d = 0.2 , μ = 0.5 , λ = 0.5 , time t = 0.5 with Δ t = 0.1 and h = 0.6 .

7 Conclusion

In summary, the system of equations studied in this article provides a useful model for understanding complex nonlinear systems, with a range of interesting behaviors such as oscillations and spatial patterns. The exact solutions were obtained using an analytical method such as the Bernoulli sub-ODE technique, which offers valuable insights into the system’s behavior as in Figure 1 through some soliton solutions like dark and kink solitons. The effects of changing the parameters b were investigated (see Figure 2). Additionally, numerical schemes based on the exponential cubic B-spline method and trigonometric cubic B-spline method were applied to the BZ reaction–diffusion model and the results demonstrated the accuracy and efficiency of the numerical schemes as shown in Tables 1–3 and Figures 3 and 4. Future research may focus on exploring the applicability of other analytical and numerical methods to the BZ system, as well as investigating the system under different conditions and parameters. The BZ system continues to be a valuable model for studying self-organization and nonlinear dynamics in chemical systems, with potential applications in materials science, environmental monitoring, and other fields.

Acknowledgments

The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through Large Research Project under grant number RGP2/222/45.

  1. Funding information: This work was funded by the Deanship of Research and Graduate Studies at King Khalid University through Large Research Project under grant number RGP2/222/45.

  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 state no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

References

[1] Murray JD. Nonlinear Differential Equations in Biology, Lectures on Models [Russian translation]. Moscow: Mir; 1983. Search in Google Scholar

[2] Shanks N. Modeling biological systems: the Belousov-Zhabotinsky reaction. Found Chem. 2001;3:33–53. 10.1023/A:1011434929814Search in Google Scholar

[3] Hou L, Dueñas-Díez M, Srivastava R, Pérez-Mercader J. Flow chemistry controls self-assembly and cargo in Belousov-Zhabotinsky driven polymerization-induced self-assembly. Commun Chem 2019;2:139. 10.1038/s42004-019-0241-1Search in Google Scholar

[4] Howell L, Osborne E, Franklin A, Hébrard EE. Pattern recognition of chemical waves: Finding the activation energy of the autocatalytic step in the Belousov-Zhabotinsky reaction. J Phys Chem B 2021;125:1667–73. 10.1021/acs.jpcb.0c11079Search in Google Scholar PubMed PubMed Central

[5] Murray JD. Mathematical biology II: spatial models and biomedical applications. Third Edition. 2002. 10.1007/b98869Search in Google Scholar

[6] Lin G, Wan-Tong L. Travelling wavefronts of Belousov-Zhabotinskii system with diffusion and delay. Appl Math Lett. 2009;22:341–6. 10.1016/j.aml.2008.04.006Search in Google Scholar

[7] Murray JD. Mathematical biology. New York: Springer; 1989. 10.1007/978-3-662-08539-4Search in Google Scholar

[8] Murray JD. On traveling wave solutions in a model for B-Z reaction. J Theor Biol. 1976;56:329–53. 10.1016/S0022-5193(76)80078-1Search in Google Scholar

[9] Tyson JJ. The Belousov-Zhabotinskii reaction. Lecture Notes in Biomathematics. vol. 10. New York: Springer; 1976. 10.1007/978-3-642-93046-1Search in Google Scholar

[10] Ye QX, Wang MX. Travelling wave front solutions of Noyes-Field system for Belousov-Zhabotinskii reaction. Nonl Anal TMA. 1987;11:1289–302. 10.1016/0362-546X(87)90046-0Search in Google Scholar

[11] Ye QX, Wang MX. A note on travelling wave solutions for Belousov-Zhabotinskii chemical reaction. J Beijing Inst Technol. 1989;9(4):5–9. Search in Google Scholar

[12] Li ZY. The existence of travelling front solutions for reaction-diffusion system. J Partial Differ Equ. 1992;5(1):17–22. Search in Google Scholar

[13] Li ZY, Ye QX. Travelling wave front solutions for reaction-diffusion systems. J Partial Differ Equ. 1991;4(3):1–14. Search in Google Scholar

[14] Trofimchuk E, Pinto M, Trofimchuk S. Traveling waves for a model of the Belousov-Zhabotinsky reaction. J Differ Equ. 2013;254(9):3690–714. 10.1016/j.jde.2013.02.005Search in Google Scholar

[15] Basavaraja C, Huh DS, Park SH, Jeon UJ, Pierson R, Vishnuvardhan TK, et al. Bifurcation phase studies of Belousov-Zhabotinsky reaction containing oxalic acid and acetone as a mixed organic substrate in an open system. Bulletin Korean Chem Soc. 2007;28:1489–92. 10.5012/bkcs.2007.28.9.1489. Search in Google Scholar

[16] Manoranjan VS, Mitchell AR. A numerical study of the Belousov-Zhabotinskii chemical reaction using Galerkin finite element method. J Math Biol. 1983;16:251–60. 10.1007/BF00276505Search in Google Scholar

[17] Wang M, Xiong S, Yeeee Q. Explicit wave front solutions of Noyes-field systems for the Belousov-Zhabotinskii reaction. J Math Anal Appl. 1994;182:705–17. 10.1006/jmaa.1994.1114Search in Google Scholar

[18] Quinney DA, On computing travelling wave solutions in a model for the Belousov Zhabotinskii reaction. J Inst Math Appl. 1979;23:193–201. 10.1093/imamat/23.2.193Search in Google Scholar

[19] Taghizadeh N, Akbari M, Nejhad PE. Application of Bernoulli sub-ODE method for finding travelling wave solutions of Schrodinger equation power law nonlinearity. Appl Math. 2017;12:596–603. Search in Google Scholar

[20] Bilal M, Ali A, Mahmoud SR, E Tag-Eldin, Balubaid M. Fractional analysis of unsteady radiative brinkman-type nanofluid flow comprised of CoFe2O3 nanoparticles across a vertical plate. J Therm Anal Calorim. 2023;148:13869–82. 10.1007/s10973-023-12705-0Search in Google Scholar

[21] Gu Y, Peng L, Huang Z, Lai Y. Soliton, breather, lump, interaction solutions and chaotic behavior for the (2+1)-dimensional KPSKR equation. Chaos Solitons Fractals. 2024;187:115351. 10.1016/j.chaos.2024.115351Search in Google Scholar

[22] Gu Y, Zhang X, Huang Z, Peng L, Lai Y, Aminakbari N. Soliton and lump and travelling wave solutions of the (3+1) dimensional KPB like equation with analysis of chaotic behaviors. Sci Rep. 2024;14:20966. 10.1038/s41598-024-71821-5Search in Google Scholar PubMed PubMed Central

[23] Dag I, Ersoy O. The exponential cubic B-spline algorithm for Fisher equation. Chaos Soliton Fractal. 2016;86:101–6. 10.1016/j.chaos.2016.02.031Search in Google Scholar

[24] Hadhoud AR, Ali KK, Shaalan MA. A septic B-spline collocation method for solving nonlinear singular boundary value problems arising in physiological models. Sci Iran E 2020;27(3):1674–84. Search in Google Scholar

[25] Raslan KR, Ali KK. Numerical study of MHD-duct flow using the two-dimensional finite difference method. Appl Math Inf Sci. 2020;14:1–5. 10.18576/amis/140417Search in Google Scholar

[26] Abbas M, Majid AA, Ismail AIM, Rashid A. The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems. Appl Math Comput. 2014;239:74–88. 10.1016/j.amc.2014.04.031Search in Google Scholar
Received: 2024-08-22
Revised: 2024-11-27
Accepted: 2025-02-17
Published Online: 2025-03-05

© 2025 the author(s), published by De Gruyter

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

Articles in the same Issue

  1. Research Articles
  2. Single-step fabrication of Ag2S/poly-2-mercaptoaniline nanoribbon photocathodes for green hydrogen generation from artificial and natural red-sea water
  3. Abundant new interaction solutions and nonlinear dynamics for the (3+1)-dimensional Hirota–Satsuma–Ito-like equation
  4. A novel gold and SiO2 material based planar 5-element high HPBW end-fire antenna array for 300 GHz applications
  5. Explicit exact solutions and bifurcation analysis for the mZK equation with truncated M-fractional derivatives utilizing two reliable methods
  6. Optical and laser damage resistance: Role of periodic cylindrical surfaces
  7. Numerical study of flow and heat transfer in the air-side metal foam partially filled channels of panel-type radiator under forced convection
  8. Water-based hybrid nanofluid flow containing CNT nanoparticles over an extending surface with velocity slips, thermal convective, and zero-mass flux conditions
  9. Dynamical wave structures for some diffusion--reaction equations with quadratic and quartic nonlinearities
  10. Solving an isotropic grey matter tumour model via a heat transfer equation
  11. Study on the penetration protection of a fiber-reinforced composite structure with CNTs/GFP clip STF/3DKevlar
  12. Influence of Hall current and acoustic pressure on nanostructured DPL thermoelastic plates under ramp heating in a double-temperature model
  13. Applications of the Belousov–Zhabotinsky reaction–diffusion system: Analytical and numerical approaches
  14. AC electroosmotic flow of Maxwell fluid in a pH-regulated parallel-plate silica nanochannel
  15. Interpreting optical effects with relativistic transformations adopting one-way synchronization to conserve simultaneity and space–time continuity
  16. Modeling and analysis of quantum communication channel in airborne platforms with boundary layer effects
  17. Theoretical and numerical investigation of a memristor system with a piecewise memductance under fractal–fractional derivatives
  18. Tuning the structure and electro-optical properties of α-Cr2O3 films by heat treatment/La doping for optoelectronic applications
  19. High-speed multi-spectral explosion temperature measurement using golden-section accelerated Pearson correlation algorithm
  20. Dynamic behavior and modulation instability of the generalized coupled fractional nonlinear Helmholtz equation with cubic–quintic term
  21. Study on the duration of laser-induced air plasma flash near thin film surface
  22. Exploring the dynamics of fractional-order nonlinear dispersive wave system through homotopy technique
  23. The mechanism of carbon monoxide fluorescence inside a femtosecond laser-induced plasma
  24. Numerical solution of a nonconstant coefficient advection diffusion equation in an irregular domain and analyses of numerical dispersion and dissipation
  25. Numerical examination of the chemically reactive MHD flow of hybrid nanofluids over a two-dimensional stretching surface with the Cattaneo–Christov model and slip conditions
  26. Impacts of sinusoidal heat flux and embraced heated rectangular cavity on natural convection within a square enclosure partially filled with porous medium and Casson-hybrid nanofluid
  27. Stability analysis of unsteady ternary nanofluid flow past a stretching/shrinking wedge
  28. Solitonic wave solutions of a Hamiltonian nonlinear atom chain model through the Hirota bilinear transformation method
  29. Bilinear form and soltion solutions for (3+1)-dimensional negative-order KdV-CBS equation
  30. Solitary chirp pulses and soliton control for variable coefficients cubic–quintic nonlinear Schrödinger equation in nonuniform management system
  31. Influence of decaying heat source and temperature-dependent thermal conductivity on photo-hydro-elasto semiconductor media
  32. Dissipative disorder optimization in the radiative thin film flow of partially ionized non-Newtonian hybrid nanofluid with second-order slip condition
  33. Bifurcation, chaotic behavior, and traveling wave solutions for the fractional (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili model
  34. New investigation on soliton solutions of two nonlinear PDEs in mathematical physics with a dynamical property: Bifurcation analysis
  35. Mathematical analysis of nanoparticle type and volume fraction on heat transfer efficiency of nanofluids
  36. Creation of single-wing Lorenz-like attractors via a ten-ninths-degree term
  37. Optical soliton solutions, bifurcation analysis, chaotic behaviors of nonlinear Schrödinger equation and modulation instability in optical fiber
  38. Chaotic dynamics and some solutions for the (n + 1)-dimensional modified Zakharov–Kuznetsov equation in plasma physics
  39. Fractal formation and chaotic soliton phenomena in nonlinear conformable Heisenberg ferromagnetic spin chain equation
  40. Single-step fabrication of Mn(iv) oxide-Mn(ii) sulfide/poly-2-mercaptoaniline porous network nanocomposite for pseudo-supercapacitors and charge storage
  41. Novel constructed dynamical analytical solutions and conserved quantities of the new (2+1)-dimensional KdV model describing acoustic wave propagation
  42. Tavis–Cummings model in the presence of a deformed field and time-dependent coupling
  43. Spinning dynamics of stress-dependent viscosity of generalized Cross-nonlinear materials affected by gravitationally swirling disk
  44. Design and prediction of high optical density photovoltaic polymers using machine learning-DFT studies
  45. Robust control and preservation of quantum steering, nonlocality, and coherence in open atomic systems
  46. Coating thickness and process efficiency of reverse roll coating using a magnetized hybrid nanomaterial flow
  47. Dynamic analysis, circuit realization, and its synchronization of a new chaotic hyperjerk system
  48. Decoherence of steerability and coherence dynamics induced by nonlinear qubit–cavity interactions
  49. Finite element analysis of turbulent thermal enhancement in grooved channels with flat- and plus-shaped fins
  50. Modulational instability and associated ion-acoustic modulated envelope solitons in a quantum plasma having ion beams
  51. Statistical inference of constant-stress partially accelerated life tests under type II generalized hybrid censored data from Burr III distribution
  52. On solutions of the Dirac equation for 1D hydrogenic atoms or ions
  53. Entropy optimization for chemically reactive magnetized unsteady thin film hybrid nanofluid flow on inclined surface subject to nonlinear mixed convection and variable temperature
  54. Stability analysis, circuit simulation, and color image encryption of a novel four-dimensional hyperchaotic model with hidden and self-excited attractors
  55. A high-accuracy exponential time integration scheme for the Darcy–Forchheimer Williamson fluid flow with temperature-dependent conductivity
  56. Novel analysis of fractional regularized long-wave equation in plasma dynamics
  57. Development of a photoelectrode based on a bismuth(iii) oxyiodide/intercalated iodide-poly(1H-pyrrole) rough spherical nanocomposite for green hydrogen generation
  58. Investigation of solar radiation effects on the energy performance of the (Al2O3–CuO–Cu)/H2O ternary nanofluidic system through a convectively heated cylinder
  59. Quantum resources for a system of two atoms interacting with a deformed field in the presence of intensity-dependent coupling
  60. Studying bifurcations and chaotic dynamics in the generalized hyperelastic-rod wave equation through Hamiltonian mechanics
  61. A new numerical technique for the solution of time-fractional nonlinear Klein–Gordon equation involving Atangana–Baleanu derivative using cubic B-spline functions
  62. Interaction solutions of high-order breathers and lumps for a (3+1)-dimensional conformable fractional potential-YTSF-like model
  63. Hydraulic fracturing radioactive source tracing technology based on hydraulic fracturing tracing mechanics model
  64. Numerical solution and stability analysis of non-Newtonian hybrid nanofluid flow subject to exponential heat source/sink over a Riga sheet
  65. Numerical investigation of mixed convection and viscous dissipation in couple stress nanofluid flow: A merged Adomian decomposition method and Mohand transform
  66. Effectual quintic B-spline functions for solving the time fractional coupled Boussinesq–Burgers equation arising in shallow water waves
  67. Analysis of MHD hybrid nanofluid flow over cone and wedge with exponential and thermal heat source and activation energy
  68. Solitons and travelling waves structure for M-fractional Kairat-II equation using three explicit methods
  69. Impact of nanoparticle shapes on the heat transfer properties of Cu and CuO nanofluids flowing over a stretching surface with slip effects: A computational study
  70. Computational simulation of heat transfer and nanofluid flow for two-sided lid-driven square cavity under the influence of magnetic field
  71. Irreversibility analysis of a bioconvective two-phase nanofluid in a Maxwell (non-Newtonian) flow induced by a rotating disk with thermal radiation
  72. Hydrodynamic and sensitivity analysis of a polymeric calendering process for non-Newtonian fluids with temperature-dependent viscosity
  73. Exploring the peakon solitons molecules and solitary wave structure to the nonlinear damped Kortewege–de Vries equation through efficient technique
  74. Modeling and heat transfer analysis of magnetized hybrid micropolar blood-based nanofluid flow in Darcy–Forchheimer porous stenosis narrow arteries
  75. Activation energy and cross-diffusion effects on 3D rotating nanofluid flow in a Darcy–Forchheimer porous medium with radiation and convective heating
  76. Insights into chemical reactions occurring in generalized nanomaterials due to spinning surface with melting constraints
  77. Review Article
  78. Examination of the gamma radiation shielding properties of different clay and sand materials in the Adrar region
  79. Special Issue on Fundamental Physics from Atoms to Cosmos - Part II
  80. Possible explanation for the neutron lifetime puzzle
  81. Special Issue on Nanomaterial utilization and structural optimization - Part III
  82. Numerical investigation on fluid-thermal-electric performance of a thermoelectric-integrated helically coiled tube heat exchanger for coal mine air cooling
  83. Special Issue on Nonlinear Dynamics and Chaos in Physical Systems
  84. Analysis of the fractional relativistic isothermal gas sphere with application to neutron stars
  85. Abundant wave symmetries in the (3+1)-dimensional Chafee–Infante equation through the Hirota bilinear transformation technique
  86. Successive midpoint method for fractional differential equations with nonlocal kernels: Error analysis, stability, and applications
https://doi.org/10.1515/phys-2025-0129
Keywords for this article
Belousov–Zhabotinsky reaction–diffusion model; Bernoulli sub-ODE technique; soliton solutions; exponential cubic B-spline method; trigonometric cubic B-spline method
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. Single-step fabrication of Ag2S/poly-2-mercaptoaniline nanoribbon photocathodes for green hydrogen generation from artificial and natural red-sea water
  3. Abundant new interaction solutions and nonlinear dynamics for the (3+1)-dimensional Hirota–Satsuma–Ito-like equation
  4. A novel gold and SiO2 material based planar 5-element high HPBW end-fire antenna array for 300 GHz applications
  5. Explicit exact solutions and bifurcation analysis for the mZK equation with truncated M-fractional derivatives utilizing two reliable methods
  6. Optical and laser damage resistance: Role of periodic cylindrical surfaces
  7. Numerical study of flow and heat transfer in the air-side metal foam partially filled channels of panel-type radiator under forced convection
  8. Water-based hybrid nanofluid flow containing CNT nanoparticles over an extending surface with velocity slips, thermal convective, and zero-mass flux conditions
  9. Dynamical wave structures for some diffusion--reaction equations with quadratic and quartic nonlinearities
  10. Solving an isotropic grey matter tumour model via a heat transfer equation
  11. Study on the penetration protection of a fiber-reinforced composite structure with CNTs/GFP clip STF/3DKevlar
  12. Influence of Hall current and acoustic pressure on nanostructured DPL thermoelastic plates under ramp heating in a double-temperature model
  13. Applications of the Belousov–Zhabotinsky reaction–diffusion system: Analytical and numerical approaches
  14. AC electroosmotic flow of Maxwell fluid in a pH-regulated parallel-plate silica nanochannel
  15. Interpreting optical effects with relativistic transformations adopting one-way synchronization to conserve simultaneity and space–time continuity
  16. Modeling and analysis of quantum communication channel in airborne platforms with boundary layer effects
  17. Theoretical and numerical investigation of a memristor system with a piecewise memductance under fractal–fractional derivatives
  18. Tuning the structure and electro-optical properties of α-Cr2O3 films by heat treatment/La doping for optoelectronic applications
  19. High-speed multi-spectral explosion temperature measurement using golden-section accelerated Pearson correlation algorithm
  20. Dynamic behavior and modulation instability of the generalized coupled fractional nonlinear Helmholtz equation with cubic–quintic term
  21. Study on the duration of laser-induced air plasma flash near thin film surface
  22. Exploring the dynamics of fractional-order nonlinear dispersive wave system through homotopy technique
  23. The mechanism of carbon monoxide fluorescence inside a femtosecond laser-induced plasma
  24. Numerical solution of a nonconstant coefficient advection diffusion equation in an irregular domain and analyses of numerical dispersion and dissipation
  25. Numerical examination of the chemically reactive MHD flow of hybrid nanofluids over a two-dimensional stretching surface with the Cattaneo–Christov model and slip conditions
  26. Impacts of sinusoidal heat flux and embraced heated rectangular cavity on natural convection within a square enclosure partially filled with porous medium and Casson-hybrid nanofluid
  27. Stability analysis of unsteady ternary nanofluid flow past a stretching/shrinking wedge
  28. Solitonic wave solutions of a Hamiltonian nonlinear atom chain model through the Hirota bilinear transformation method
  29. Bilinear form and soltion solutions for (3+1)-dimensional negative-order KdV-CBS equation
  30. Solitary chirp pulses and soliton control for variable coefficients cubic–quintic nonlinear Schrödinger equation in nonuniform management system
  31. Influence of decaying heat source and temperature-dependent thermal conductivity on photo-hydro-elasto semiconductor media
  32. Dissipative disorder optimization in the radiative thin film flow of partially ionized non-Newtonian hybrid nanofluid with second-order slip condition
  33. Bifurcation, chaotic behavior, and traveling wave solutions for the fractional (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili model
  34. New investigation on soliton solutions of two nonlinear PDEs in mathematical physics with a dynamical property: Bifurcation analysis
  35. Mathematical analysis of nanoparticle type and volume fraction on heat transfer efficiency of nanofluids
  36. Creation of single-wing Lorenz-like attractors via a ten-ninths-degree term
  37. Optical soliton solutions, bifurcation analysis, chaotic behaviors of nonlinear Schrödinger equation and modulation instability in optical fiber
  38. Chaotic dynamics and some solutions for the (n + 1)-dimensional modified Zakharov–Kuznetsov equation in plasma physics
  39. Fractal formation and chaotic soliton phenomena in nonlinear conformable Heisenberg ferromagnetic spin chain equation
  40. Single-step fabrication of Mn(iv) oxide-Mn(ii) sulfide/poly-2-mercaptoaniline porous network nanocomposite for pseudo-supercapacitors and charge storage
  41. Novel constructed dynamical analytical solutions and conserved quantities of the new (2+1)-dimensional KdV model describing acoustic wave propagation
  42. Tavis–Cummings model in the presence of a deformed field and time-dependent coupling
  43. Spinning dynamics of stress-dependent viscosity of generalized Cross-nonlinear materials affected by gravitationally swirling disk
  44. Design and prediction of high optical density photovoltaic polymers using machine learning-DFT studies
  45. Robust control and preservation of quantum steering, nonlocality, and coherence in open atomic systems
  46. Coating thickness and process efficiency of reverse roll coating using a magnetized hybrid nanomaterial flow
  47. Dynamic analysis, circuit realization, and its synchronization of a new chaotic hyperjerk system
  48. Decoherence of steerability and coherence dynamics induced by nonlinear qubit–cavity interactions
  49. Finite element analysis of turbulent thermal enhancement in grooved channels with flat- and plus-shaped fins
  50. Modulational instability and associated ion-acoustic modulated envelope solitons in a quantum plasma having ion beams
  51. Statistical inference of constant-stress partially accelerated life tests under type II generalized hybrid censored data from Burr III distribution
  52. On solutions of the Dirac equation for 1D hydrogenic atoms or ions
  53. Entropy optimization for chemically reactive magnetized unsteady thin film hybrid nanofluid flow on inclined surface subject to nonlinear mixed convection and variable temperature
  54. Stability analysis, circuit simulation, and color image encryption of a novel four-dimensional hyperchaotic model with hidden and self-excited attractors
  55. A high-accuracy exponential time integration scheme for the Darcy–Forchheimer Williamson fluid flow with temperature-dependent conductivity
  56. Novel analysis of fractional regularized long-wave equation in plasma dynamics
  57. Development of a photoelectrode based on a bismuth(iii) oxyiodide/intercalated iodide-poly(1H-pyrrole) rough spherical nanocomposite for green hydrogen generation
  58. Investigation of solar radiation effects on the energy performance of the (Al2O3–CuO–Cu)/H2O ternary nanofluidic system through a convectively heated cylinder
  59. Quantum resources for a system of two atoms interacting with a deformed field in the presence of intensity-dependent coupling
  60. Studying bifurcations and chaotic dynamics in the generalized hyperelastic-rod wave equation through Hamiltonian mechanics
  61. A new numerical technique for the solution of time-fractional nonlinear Klein–Gordon equation involving Atangana–Baleanu derivative using cubic B-spline functions
  62. Interaction solutions of high-order breathers and lumps for a (3+1)-dimensional conformable fractional potential-YTSF-like model
  63. Hydraulic fracturing radioactive source tracing technology based on hydraulic fracturing tracing mechanics model
  64. Numerical solution and stability analysis of non-Newtonian hybrid nanofluid flow subject to exponential heat source/sink over a Riga sheet
  65. Numerical investigation of mixed convection and viscous dissipation in couple stress nanofluid flow: A merged Adomian decomposition method and Mohand transform
  66. Effectual quintic B-spline functions for solving the time fractional coupled Boussinesq–Burgers equation arising in shallow water waves
  67. Analysis of MHD hybrid nanofluid flow over cone and wedge with exponential and thermal heat source and activation energy
  68. Solitons and travelling waves structure for M-fractional Kairat-II equation using three explicit methods
  69. Impact of nanoparticle shapes on the heat transfer properties of Cu and CuO nanofluids flowing over a stretching surface with slip effects: A computational study
  70. Computational simulation of heat transfer and nanofluid flow for two-sided lid-driven square cavity under the influence of magnetic field
  71. Irreversibility analysis of a bioconvective two-phase nanofluid in a Maxwell (non-Newtonian) flow induced by a rotating disk with thermal radiation
  72. Hydrodynamic and sensitivity analysis of a polymeric calendering process for non-Newtonian fluids with temperature-dependent viscosity
  73. Exploring the peakon solitons molecules and solitary wave structure to the nonlinear damped Kortewege–de Vries equation through efficient technique
  74. Modeling and heat transfer analysis of magnetized hybrid micropolar blood-based nanofluid flow in Darcy–Forchheimer porous stenosis narrow arteries
  75. Activation energy and cross-diffusion effects on 3D rotating nanofluid flow in a Darcy–Forchheimer porous medium with radiation and convective heating
  76. Insights into chemical reactions occurring in generalized nanomaterials due to spinning surface with melting constraints
  77. Review Article
  78. Examination of the gamma radiation shielding properties of different clay and sand materials in the Adrar region
  79. Special Issue on Fundamental Physics from Atoms to Cosmos - Part II
  80. Possible explanation for the neutron lifetime puzzle
  81. Special Issue on Nanomaterial utilization and structural optimization - Part III
  82. Numerical investigation on fluid-thermal-electric performance of a thermoelectric-integrated helically coiled tube heat exchanger for coal mine air cooling
  83. Special Issue on Nonlinear Dynamics and Chaos in Physical Systems
  84. Analysis of the fractional relativistic isothermal gas sphere with application to neutron stars
  85. Abundant wave symmetries in the (3+1)-dimensional Chafee–Infante equation through the Hirota bilinear transformation technique
  86. Successive midpoint method for fractional differential equations with nonlocal kernels: Error analysis, stability, and applications
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 9.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/phys-2025-0129/html
Scroll to top button