
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

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69445
10.1038/s41598-024-69445-w
Article
Regularity and wave study of an advection–diffusion–reaction equation
Akgül Ali aliakgul00727@gmail.com

15
Ahmed Nauman 25
Shahzad Muhammad 2
Baber Muhammad Zafarullah 2
Iqbal Muhammad Sajid 347
Chan Choon Kit 6
1 https://ror.org/05ptwtz25 grid.449212.8 0000 0004 0399 6093 Department of Mathematics, Art and Science Faculty, Siirt University, 56100 Siirt, Turkey
2 https://ror.org/051jrjw38 grid.440564.7 0000 0001 0415 4232 Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
3 grid.412117.0 0000 0001 2234 2376 Department of Humanities & Basic Science, Military College of Signals, NUST, Islamabad, Pakistan
4 https://ror.org/04zfme737 grid.4425.7 0000 0004 0368 0654 Department of Academic Affairs, School of Leadership and Business, Oryx Universal College with Liverpool John Moores University (UK), 12253 Doha, Qatar
5 https://ror.org/00hqkan37 grid.411323.6 0000 0001 2324 5973 Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon
6 https://ror.org/03fj82m46 grid.444479.e 0000 0004 1792 5384 Faculty of Engineering and Quantity Surveying, INTI International University, 71800 Nilai, Malaysia
7 https://ror.org/059bgad73 grid.449114.d 0000 0004 0457 5303 MEU Research Unit, Faculty of Information Technology, Middle East University, Amman, 11831 Jordan
6 9 2024
6 9 2024
2024
14 207765 1 2024
5 8 2024
© The Author(s) 2024
2024
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In this paper, we investigate the optimal conditions to the boundaries where the unique existence of the solutions to an advection-diffusion-reaction equation is secured by applying the contraction mapping theorem from the study of fixed points. Also, we extract, traveling wave solutions of the underlying equation. To this purpose, a new extended direct algebraic method with traveling wave transformation has been used. Achieved soliton solutions are different functions which are hyperbolic, trigonometric, exponential, and some mixed trigonometric functions. These functions show the nature of solitons. Two and three-dimensional plots are drawn using different values of parameters and coefficients for the comparison and behavior of solitons as combined bright-dark, dark, and bright solitons.

Keywords

Advection–diffusion–reaction
Extinction wave
Contraction
Lipschitz
Solitons
New MEDA method
Subject terms

Biochemistry
Cell biology
Mathematics and computing
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Several nonlinear natural phenomena and physical processes are analyzed by nonlinear partial differential equations (NPDEs). Indeed, these equations play a significant role in different dynamic and physical problems in many fields of science such as physics, geophysics, plasma physics, engineering, chemistry, geochemistry, optical fibers, fluid dynamics, and others1–9. In these days, nonlinear systems become an important and intrusting field for modern researchers to find the exact and analytical solutions for the problems because of their research contributions to the investigation of the actual characteristics. There are some contributions for the nonlinear systems are discussed. A comprehensive study for the soliton solutions to the partial differential equations is carried out by Helal10. He introduced various techniques for analytical and numerical treatment. He described that a localized wave of a special nonlinear type is represented by solitons. The solutions of a nonlinear system of equations or a single equation are described as the solitons. Samir et al.11 used a modified extended mapping approach to extract traveling wave results of a nonlinear system. Rezazadeh et al.12 implemented an exponential rational function and the Jacobi elliptic functions schemes to investigate wave surfaces of the same nonlinear model. Further, Yokus et al.13 obtained the solitary wave solutions of the nonlinear process with the use of the sinh-Gordon function.

In this paper, we consider a nonlinear advection reaction and diffusion equations for the analysis of the optimization in function spaces and to extract the solitary wave solutions using an analytical technique known as the new extended direct algebraic method14, the proposed equation is given by the formula1 Θt(x,t)=κ1Θ(x,t)-κ2Θμ(x,t)-η1Θx(x,t)+η2Δx[Θ(x,t)],

where Θ(x,t) represent the main substance and Δx[Θ(x,t)]=Θxx(x,t). Also, η1>0 represents the velocity of wave movement, η2>0 is the coefficient of diffusion while propagating, κ1>0,κ2>0 are coefficients in the term of reaction, and the parameter μ>1. The nonlinear equation (2) arises in different fields of engineering and modern science, such as fluid dynamics, acoustics, electricity, and population dynamics. Where μ is the nonlinearity of the underlying model because the behavior and physical effects of a system can be substantially changed by adding a nonlinearity term (Θ2) to nonlinear advection, reaction, and diffusion equations. The addition of an Θ2 component to the equation’s reaction portion suggests that the response rate is now reliant on the quantity of interest squared. This may result in oscillations, more complex dynamics, and bistability-the occurrence of multiple stable states-in the system’s history. It’s possible that the behavior of the system deviates from first-order or basic linear reaction kinetics. Rich and intricate dynamics can result from combining nonlinear advection, response, and diffusion, particularly when a component like Θ2 is included. Pattern creation, chaos, bifurcations, and the evolution of coherent structures are examples of phenomena that these systems may display. Numerical simulations are usually necessary for the analysis of such systems, which may use methods from bifurcation theory and nonlinear dynamics. So, we consider the underlying model as2 Θt(x,t)=κ1Θ(x,t)-κ2Θ2(x,t)-η1Θx(x,t)+η2Δx[Θ(x,t)].

Many scientists have already worked on the study of the advection-diffusion-reaction equation with several approaches in different fields and with different theories. Some work is under study are given. In 2001, Hauke et al. investigated the advection-diffusion-reaction equation with variational sub-grid scale to the formulation of the underlying equation15. In 2002, Hauke investigated the advection-diffusion-reaction equation using a simple subgrid scale stabilized technique for numerical solutions16. In 2003, Hundsdorfer and Verwer wrote an interesting book on the application and numerical results of dynamical advection-diffusion-reaction equations. In this book, the proposed time-dependent equation is analyzed with several numerical techniques and schemes17. In 2006, Spiegelman and Katz worked on an advection-diffusion problem and found the numerical solution with the help of a semi-Lagrangian Crank-Nicolson algorithm18. In 2011, Boonkkamp and Anthonissen worked on advection-diffusion-reaction equations using a finite volume-complete flux scheme to analyze the numerical approximation19. In 2014, Kaya and Gharehbaghi extracted the implicit results of the investigation of an advection-diffusion equation with many numerical approaches20. In 2017, Mirza et al. found the basic results for advection-diffusion equation with different derivatives and nonlocal techniques using non-integer differentiation with nonsingular kernel21. In 2019, Jannelli et al. Obtained different analytical solutions and some numerical results of both space and time advection-diffusion-reaction equation22. In the same year 2019, Singh et al. also worked on the numerical solutions using different numerical techniques and achieved important numerical results to the reaction-advection-diffusion equation23. In 2022, Savovic and co-authors wrote an article on the investigation of a comparative study to the advection-diffusion-reaction equation to model the exponential type traveling wave in the process of heat transformation and mass transformation24.

Our aim in this research is to analyze the unique existence of the solutions in the optimal closed and convex subset of the Banach space. We apply some fundamental results from fixed point theory for uniqueness. Fixed point study is an intrusting and important mathematical approach to analyze the existence of unique solutions for different physical problems. As a consequence, the existence of unique results for any system of differential equations is analyzed with the help of some basic theorems of fixed point theory. For many years, fixed point theory has been growing up to achieve the structure and promoted very fast. Fixed point study is appreciable and plays a wonderful role in mathematical sciences and makes its own space in research areas due to its physical applications25,26. Several theorems of fixed point analysis are developed for high mappings of metric spaces. In Particular, fixed point theorems for group mappings are rather advantageous in the analysis of optimum control and have been regularly used to compute several physical problems in economic studies and the study of games27. The fixed point approach can be applied in different spaces. These characteristics of this study make it strong and valuable in the analysis of many problems of practical sciences modeled with differential equations or partial differential equations. The main result known as Banach fixed point theorem for the existence of unique solutions with the help of the Lipschitz condition for initial value problems of many differential equations gives the condition for the unique existence of the solutions in these initial value problems and we ensure the uniqueness of the achieved solutions28,29. In this literature, Iqbal and co-authors investigated the existence and uniqueness in the analysis of the fractional diarrhea model with the Mittag-Leffler kernel30. Ahmed and co-authors worked on a novel time-efficient structure-preserving splitting method for the solution of two-dimensional reaction-diffusion systems31. Furthermore, this paper studied the exact traveling wave results for an advection-diffusion-reaction equation. These exact soliton results are in the form of different functions like trigonometric, hyperbolic, exponential, and some mixed functions extracted by using two forms of integration. For this purpose, we apply a new extended direct algebraic method with the use of traveling wave transformation. The new extended direct algebraic method is a convincing and direct process to find solitons. This method is used by different researchers to obtain the exact results for a family of differential equations32–34. Younis et al. used this technique in their paper on nonlinear dynamical exact and solitary wave structures in the separation phase of iron to the ternary alloys35. Hussain et al. apply this technique in their research on optical solitons of NLS-type differential equations36. Malik et al. showed the integrability of the model in 2022 and used a combination of Lie classical technique with polynomial type assumption to find its series solutions. They derived some soliton results using the tanh-expansion analytical strategy and Kudryashov technique and they constructed local conservation laws with the Ibragimov approach37. Kumar et al. find the analytical wave results to Kudryashov-Sinelshchikov (KS) in 2022 by applying the generalized exponential rational function technique38. In 2023 Kumar and Niwas applied a novel technique namely the new inverse expansion method to calculate the wave solution to the (2+1)-dimensional nonlinear Heisenberg ferromagnetic spin chain (HFSC) equation39, also they used Lie classical method and unified method for (2+1)-dimensional variable coefficient Boiti Leon Manna Pempinelli equation40 and they apply inverse (G’/G)-expansion method to get the soliton results for (2+ 1)-dimensional generalized Benjamin-Ono equation41. Further, many professionals apply this method in their different research articles42. The methodology established given for a technical way to earn an infinitely stable combined bright and dark, bright and dark soliton results in solitary waves of an advection-diffusion-reaction equation with non-discrete dispersion term of the equation, nonlinearity term of the equation, and absorption of this model. It is given that solitons exist in the solution only under specific conditions and the unique parameter functions shown in dispersion, nonlinearity, and absorption inhomogeneities cannot be chosen independently. Fundamental soliton management regimes are discovered43–47.

This article is classified into different sections. In “Preliminaries” we discuss some important basic definitions which are helpful for our work. In “Unique results” we use the fundamental technique and some important results and theorems from the fixed point theory of the Banach space for this important advection equation. We include two significant theorems in this section to a final decision about the existence of the unique results. The first theorem shows the self-mapping of the fixed point operator and the second theorem shows the contraction mapping of the fixed point operator derived from an advection–diffusion–reaction equation. “Key points for new extended direct algebraic method” is specified to obtain the exact soliton results and soliton behavior of the equation and show the achieved solutions are in the different types of functions. The constraints and conditions for the exact solution solutions also emerged during the derivation of the solutions. “Analytical application” is devoted to the simulation and in this section three and two-dimensional plots are drawn with the help of computer software. These figures show the behavior of the solitons in different ways like dark and bright solitons. In “Graphical analysis” we conclude the final analysis of our paper and discuss the complete paper from a moral view.

Preliminaries

Definition 1

A space C=(X,d) has the fixed point property if for any continuous function, Θ:X→X there exists x∈X such that Θ(x)=x48.

Definition 2

Under the given transformation a value that does not change is called a fixed point or invariant point. For a function, an element that is mapped to itself by the given function is called the fixed point of such function. Such that, if a function Θ(x)=sin(x) then 0 is a fixed point because Θ(0)=0.49

Definition 3

Let C=(X,d) is Banach space. A self mapping Θ:X→X is said to be a contraction mapping on C if there exists a positive real constant λ<1 such that ||Θ(x1),Θ(x2)||=λ||x1,x2|| for all x1,x2∈X50.

Theorem 1

Let C=(X,d) be a Banach space and suppose that Θ:X→X is a contraction mapping on the space C. Then Θ has a unique fixed point x∈X such that Θ(x)=x51.

Unique results

This portion is devoted to achieving the first and main goal of this article, where we describe the basic terms or conditions for the existence of the unique solutions to the underlying nonlinear equation (2). To this purpose, we apply different technical approaches from the fundamental theory of fixed points. In an initial step, we construct a fixed point operator from the Eq. (2) in the form of a nonhomogeneous first kind of Volterra type integral equation52. Further, this Volterra-type integral equation investigates the description of the self-mapping of a fixed point operator then we apply Schauder’s fixed point theorem for the relative compactness in the given mapping with the help of Arzela Ascoli’s theorem to the observation of the existence of solutions and further we apply Banach fixed point theorem known as contraction mapping principle to the purpose of unique existence. Some meaningful results are established for analysis. Consider Eq. (2) with initial values, as follows3 Θt=κ1Θ-κ2Θ2-η1Θx+η2Δx[Θ],

subject to the initial conditionΘ(x,t)=Θ(t)=Θ(0)=Θ0≥0.

Integrate the Eq. (3) with respect to substitute variable of time from t0=0→t, such thatΘ(x,t)-Θ(x,0)=∫0tκ1Θ(x,τ)-κ2Θ2(x,τ)-η1Θx+η2Δx[Θ(x,τ)]dτ,

using initial conditions, we get4 MΘ(t)=Θ0+∫0tκ1Θ(τ)-κ2Θ2(τ)-η1Θx+η2Δx[Θ(τ)]dτ,

where we use MΘ(t)=Θ(x,t) to just represent this integral equation as a fixed point operator hence, Eq. (4) is the first kind of Volterra-type integral equation and is known as a fixed point operator. Equation (4) satisfies the Eq. (3) which shows that the Eq. (4) is a solution for the Eq. (3). For further analysis, we let a complete norm space (Banach space) of all continuous functions, denoted by C, and draw a closed ball Ω in C with radius r and center at initial value Θ0. So, Ω is a closed and convex subset of C such that5 Ω=Ωr(Θ0)={Θ,Θ∈C[0,L]:||Θ-Θ0||≤r},||Θ||≤(r+Θ0)∵Θ0>0,

where L is a representation of time length from initial time 0 to maximum time tm.

Theorem 2

Suppose that MΘ is a continuous operator. L represent the time length from initial time 0 to maximum time tm and Ω is a closed and convex subset in Banach space C for operator (4) which is mapped itself under the inequalityLMΘ(r)≤rR(|κ1|+|κ2|R2-1+η1∗+η2∗).

Proof

Consider the Eq. (4)MΘ(t)-Θ0=∫0mκ1Θ(τ)-κ2Θ2(τ)-η1Θx+η2Δx[Θ(τ)]dτ,

taking norm on both sides and proceeding as follows||MΘ(t)-Θ0||≤∫0m||κ1Θ(τ)-κ2Θ2(τ)-η1Θx+η2Δx[Θ(τ)]||dτ,

6 ||MΘ(t)-Θ0||≤∫0mH(τ)dτ,

whereH(τ)=||κ1Θ(τ)-κ2Θ2(τ)-η1Θx+η2Δx[Θ(τ)]||,H(τ)≤|κ1|;||Θ||+|κ2|;||Θ||2+|η1|;||Θx||+|η2|;||ΔxΘ||,

now, apply embedding theorem in both ways one time and two time embedding (see Theorem 1.34 from53) we get,H(τ)≤|κ1|;||Θ||+|κ2|;||Θ||2+η1∗;||Θ||+η2∗;||Θ||,

using (5)H(τ)≤|κ1|(r-Θ0)+|κ2|(r-Θ0)2+η1∗(r-Θ0)+η2∗(r-Θ0),

suppose that R is distance from Θ0 to the boundary of Ω which gives R=(r-Θ0) so thatH(τ)≤R(|κ1|+|κ2|R2-1+η1∗+η2∗),

putting in (6)7 ||MΘ(t)-Θ0||≤RL(|κ1|+|κ2|R2-1+η1∗+η2∗),

for self mapping we know that the distance between the function values at initial to final stage is always less or equal to the radius of the given closed ball therefore we haveRL(|κ1|+|κ2|R2-1+η1∗+η2∗)≤r

which gives the condition LΘ(r) for the operator (4) such that8 LMΘ(r)≤rR(|κ1|+|κ2|R2-1+η1∗+η2∗).

Hence, operator (4) mapped itself under the inequality (8). □

Theorem 3

Let the operator MΘ(t) is at least a locally Lipschitz continuous function and L is a time duration from initial time 0 to the final time tf. Then the operator (4) mapped as a contraction mapping within the closed, bounded and convex subset Ω in C under the inequality conditionLM<1|κ1|+|κ2|2c2-1+η1′+η2′.

Proof

Consider the Eq. (4) and assuming two different pre-images Θ1 and Θ2 corresponding to their images MΘ1 and MΘ2 as follows,9 MΘ1=Θ0+∫0nκ1Θ1-κ2Θ12-η1Θ1x+η2Δx(Θ1)dτ,

10 MΘ2=Θ0+∫0nκ1Θ2-κ2Θ22-η1Θ2x+η2Δx(Θ2)dτ,

subtract the Eq. (10) from the Eq. (9) we have,MΘ1-MΘ2=∫0nκ1Θ1-κ2Θ12-η1Θ1x+η2Δx(Θ1)dτ-∫0nκ1Θ2-κ2Θ22-η1Θ2x+η2Δx(Θ2)dτ,MΘ1-MΘ2=∫0nκ1(Θ1-Θ2)+κ2(Θ22-Θ12)+η1(Θ2x-Θ1x)+η2(Δx(Θ1)-Δx(Θ2))dτ,

apply norm on both sides||MΘ1-MΘ2||≤∫0n||κ1(Θ1-Θ2)+κ2(Θ22-Θ12)+η1(Θ2x-Θ1x)+η2(Δx(Θ1)-Δx(Θ2))||dτ,

11 ||MΘ1-MΘ2||≤∫0nG(τ)dτ,

whereG(τ)=||κ1(Θ1-Θ2)+κ2(Θ22-Θ12)+η1(Θ2x-Θ1x)+η2(Δx(Θ1)-Δx(Θ2))||,G(τ)≤|κ1|·.||Θ1-Θ2||+|κ2|;||Θ22-Θ12||+|η1|;||Θ2x-Θ1x||+|η2|;||Δx(Θ1)-Δx(Θ2))||,

again apply single and double embedding on the terms involving partial derivatives we getG(τ)≤|κ1|;||Θ1-Θ2||+|κ2|;||Θ22-Θ12||+η1′||Θ2-Θ1||+η2′||Θ1-Θ2||,

by mean value theorem we haveG(τ)≤|κ1|;||Θ1-Θ2||+|κ2|;||Θ1-Θ2||2c2-1+η1′||Θ1-Θ2||+η2′||Θ1-Θ2||,

so,G(τ)≤|κ1|+|κ2|2c2-1+η1′+η2′||Θ1-Θ2||,

from Eq. (11) we get the final form as follows12 ||MΘ1-MΘ2||≤L|κ1|+|κ2|2c2-1+η1′+η2′||Θ1-Θ2||,

since for contraction mapping, lipschitz constant is always less than one (see Definition 3). Here Eq. (12) fulfill lipschitz condition so that the lipschitz constant in Eq. (12) as followsL|κ1|+|κ2|2c2-1+η1′+η2′<1,

which gives the final condition13 LM<1|κ1|+|κ2|2c2-1+η1′+η2′.

Hence, under the inequality (13) operator (4) mapped as a contraction mapping. □

Key points for new extended direct algebraic method

It is a direct and convincing procedure to find solitons and different wave results of propagation. This method applied by numerous researchers to earn the soliton results for a number of nonlinear partial differential equation. The underlying system includes such procedure35.

Step-1: Consider the NPDE such as14 Γ(Θ,Θt,Θx,Θtt,Θxt,Θxx,...)=0,

here Θ=Θ(x,t) be an unknown function, Γ be a polynomial with Θ(x,t) and their derivatives with respect to time and space variable.

Step-2: Put a compound variable ρ against the combination of both real variables t and x as follows,15 Θ(x,t)=Φ(ρ),ρ=lx-mt,

here l is a number of wave propagation and m describe the speed of the waves propagation. Using this traveling wave transformation as Eq. (15) to Eq. (14), reduce it as an ordinary differential equation, such that16 Ψ(Φ,Φ′,Φ″,Φ″′,...)=0,

here Ψ known as a new polynomial of Φ along with the derivatives of Φ.

Step-3: Suppose that the general solution by the following polynomial17 Φ(ρ)=∑j=0Nαjg(ρ)j,

here αj,(j=0,1,2,3,......N) are the constants to be find and g(ρ) is a function which satisfy the following ordinary differential equation18 g′(ρ)=log(b)a1+a2g(ρ)+a3g(ρ)2,b≠0,1,

then the Eq. (14) secure the following solutions.

Set-1: For a22-4a1a3<0, and a3≠0,19 g1(ρ)=-a22a3+-(a22-4a1a3)2a3tanb-(a22-4a1a3)2ρ,

20 g2(ρ)=-a22a3--(a22-4a1a3)2a3cotb-(a22-4a1a3)2ρ,

21 g3(ρ)=-a22a3+-(a22-4a1a3)2a3tanb-(a22-4a1a3)ρ±pqsecb-(a22-4a1a3)ρ,

22 g4(ρ)=-a22a3--(a22-4a1a3)2a3cotb-(a22-4a1a3)ρ±pqcscb-(a22-4a1a3)ρ,

23 g5(ρ)=-a22a3--(a22-4a1a3)4a3tanb-(a22-4a1a3)4ρ-cotb-(a22-4a1a3)4ρ.

Set-2: For a22-4a1a3>0, and a3≠0,24 g6(ρ)=-a22a3-(a22-4a1a3)2a3tanhb(a22-4a1a3)2ρ,

25 g7(ρ)=-a22a3-(a22-4a1a3)2a3cothb(a22-4a1a3)2ρ,

26 g8(ρ)=-a22a3-(a22-4a1a3)2a3tanhb(a22-4a1a3)ρ±ιpqsechb(a22-4a1a3)ρ,

27 g9(ρ)=-a22a3--(a22-4a1a3)2a3cothb(a22-4a1a3)ρ±pqcschb(a22-4a1a3)ρ,

28 g10(ρ)=-a22a3+-(a22-4a1a3)4a3tanhb(a22-4a1a3)4ρ-cothb(a22-4a1a3)4ρ.

Set-3: For a1a3>0, and a2=0,29 g11(ρ)=a1a3tanba1a3ρ,

30 g12(ρ)=-a1a3cotba1a3ρ,

31 g13(ρ)=a1a3tanb2a1a3ρ±pqsecb2a1a3ρ,

32 g14(ρ)=-a1a3cotb2a1a3ρ±pqcscb2a1a3ρ,

33 g15(ρ)=12a1a3tanba1a32ρ-cotba1a32ρ.

Set-4: For a1a3<0, and a2=0,34 g16(ρ)=--a1a3tanhb-a1a3ρ,

35 g17(ρ)=--a1a3cothb-a1a3ρ,

36 g18(ρ)=--a1a3tanhb2-a1a3ρ±ιsqrt-pqsechb2-a1a3ρ,

37 g19(ρ)=--a1a3cothb2-a1a3ρ±-pqcschb2-a1a3ρ,

38 g20(ρ)=-12-a1a3tanhb-a1a32ρ-cothb-a1a32ρ.

Set-5: For a2=0 and a1=a3,39 g21(ρ)=tanba1ρ,

40 g22(ρ)=-cotba1ρ,

41 g23(ρ)=tanb2a1ρ±pqsecb2a1ρ,

42 g24(ρ)=-cotb2a1ρ±pqcscb2a1ρ,

43 g25(ρ)=12tanba12ρ-cotba12ρ.

Set-6: For a2=0 and a1=a3,44 g26(ρ)=-tanba1ρ,

45 g27(ρ)=-cotba1ρ,

46 g28(ρ)=-tanb2a1ρ±pqsecb2a1ρ,

47 g29(ρ)=-cotb2a1ρ±pqcscb2a1ρ,

48 g30(ρ)=-12tanba12ρ+cotba12ρ.

Set-7: For a2=4a1a3, a single solution49 g31(ρ)=--2a1(a2ρln(b)+2)a22ln(b).

Set-8: For a2=χ,a1=rχ(r≠0) and a3=0 a single solution50 g32(ρ)=bχρ-r.

Set-9: For a2=a3=0 a single solution51 g33(ρ)=a1ρln(b).

Set-10: For a2=a1=0 a single solution52 g34(ρ)=-1a3ρln(b).

Set-11: For a2≠0 and a1=0 double solutions53 g35(ρ)=a2pa3(coshb(a2ρ)-sinh(a2ρ)+p),

54 g36(ρ)=a2(coshb(a2ρ)+sinh(a2ρ))a3(coshb(a2ρ)+sinh(a2ρ)+q).

Set-12: For a2=χ,a3=rχ(r≠0) and a1=0 a single solution55 g32(ρ)=pbχρp-rqbχρ.

In these sets, trigonometric and generalized hyperbolic functions defined as follow35,sinhb(ρ)=pbρ-qb-ρ2,coshb(ρ)=pbρ+qb-ρ2,tanhb(ρ)=pbρ-qb-ρpbρ+qb-ρ,cothb(ρ)=pbρ+qb-ρpbρ-qb-ρ,sechb(ρ)=2pbρ+qb-ρ,cschb(ρ)=2pbρ-qb-ρ,sinb(ιρ)=pbιρ-qb-ιρ2,cosb(ιρ)=pbιρ+qb-ιρ2,tanb(ιρ)=-ιpbιρ-qb-ιρpbιρ+qb-ιρ,ιcotb(ιρ)=pbιρ+qb-ιρpbιρ-qb-ιρ,secb(ιρ)=2pbιρ+qb-ιρ,cscb(ιρ)=2pbιρ-qb-ιρ,

here p, q are always positive and ρ is independent.

Step 4: Evaluate N for (17) from balancing between higher derivative and higher order nonlinear term of the nonlinear equation (16).

Step 5: Substituting the Eq. (17) along with its all relevant derivatives to the Eq. (16) and putting its coefficients equal to zero with the same exponent number of g(ρ), easily obtained the related strategic equations and by solving these strategic equations using a computer software we achieve the required wave results.

Analytical application

In this section, various exact and solitary wave results of an advection-diffusion-reaction partial differential equation are obtained in different kind of shapes like hyperbolic in nature, rational type, trigonometric function, and exponential and we secure some mixed or combined results and singular periodic wave results with different unknown parameters with the use of a sound computational integration approach known as a new extended direct algebraic method54,55. The outputs described that the governing equation theoretically possesses an extremely rich model of exact solitary wave results. Consider the nonlinear PDEs, Eq. (2) and apply traveling wave transformationΘ(x,t)=Φ(ρ),∵ρ=lx-mt,

where m is a velocity of traveling wave in Eq. (2). On calculating and applying 2-times integration by taking integration constant is zero. we get56 η2l2Φ″+(m-η1l)Φ′+κ1Φ-κ2Φ2=0,

applying the balance principle homogenous rule in this achieved an ordinary differential equation (56) we get N = 2, and corresponding to N = 2, the proposed method has the results to Eq. (56) as follow:57 Φ(ρ)=α0+α1g(ρ)+α2g(ρ)2,

where,g′(ρ)=log(b)a1+a2g(ρ)+a3g(ρ)2,

putting the Eq. (57) in the Eq. (56) and taking coefficient of same powers of Θ(ρ) equal to zero, the strategic required equations are obtained easily. After some computations on this resultant equation with the use of computer software, we finally summarize the following achieved solutions sets of hyperbolic, logarithm, trigonometric, mixed, and some generalized functions.

Set-1

For a22-4a1a3<0, and a3≠0, there are 5-mixed trigonometric results for 2 and these results expressed the combined trigonometric behavior of solitons in Eq. (2) as orderedΘ1=α1a22α12κ2-a32κ1α12κ2-a22TanB12a22α12κ2-a32κ1α12κ2-a22lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)2a3-α1a22a3+a2α1κ2+a3κ12a3κ2,Θ2=-α1a22α12κ2-a32κ1α12κ2-a22CotB12a22α12κ2-a32κ1α12κ2-a22lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)2a3-α1a22a3+a2α1κ2+a3κ12a3κ2,

Θ3=a22α12κ2-a32κ1α12κ2-a22TanBa22α12κ2-a32κ1α12κ2-a22lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)α1±pqCscBa22α12κ2-a32κ1α12κ2-a22lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)2a3-a22a3+a2α1κ2+a3κ12a3κ2,Θ4=-a22α12κ2-a32κ1α12κ2-a22CotBa22α12κ2-a32κ1α12κ2-a22lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)α1±pqCscBa22α12κ2-a32κ1α12κ2-a22lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)2a3-a22a3+a2α1κ2+a3κ12a3κ2,

Θ5=TanB14a22α12κ2-a32κ1α12κ2-a22lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)-CotB14a22α12κ2-a32κ1α12κ2-a22lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)4a3α1a22α12κ2-a32κ1α12κ2-a22-a22a3+a2α1κ2+a3κ12a3κ2.

Set-2 For a22-4a1a3>0, and a3≠0, many soliton solutions for Eq. (2) are obtained, the dark solutions to the Eq. (2) is described as follow: The dark soliton solution is shown asΘ6=-α1a22-a22α12κ2-a32κ1α12κ2TanhB12a22-a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)2a3-α1a22a3+a2α1κ2+a3κ12a3κ2.

The singular soliton solution is shown asΘ7=-α1a22-a22α12κ2-a32κ1α12κ2CothB12a22-a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)2a3-α1a22a3+a2α1κ2+a3κ12a3κ2.

The complex dark-bright soliton solution is given asΘ8=-α1a22-a22α12κ2-a32κ1α12κ2TanBa22-a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)2a3±ipqSecBa22-a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)-α1a22a3+a2α1κ2+a3κ12a3κ2.

The mixed singular soliton solution is shown asΘ9=-α1a22α12κ2-a32κ1α12κ2-a22CotBa22-a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)2a3±pqCscBa22-a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)-α1a22a3+a2α1κ2+a3κ12a3κ2.

The dark-singular soliton solution is attained asΘ10=-α1a22α12κ2-a32κ1α12κ2-a22CotB14a22-a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)4a3+TanB14a22-a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)-α1a22a3+a2α1κ2+a3κ12a3κ2.

Set-3

For a1a3>0 and a2=0, we get trigonometric results as follows.Θ11=TanB12a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)12α1a22α12κ2-a32κ1a32α12κ2+a2α1κ2+a3κ12a3κ2,Θ12=-CotB12a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)12α1a22α12κ2-a32κ1a32α12κ2+a2α1κ2+a3κ12a3κ2,

now the mixed-trigonometric soliton results are gained asΘ13=TanBa22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)±pqSecBa22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)12α1a22α12κ2-a32κ1a32α12κ2+a2α1κ2+a3κ12a3κ2,Θ14=-CotBa22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)±pqCscBa22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)12α1a22α12κ2-a32κ1a32α12κ2+a2α1κ2+a3κ12a3κ2,Θ15=TanB14a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)-CotB14a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)14α1a22α12κ2-a32κ1a32α12κ2+a2α1κ2+a3κ12a3κ2..

Set-4

For a1a3<0 and a2=0, we have the dark solutions asΘ16=-TanB12-a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)12α1-a22α12κ2-a32κ1a32α12κ2+a2α1κ2+a3κ12a3κ2.

The singular soliton solution is derived asΘ17=-CotB12-a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)12α1-a22α12κ2-a32κ1a32α12κ2+a2α1κ2+a3κ12a3κ2.

The different complex combo soliton results are gained asΘ18=-TanB-a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)±ipqSecB-a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)12α1-a22α12κ2-a32κ1a32α12κ2+a2α1κ2+a3κ12a3κ2.Θ19=-CotB-a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)±pqCscB-a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)12α1-a22α12κ2-a32κ1a32α12κ2+a2α1κ2+a3κ12a3κ2,Θ20=-TanB14-a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)-CotB14-a22α12κ2-a32κ1α12κ2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)14α1-a22α12κ2-a32κ1a32α12κ2+a2α1κ2+a3κ12a3κ2.

Set-5

For a2=0 and a1=a3, the periodic and mixed periodic solutions are found asΘ21=α1TanBa22α12κ2-a32κ1lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)4a3α12κ2+a2α1κ2+a3κ12a3κ2,Θ22=-α1CotBa22α12κ2-a32κ1lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)4a3α12κ2+a2α1κ2+a3κ12a3κ2,Θ23=a2α1κ2+a3κ12a3κ2+α1TanBa22α12κ2-a32κ1lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)2a3α12κ2±pqSecBa22α12κ2-a32κ1lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)2a3α12κ2,Θ24=a2α1κ2+a3κ12a3κ2+α1-CotBa22α12κ2-a32κ1lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)2a3α12κ2±pqCscBa22α12κ2-a32κ1lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)2a3α12κ2,Θ25=a2α1κ2+a3κ12a3κ2+12α1TanBa22α12κ2-a32κ1lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)8a3α12κ2-CotBa22α12κ2-a32κ1lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)8a3α12κ2.

Set-6

For a2=0 and a1=-a3, we acquire different types of soliton results.Θ26=a2α1κ2+a3κ12a3κ2-α1TanhBa22α12κ2-a32κ1lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)4a3α12κ2,Θ27=a2α1κ2+a3κ12a3κ2-α1CothBa22α12κ2-a32κ1lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)4a3α12κ2,Θ28=a2α1κ2+a3κ12a3κ2+α1-TanhBa22α12κ2-a32κ1lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)2a3α12κ2±ipqSechBa22α12κ2-a32κ1lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)2a3α12κ2,Θ29=a2α1κ2+a3κ12a3κ2+α1-CothBa22α12κ2-a32κ1lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)2a3α12κ2±pqCschBa22α12κ2-a32κ1lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)2a3α12κ2,Θ30=a2α1κ2+a3κ12a3κ2+12α1-CothBa22α12κ2-a32κ1lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)8a3α12κ2-TanhBa22α12κ2-a32κ1lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)8a3α12κ2.

Set-7

For a22=4a1a3, there is one soliton resultΘ31=a2α1κ2+a3κ12a3κ2-a22α12κ2-a32κ1a2log(B)lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)+22a22a3α1κ2log(B)lx-t2a3η1llog(b)+3α1κ1κ22a3log(b).

Set-8

For a2=χ,a1=rχ(r≠0) and a3=0, there is only one soliton result,Θ32=a2α1κ2+a3κ12a3κ2+α1Bχlx-t2a3η1llog(b)+3α1κ1κ22a3log(b)-r.

Set-9

For a2=a3=0, there is also one soliton result,Θ33=a2α1κ2+a3κ12a3κ2+log(B)a22α12κ2-a32κ1lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)4a3α1κ2.

Set-10

For a2=a1=0, there is also one soliton solution,Θ34=a2α1κ2+a3κ12a3κ2-α1a3log(B)lx-t2a3η1llog(b)+3α1κ1κ22a3log(b).

Set-11

For a1=0 and a2≠0, there are two different soliton solutions,Θ35=a2α1κ2+a3κ12a3κ2-a2α1pa3CoshBa2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)1-SinhBa2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)+p,

Θ36=a2α1-CoshBa2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)-SinhBa2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)a3CoshBa2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)+SinhBa2lx-t2a3η1llog(b)+3α1κ1κ22a3log(b)+p+a2α1κ2+a3κ12a3κ2.

Set-12

For a2=χ,a3=rχ(r≠0) and a1=0, there is also only one soliton solution,Θ37=a2α1κ2+a3κ12a3κ2+α1pBχlx-t2a3η1llog(b)+3α1κ1κ22a3log(b)p-qrBχlx-t2a3η1llog(b)+3α1κ1κ22a3log(b).

Graphical analysis

In this portion, the graphical explanation and thorough analysis are described to show the novelty of the underlying article. In this connection, the recovered results have been highlighted in the given different figures by choosing the appropriate numeric values of the given parameters. Figures 1, 2 and 3 show the trigonometric behavior of solitons. Figures 4 and 5 describe the dark trigonometric solitons. Figure 6 highlights the dark behavior of hyperbolic solitons. Figures 7, 8, 9 and 10 shows the singular nature of hyperbolic solitons. Figures 11, 12 and 13 represents the complex dark-bright view of trigonometrical solitons. Figures 14, 15, 16 and 17 show the mixed-singular structure of solitons. Figures 18, 19, 20, 21 and 22 represents the graphical image view of the trigonometric nature of solitons. Mixed trigonometric view of Θ described by Figs. 23, 24 and 25. Figures 26 and 27 show complex mixed trigonometric behavior of Θ in different views.Figure 1 Three dimensional plot of Θ1(x,t) for the different suitable values of parameters and coefficients as, α1=8.3,a1=0.3,α2=1.1,a2=0.9,a3=1.44,b=10,η1=4.3,η2=0.5,κ1=2,κ2=9.1,l=0.1.

Figure 2 Three dimensional plot of Θ2(x,t) for the different suitable values of parameters and coefficients as, α1=0.9,a1=0.5,α2=1.1,a2=0.9,a3=1.44,b=10,η1=4.3,η2=0.5,κ1=2,κ2=0.01,l=0.1.

Figure 3 Three dimensional plot of Θ3(x,t) for the different suitable values of parameters and coefficients as, α1=0.3,a1=0.5,α2=0.1,a2=0.9,a3=1.44,b=10,η1=4.3,η2=0.5,κ1=2,κ2=9.1,l=0.1.

Figure 4 Three dimensional plot of Θ4(x,t) for the different suitable values of parameters and coefficients as, α1=0.9,a1=0.5,α2=0.1,a2=1.9,a3=1.44,b=4.1,η1=1.3,η2=1.5,κ1=1,κ2=1.1,l=2.1.

Figure 5 Three dimensional plot of Θ5(x,t) for the different suitable values of parameters and coefficients as, α1=2.3,a1=0.5,α2=0.1,a2=2.9,a3=2.44,b=2.1,η1=2.3,η2=2.5,κ1=1,κ2=2.1,l=2.1.

Figure 6 Three dimensional plot of Θ6(x,t) for the different suitable values of parameters and coefficients as, α1=1.3,a1=-0.5,α2=2.1,a2=2.9,a3=0.44,b=1.9,η1=3.3,η2=1.5,κ1=1,κ2=0.1,l=2.1.

Figure 7 Three dimensional plot of Θ11(x,t) for the different suitable values of parameters and coefficients as, α1=2.3,a1=3.5,α2=0.1,a2=1.9,a3=2.44,b=2.1,η1=2.3,η2=0.5,κ1=3,κ2=2.1,l=0.1.

Figure 8 Three dimensional plot of Θ12(x,t) for the different suitable values of parameters and coefficients as, α1=2.3,a1=3.5,α2=2.1,a2=1.9,a3=2.44,b=2.,η1=2.3,η2=1.5,κ1=2.1,κ2=0.1,l=0.1.

Figure 9 Three dimensional plot of Θ13(x,t) for the different suitable values of parameters and coefficients as, α1=0.3,a1=0.5,α2=1.1,a2=1.9,a3=2.44,b=2.,η1=2.3,η2=1.5,κ1=3,κ2=0.5,l=-4.1.

Figure 10 Three dimensional plot of Θ14(x,t) for the different suitable values of parameters and coefficients as, α1=0.3,a1=0.5,α2=2.1,a2=1.9,a3=2.44,b=3.,η1=0.3,η2=1.5,κ1=3,κ2=0.1,l=0.1.

Figure 11 Three dimensional plot of Θ15(x,t) for the different suitable values of parameters and coefficients as, α1=0.9,a1=0.5,α2=2.1,a2=1.9,a3=2.44,b=3.,η1=0.3,η2=1.5,κ1=3,κ2=0.1,l=0.1.

Figure 12 Three dimensional plot of Θ21(x,t) for the different suitable values of parameters and coefficients as, α1=1.3,a1=3.5,α2=2.1,a2=1.9,a3=4.44,b=3.,η1=3.3,η2=1.5,κ1=3,κ2=2.1,l=-1.1.

Figure 13 Three dimensional plot of Θ22(x,t) for the different suitable values of parameters and coefficients as, α1=1.3,a1=3.5,α2=2.1,a2=1.9,a3=4.44,b=3.,η1=3.3,η2=1.5,κ1=3,κ2=2.1,l=-1.1.

Figure 14 Three dimensional plot of Θ23(x,t) for the different suitable values of parameters and coefficients as, α1=1.3,a1=3.5,α2=3.1,a2=1.9,a3=3.44,b=3.,η1=2.3,η2=1.5,κ1=3,κ2=2.1,l=-1.5.

Figure 15 Three dimensional plot of Θ24(x,t) for the different suitable values of parameters and coefficients as, α1=1.3,a1=1.5,α2=2.1,a2=1.9,a3=4.44,b=2.,η1=1.3,η2=1.5,κ1=3,κ2=2.1,l=-2.1.

Figure 16 Three dimensional plot of Θ25(x,t) for the different suitable values of parameters and coefficients as, α1=0.34,a1=1.5,α2=2.1,a2=1.9,a3=1.44,b=2.,η1=1.3,η2=1.5,κ1=3,κ2=2.1,l=-0.5.

Figure 17 Three dimensional plot of Θ26(x,t) for the different suitable values of parameters and coefficients as, α1=0.3,a1=0.5,α2=1.1,a2=2.9,a3=0.2,b=1.5,η1=0.3,η2=1.5,κ1=1,κ2=1.1,l=1.1.

Figure 18 Three dimensional plot of Θ27(x,t) for the different suitable values of parameters and coefficients as, α1=0.3,a1=0.5,α2=1.1,a2=2.9,a3=1.2,b=8.,η1=0.3,η2=1.5,κ1=1,κ2=1.1,l=1.1.

Figure 19 Three dimensional plot of Θ28(x,t) for the different suitable values of parameters and coefficients as, α1=0.3,a1=0.5,α2=1.1,a2=2.9,a3=1.2,b=8.,η1=0.3,η2=1.5,κ1=1,κ2=1.1,l=1.1.

Figure 20 Three dimensional plot of Θ30(x,t) for the different suitable values of parameters and coefficients as, α1=0.3,a1=0.5,α2=1.1,a2=2.9,a3=1.2,b=8.,η1=0.3,η2=1.5,κ1=1,κ2=1.1,l=1.1.

Figure 21 Three dimensional plot of Θ31(x,t) for the different suitable values of parameters and coefficients as, α1=1.3,a1=1.5,α2=2.1,a2=1.9,a3=1.44,b=2.,η1=1.3,η2=1.5,κ1=3,κ2=2.1,l=1.1.

Figure 22 Three dimensional plot of Θ32(x,t) for the different suitable values of parameters and coefficients as, α1=1.3,a1=1.5,α2=0.1,a2=1.9,a3=1.44,b=2.,η1=1.3,η2=0.5,κ1=3,κ2=2.1,l=-2.1,r=1.2,χ=1.5.

Figure 23 Three dimensional plot of Θ33(x,t) for the different suitable values of parameters and coefficients as, α1=0.3,a1=1.5,α2=0.1,a2=1.9,a3=0.44,b=2.,η1=1.3,η2=0.5,κ1=3,κ2=0.1,l=-0.1,r=0.2,χ=0.5.

Figure 24 Three dimensional plot of Θ34(x,t) for the different suitable values of parameters and coefficients as, α1=0.3,a1=1.5,α2=0.1,a2=1.9,a3=1.44,b=2.,η1=1.3,η2=0.5,κ1=3,κ2=1.1,l=-2.1,r=1.2,χ=1.5.

Figure 25 Three dimensional plot of Θ35(x,t) for the different suitable values of parameters and coefficients as, α1=0.3,a1=0.5,α2=1.1,a2=0.9,a3=1.44,b=2.8,η1=0.3,η2=0.5,κ1=3,κ2=0.01,l=-5.8,r=0.2,χ=2.5.

Figure 26 Three dimensional plot of Θ36(x,t) for the different suitable values of parameters and coefficients as, α1=1.3,a1=0.5,α2=1.1,a2=0.9,a3=1.44,b=3.,η1=0.3,η2=1.5,κ1=1,κ2=5.1,l=-0.1,r=0.2,χ=2.5.

Figure 27 Three dimensional plot of Θ37(x,t) for the different suitable values of parameters and coefficients as, α1=1.3,a1=0.5,α2=1.1,a2=0.9,a3=1.44,b=3.,B=2.5,η1=0.3,η2=1.5,κ1=1,κ2=1.1,l=-2.1,p=5.2,q=7.01,r=1,χ=0.005.

Conclusion

In this article, traveling wave results and the unique existence of the solutions are analyzed for the proposed advection–diffusion–reaction-equation. The unique results are investigated within the closed and convex optimal subset of the Banach space of all continuous functions using the Banach fixed point theorem (contraction mapping principle) following the Lipschitz condition. This optimal analysis describes the conditions in which the unique results exist. Traveling wave solutions are obtained using a new extended direct algebraic analytical technique. These solutions are in the form of different trigonometric, hyperbolic, rational, periodic, exponential, and some mixed trigonometric functions. These solutions are observed as combined bright-dark, bright, and dark soliton solutions. These earned exact soliton solutions are a new type of outcome in the latest literature. The main accomplishment of the underlying analytical method exists in the direction that, we must have succeeded in one propagation to obtain maximum results which shows it separate from others. Furthermore, with the different values of the parameters, the physical propagations of some earned results are described via drawing different two and three-dimensional graphical plots. These analytical solutions play a wonderful role in the area of scientific research on physical problems. We have explored the unique existence, and wave properties of advection–diffusion equations. Notably, our investigation has led to the discovery of soliton solutions, which represents a significant advancement in understanding the dynamics of transport phenomena in the context of advection–diffusion systems. The soliton solutions offer valuable insights into the behavior of advection-diffusion equations. This finding holds promise for applications in diverse fields. Moving forward, further analysis and exploration of the soliton solutions, along with investigation into its stability, interaction properties, and potential extensions to multidimensional systems, represent promising avenues for future research in this area.

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References

1. Katzourakis, N. An Introduction to Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in (Springer, 2014).
2. Showalter, R. E. Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Vol. 49. (American Mathematical Society, 2013).
3. Logan, J. D. An Introduction to Nonlinear Partial Differential Equations. Vol. 89. (Wiley, 2008).
4. Helal MA Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics Chaos Solitons Fractals 2002 13 9 1917 1929 10.1016/S0960-0779(01)00189-8
Helal, M. A. Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics. Chaos Solitons Fractals 13(9), 1917–1929 (2002).10.1016/S0960-0779(01)00189-8
5. Ambrosio L Caffarelli L Crandall MG Evans LC Fusco N Calculus of Variations and Nonlinear Partial Differential Equations: With a Historical Overview by Elvira Mascolo 2008 Springer
Ambrosio, L., Caffarelli, L., Crandall, M. G., Evans, L. C. & Fusco, N. Calculus of Variations and Nonlinear Partial Differential Equations: With a Historical Overview by Elvira Mascolo (Springer, 2008).
6. Thomée, V. From finite differences to finite elements A short history of numerical analysis of partial differential equations. In Numerical Analysis: Historical Developments in the 20th Century. 361–414. (Elsevier, 2001).
7. Davis, H. T. Introduction to Nonlinear Differential and Integral Equations. (US Atomic Energy Commission, 1960).
8. Roubícek, T. Nonlinear Partial Differential Equations with Applications. Vol. 153. (Springer, 2013).
9. Girma T., Chala M. I. N., Ma’arof Fiseha M., Guangul Shaharin A., Sulaiman Clean Energy Opportunities in Tropical Countries Tidal and Wave Energy Potential Assessment 217–236. 10.1007/978-981-15-9140-2 (Springer Singapore, Singapore, 2021).
10. Helal MA Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics Chaos Solitons Fractals 2002 13 9 1917 1929 10.1016/S0960-0779(01)00189-8
Helal, M. A. Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics. Chaos Solitons Fractals 13(9), 1917–1929 (2002).10.1016/S0960-0779(01)00189-8
11. Samir I Abd-Elmonem A Ahmed HM General solitons for eighth-order dispersive nonlinear Schrödinger equation with ninth-power law nonlinearity using improved modified extended tanh method Opt. Quantum Electron. 2023 55 5 470 10.1007/s11082-023-04753-5
Samir, I., Abd-Elmonem, A. & Ahmed, H. M. General solitons for eighth-order dispersive nonlinear Schrödinger equation with ninth-power law nonlinearity using improved modified extended tanh method. Opt. Quantum Electron. 55(5), 470 (2023).10.1007/s11082-023-04753-5
12. Rezazadeh H Korkmaz A Achab AE Adel W Bekir A New travelling wave solution-based new Riccati Equation for solving KdV and modified KdV Equations Appl. Math. Nonlinear Sci. 2021 6 1 447 458 10.2478/amns.2020.2.00034
Rezazadeh, H., Korkmaz, A., Achab, A. E., Adel, W. & Bekir, A. New travelling wave solution-based new Riccati Equation for solving KdV and modified KdV Equations. Appl. Math. Nonlinear Sci. 6(1), 447–458 (2021).10.2478/amns.2020.2.00034
13. Yokus A Durur H Nofal TA Abu-Zinadah H Tuz M Ahmad H Study on the applications of two analytical methods for the construction of traveling wave solutions of the modified equal width equation Open Phys. 2020 18 1 1003 1010 10.1515/phys-2020-0207
Yokus, A. et al. Study on the applications of two analytical methods for the construction of traveling wave solutions of the modified equal width equation. Open Phys. 18(1), 1003–1010 (2020).10.1515/phys-2020-0207
14. Vahidi J Zabihi A Rezazadeh H Ansari R New extended direct algebraic method for the resonant nonlinear Schrödinger equation with Kerr law nonlinearity Optik 2021 227 165936 10.1016/j.ijleo.2020.165936
Vahidi, J., Zabihi, A., Rezazadeh, H. & Ansari, R. New extended direct algebraic method for the resonant nonlinear Schrödinger equation with Kerr law nonlinearity. Optik 227, 165936 (2021).10.1016/j.ijleo.2020.165936
15. Hauke G Garcia-Olivares A Variational subgrid scale formulations for the advection–diffusion–reaction equation Comput. Methods Appl. Mech. Eng. 2001 190 51–52 6847 6865 10.1016/S0045-7825(01)00262-6
Hauke, G. & Garcia-Olivares, A. Variational subgrid scale formulations for the advection–diffusion–reaction equation. Comput. Methods Appl. Mech. Eng. 190(51–52), 6847–6865 (2001).10.1016/S0045-7825(01)00262-6
16. Hauke G A simple subgrid scale stabilized method for the advection–diffusion–reaction equation Comput. Methods Appl. Mech. Eng. 2002 191 27–28 2925 2947 10.1016/S0045-7825(02)00217-7
Hauke, G. A simple subgrid scale stabilized method for the advection–diffusion–reaction equation. Comput. Methods Appl. Mech. Eng. 191(27–28), 2925–2947 (2002).10.1016/S0045-7825(02)00217-7
17. Hundsdorfer, W. H., Verwer, J. G., & Hundsdorfer, W. H. Numerical Solution of Time-Dependent Advection–Diffusion–Reaction Equations. Vol. 33. x+-471. (Springer, 2003).
18. Spiegelman, M., & Katz, R. F. A semi-Lagrangian Crank–Nicolson algorithm for the numerical solution of advection–diffusion problems. Geochem. Geophys. Geosyst. 7(4) (2006).
19. ten Thije Boonkkamp JHM Anthonissen MJH The finite volume-complete flux scheme for advection–diffusion–reaction equations J. Sci. Comput. 2011 46 47 70 10.1007/s10915-010-9388-8
ten Thije Boonkkamp, J. H. M. & Anthonissen, M. J. H. The finite volume-complete flux scheme for advection–diffusion–reaction equations. J. Sci. Comput. 46, 47–70 (2011).10.1007/s10915-010-9388-8
20. Kaya B Gharehbaghi A Implicit solutions of advection diffusion equation by various numerical methods Aust. J. Basic Appl. Sci. 2014 8 1 381 391
Kaya, B. & Gharehbaghi, A. Implicit solutions of advection diffusion equation by various numerical methods. Aust. J. Basic Appl. Sci. 8(1), 381–391 (2014).
21. Mirza IA Vieru D Fundamental solutions to advection–diffusion equation with time-fractional Caputo–Fabrizio derivative Comput. Math. Appl. 2017 73 1 1 10 10.1016/j.camwa.2016.09.026
Mirza, I. A. & Vieru, D. Fundamental solutions to advection–diffusion equation with time-fractional Caputo–Fabrizio derivative. Comput. Math. Appl. 73(1), 1–10 (2017).10.1016/j.camwa.2016.09.026
22. Jannelli A Ruggieri M Speciale MP Analytical and numerical solutions of time and space fractional advection–diffusion–reaction equation Commun. Nonlinear Sci. Numer. Simul. 2019 70 89 101 10.1016/j.cnsns.2018.10.012
Jannelli, A., Ruggieri, M. & Speciale, M. P. Analytical and numerical solutions of time and space fractional advection–diffusion–reaction equation. Commun. Nonlinear Sci. Numer. Simul. 70, 89–101 (2019).10.1016/j.cnsns.2018.10.012
23. Singh A Das S Ong SH Jafari H Numerical solution of nonlinear reaction–advection–diffusion equation J. Comput. Nonlinear Dyn. 2019 14 4 041003 10.1115/1.4042687
Singh, A., Das, S., Ong, S. H. & Jafari, H. Numerical solution of nonlinear reaction–advection–diffusion equation. J. Comput. Nonlinear Dyn. 14(4), 041003 (2019).10.1115/1.4042687
24. Savovic, S., Drljaca, B., & Djordjevich, A. A comparative study of two different finite difference methods for solving advection–diffusion reaction equation for modeling exponential traveling wave in heat and mass transfer processes. Ricerche Mat. 1–8 (2022).
25. Chang, S. S. Fixed Point Theory and Application (1984).
26. Agarwal, R. P., O’Regan, D., & Sahu, D. R. Fixed Point Theory for Lipschitzian-Type Mappings with Applications. Vol. 6. x+-368 (Springer, 2009).
27. Anley EF Basha M Hussain A Dai B Numerical simulation for nonlinear space-fractional reaction convection–diffusion equation with its application Alex. Eng. J. 2023 65 245 261 10.1016/j.aej.2022.10.047
Anley, E. F., Basha, M., Hussain, A. & Dai, B. Numerical simulation for nonlinear space-fractional reaction convection–diffusion equation with its application. Alex. Eng. J. 65, 245–261 (2023).10.1016/j.aej.2022.10.047
28. Ignat LI Rossi JD A nonlocal convection–diffusion equation J. Funct. Anal. 2007 251 2 399 437 10.1016/j.jfa.2007.07.013
Ignat, L. I. & Rossi, J. D. A nonlocal convection–diffusion equation. J. Funct. Anal. 251(2), 399–437 (2007).10.1016/j.jfa.2007.07.013
29. Haque M Existence of weak solutions to a convection–diffusion equation in amalgam spaces J. Egypt. Math. Soc. 2022 30 1 1 19 10.1186/s42787-022-00156-9
Haque, M. Existence of weak solutions to a convection–diffusion equation in amalgam spaces. J. Egypt. Math. Soc. 30(1), 1–19 (2022).10.1186/s42787-022-00156-9
30. Iqbal, M. S., Ahmed, N., Akgül, A., Raza, A., Shahzad, M., Iqbal, Z., & Jarad, F. Analysis of the fractional diarrhea model with Mittag–Leffler kernel. AIMS Math. 7, 13000–13018 (2022).
31. Abbasbandy S Kazem S Alhuthali MS Alsulami HH Application of the operational matrix of fractional-order Legendre functions for solving the time-fractional convection–diffusion equation Appl. Math. Comput. 2015 266 31 40
Abbasbandy, S., Kazem, S., Alhuthali, M. S. & Alsulami, H. H. Application of the operational matrix of fractional-order Legendre functions for solving the time-fractional convection–diffusion equation. Appl. Math. Comput. 266, 31–40 (2015).
32. Gill WN Sankarasubramanian R Exact analysis of unsteady convective diffusion Proc. R. Soc. Lond. A. Math. Phys. Sci. 1970 316 1526 341 350 10.1098/rspa.1970.0083
Gill, W. N. & Sankarasubramanian, R. Exact analysis of unsteady convective diffusion. Proc. R. Soc. Lond. A. Math. Phys. Sci. 316(1526), 341–350 (1970).10.1098/rspa.1970.0083
33. Kalita JC Dalal DC Dass AK A class of higher order compact schemes for the unsteady two-dimensional convection-diffusion equation with variable convection coefficients International Journal for Numerical Methods in Fluids 2002 38 12 1111 1131 10.1002/fld.263
Kalita, J. C., Dalal, D. C. & Dass, A. K. A class of higher order compact schemes for the unsteady two-dimensional convection-diffusion equation with variable convection coefficients. International Journal for Numerical Methods in Fluids 38(12), 1111–1131 (2002).10.1002/fld.263
34. Ghayad MS Badra NM Ahmed HM Rabie WB Derivation of optical solitons and other solutions for nonlinear Schrödinger equation using modified extended direct algebraic method Alex. Eng. J. 2023 64 801 811 10.1016/j.aej.2022.10.054
Ghayad, M. S., Badra, N. M., Ahmed, H. M. & Rabie, W. B. Derivation of optical solitons and other solutions for nonlinear Schrödinger equation using modified extended direct algebraic method. Alex. Eng. J. 64, 801–811 (2023).10.1016/j.aej.2022.10.054
35. Younas U Rezazadeh H Ren J Bilal M Propagation of diverse exact solitary wave solutions in separation phase of iron (Fe-Cr- X (X= Mo, Cu)) for the ternary alloys Int. J. Mod. Phys. B 2022 36 04 2250039 10.1142/S0217979222500394
Younas, U., Rezazadeh, H., Ren, J. & Bilal, M. Propagation of diverse exact solitary wave solutions in separation phase of iron (Fe-Cr- X (X= Mo, Cu)) for the ternary alloys. Int. J. Mod. Phys. B 36(04), 2250039 (2022).10.1142/S0217979222500394
36. Hussain A Junaid-U-Rehman M Jabeen F Khan I Optical solitons of NLS-type differential equations by extended direct algebraic method Int. J. Geom. Methods Mod. Phys. 2022 19 05 2250075 10.1142/S021988782250075X
Hussain, A., Junaid-U-Rehman, M., Jabeen, F. & Khan, I. Optical solitons of NLS-type differential equations by extended direct algebraic method. Int. J. Geom. Methods Mod. Phys. 19(05), 2250075 (2022).10.1142/S021988782250075X
37. Malik S Kumar S Kumari P Nisar KS Some analytic and series solutions of integrable generalized Broer–Kaup system Alex. Eng. J. 2022 61 9 7067 7074 10.1016/j.aej.2021.12.051
Malik, S., Kumar, S., Kumari, P. & Nisar, K. S. Some analytic and series solutions of integrable generalized Broer–Kaup system. Alex. Eng. J. 61(9), 7067–7074 (2022).10.1016/j.aej.2021.12.051
38. Kumar S Niwas M Dhiman SK Abundant analytical soliton solutions and different wave profiles to the Kudryashov–Sinelshchikov equation in mathematical physics J. Ocean Eng. Sci. 2022 7 6 565 577 10.1016/j.joes.2021.10.009
Kumar, S., Niwas, M. & Dhiman, S. K. Abundant analytical soliton solutions and different wave profiles to the Kudryashov–Sinelshchikov equation in mathematical physics. J. Ocean Eng. Sci. 7(6), 565–577 (2022).10.1016/j.joes.2021.10.009
39. Kumar S Niwas M Exploring lump soliton solutions and wave interactions using new Inverse (G’/G)-expansion approach: Applications to the (2+ 1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation Nonlinear Dyn. 2023 111 21 20257 20273 10.1007/s11071-023-08937-2
Kumar, S. & Niwas, M. Exploring lump soliton solutions and wave interactions using new Inverse (G’/G)-expansion approach: Applications to the (2+ 1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation. Nonlinear Dyn. 111(21), 20257–20273 (2023).10.1007/s11071-023-08937-2
40. Kumar S Niwas M Analyzing multi-peak and lump solutions of the variable-coefficient Boiti–Leon–Manna–Pempinelli equation: A comparative study of the Lie classical method and unified method with applications Nonlinear Dyn. 2023 111 24 22457 22475 10.1007/s11071-023-09012-6
Kumar, S. & Niwas, M. Analyzing multi-peak and lump solutions of the variable-coefficient Boiti–Leon–Manna–Pempinelli equation: A comparative study of the Lie classical method and unified method with applications. Nonlinear Dyn. 111(24), 22457–22475 (2023).10.1007/s11071-023-09012-6
41. Niwas M Kumar S Multi-peakons, lumps, and other solitons solutions for the (2+ 1)-dimensional generalized Benjamin–Ono equation: An inverse (G’/G)-expansion method and real-world applications Nonlinear Dyn. 2023 111 24 22499 22512 10.1007/s11071-023-09023-3
Niwas, M. & Kumar, S. Multi-peakons, lumps, and other solitons solutions for the (2+ 1)-dimensional generalized Benjamin–Ono equation: An inverse (G’/G)-expansion method and real-world applications. Nonlinear Dyn. 111(24), 22499–22512 (2023).10.1007/s11071-023-09023-3
42. Rehman HU Ullah N Asjad MI Akgül A Exact solutions of convective–diffusive Cahn–Hilliard equation using extended direct algebraic method Numer. Methods Partial Differ. Equ. 2023 39 6 4517 4532 10.1002/num.22622
Rehman, H. U., Ullah, N., Asjad, M. I. & Akgül, A. Exact solutions of convective–diffusive Cahn–Hilliard equation using extended direct algebraic method. Numer. Methods Partial Differ. Equ. 39(6), 4517–4532 (2023).10.1002/num.22622
43. Serkin VN Hasegawa A Soliton management in the nonlinear Schrödinger equation model with varying dispersion, nonlinearity, and gain J. Exp. Theor. Phys. Lett. 2000 72 89 92 10.1134/1.1312019
Serkin, V. N. & Hasegawa, A. Soliton management in the nonlinear Schrödinger equation model with varying dispersion, nonlinearity, and gain. J. Exp. Theor. Phys. Lett. 72, 89–92 (2000).10.1134/1.1312019
44. Serkin VN Matsumoto M Belyaeva TL Bright and dark solitary nonlinear Bloch waves in dispersion managed fiber systems and soliton lasers Opt. Commun. 2001 196 1–6 159 171 10.1016/S0030-4018(01)01365-7
Serkin, V. N., Matsumoto, M. & Belyaeva, T. L. Bright and dark solitary nonlinear Bloch waves in dispersion managed fiber systems and soliton lasers. Opt. Commun. 196(1–6), 159–171 (2001).10.1016/S0030-4018(01)01365-7
45. Malomed BA Soliton Management in Periodic Systems 2006 Springer
Malomed, B. A. Soliton Management in Periodic Systems (Springer, 2006).
46. Serkin VN Hasegawa A Novel soliton solutions of the nonlinear Schrödinger equation model Phys. Rev. Lett. 2000 85 21 4502 10.1103/PhysRevLett.85.4502 11082581
Serkin, V. N. & Hasegawa, A. Novel soliton solutions of the nonlinear Schrödinger equation model. Phys. Rev. Lett. 85(21), 4502 (2000).11082581 10.1103/PhysRevLett.85.4502
47. Liu WJ Tian B Xu T Sun K Jiang Y Bright and dark solitons in the normal dispersion regime of inhomogeneous optical fibers: Soliton interaction and soliton control Ann. Phys. 2010 325 8 1633 1643 10.1016/j.aop.2010.02.012
Liu, W. J., Tian, B., Xu, T., Sun, K. & Jiang, Y. Bright and dark solitons in the normal dispersion regime of inhomogeneous optical fibers: Soliton interaction and soliton control. Ann. Phys. 325(8), 1633–1643 (2010).10.1016/j.aop.2010.02.012
48. Liu D Yang Z The topological structures of the spaces of copulas and subcopulas Fuzzy Sets Syst. 2023 467 108485 10.1016/j.fss.2023.02.006
Liu, D. & Yang, Z. The topological structures of the spaces of copulas and subcopulas. Fuzzy Sets Syst. 467, 108485 (2023).10.1016/j.fss.2023.02.006
49. Agarwal, R. P., Meehan, M. & O’regan, D. Fixed Point Theory and Applications. Vol. 141. (Cambridge University Press , 2001).
50. Wang X Convergence of Newton’s method and uniqueness of the solution of equations in Banach space IMA J. Numer. Anal. 2000 20 1 123 134 10.1093/imanum/20.1.123
Wang, X. Convergence of Newton’s method and uniqueness of the solution of equations in Banach space. IMA J. Numer. Anal. 20(1), 123–134 (2000).10.1093/imanum/20.1.123
51. Ahmed N Yasin MW Iqbal MS Akgül A Rafiq M Raza A Baber MZ Numerical investigations of stochastic Newell–Whitehead–Segel equation in (2+ 1) dimensions Int. J. Mod. Phys. B 2023 37 30 2350261 10.1142/S0217979223502612
Ahmed, N. et al. Numerical investigations of stochastic Newell–Whitehead–Segel equation in (2+ 1) dimensions. Int. J. Mod. Phys. B 37(30), 2350261 (2023).10.1142/S0217979223502612
52. Iqbal MS Boundary value problems for non-linear first order systems of partial differential equations in higher dimensions, especially in three dimensions Adv. Appl. Clifford Algebras 2019 29 5 98 10.1007/s00006-019-1019-3
Iqbal, M. S. Boundary value problems for non-linear first order systems of partial differential equations in higher dimensions, especially in three dimensions. Adv. Appl. Clifford Algebras 29(5), 98 (2019).10.1007/s00006-019-1019-3
53. Dowker CH Mapping theorems for non-compact spaces Am. J. Math. 1947 69 2 200 242 10.2307/2371848
Dowker, C. H. Mapping theorems for non-compact spaces. Am. J. Math. 69(2), 200–242 (1947).10.2307/2371848
54. Iqbal MS Baber MZ Inc M Younis M Ahmed N Qasim M On multiple solitons of glycolysis reaction–diffusion system for the chemical concentration Int. J. Mod. Phys. B 2024 38 04 2450055 10.1142/S0217979224500553
Iqbal, M. S. et al. On multiple solitons of glycolysis reaction–diffusion system for the chemical concentration. Int. J. Mod. Phys. B 38(04), 2450055 (2024).10.1142/S0217979224500553
55. Baber MZ Seadway AR Ahmed N Iqbal MS Yasin MW Selection of solitons coinciding the numerical solutions for uniquely solvable physical problems: A comparative study for the nonlinear stochastic Gross-Pitaevskii equation in dispersive media Int. J. Mod. Phys. B 2023 37 20 2350191 10.1142/S0217979223501916
Baber, M. Z., Seadway, A. R., Ahmed, N., Iqbal, M. S. & Yasin, M. W. Selection of solitons coinciding the numerical solutions for uniquely solvable physical problems: A comparative study for the nonlinear stochastic Gross-Pitaevskii equation in dispersive media. Int. J. Mod. Phys. B 37(20), 2350191 (2023).10.1142/S0217979223501916
