EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 945-955 ISSN 1307-5543 – ejpam.com Published by New York Business Global Exact solutions for the modified Burgers equation with additional time-dependent variable coefficient Bazar Babajanov1,4, Fakhriddin Abdikarimov2,∗, Sarbinaz Bazarbaeva3 1 Department of Applied Mathematics and Mathematical Physics, Urgench State University, Urgench, Uzbekistan 2 Khorezm Mamun Academy, Khiva, Uzbekistan 3 Karakalpak State University, Nukus, Uzbekistan 4 Khorezm Branch of Uzbekistan Academy of Sciences V. I. Romanovskiy Institute of Mathematics, Urgench, Uzbekistan Abstract. In this article, we investigated new travelling wave solutions for the modified Burgers equation with additional time-dependent variable coefficient via the functional variable method. The performance of this method is reliable and effective and gives the exact solitary wave solutions. All solutions of this equation have been examined and three dimensional graphics of the obtained solutions have been drawn by using the Matlab program. The exact solutions have its great importance to reveal the internal mechanism of the physical phenomena. This method presents a wider applicability for handling nonlinear wave equations. 2020 Mathematics Subject Classifications: 34A34, 34B15, 35Q51, 35J60, 35J66, 35L05 Key Words and Phrases: modified Burgers equation, solitary wave solutions, kinematics vis- cosity, variable coefficient, velocity, functional variable method, viscosity parameter, ordinary dif- ferential equation, continuous differentiable functions, internal mechanism, distinct variables 1. Introduction Burgers equation was first given by Bateman and later was studied by Burgers as a mathematical model for turbulence[12, 15]. The Burgers equation has applications in various fields such as convection and diffusion, number theory, gas dynamics, heat conduction, elasticity, engineering and other scientific fields[29]. The Burgers equation is in the form ut + uux − νuxx = 0, where u(x, t) denotes the velocity for space x and time t and ν > 0 is a constant repre- senting the kinematics viscosity of the fluid. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5064 Email addresses: a.murod@mail.ru (B. Babajanov), goodluck 0714@mail.ru (F. Abdikarimov), sarbinazbazarbaeva6@gmail.com (S. Bazarbaeva) https://www.ejpam.com 945 © 2024 EJPAM All rights reserved. B. Babajanov, F. Abdikarimov, S. Bazarbaeva / Eur. J. Pure Appl. Math, 17 (2) (2024), 945-955 946 The one-dimensional modified Burgers equation is in the form ut + u2ux − νuxx = 0, where u(x, t) is the dependent variable, ν is the viscosity parameter, t and x are the independent parameters. This equation describes in several areas of applied mathematics such as various practical transport problems, nonlinear waves in a medium with low- frequency pumping or absorption, ion reflection at quasi-perpendicular shocks, turbulence transport, the transport and dispersion of pollutants in rivers[14]. In the literature many numerical method was applied to approximate the solution of the modified Burgers equation by several authors. The collocation method with quintic splines[14], the colocation method with septic splines[31], the sextic B-spline collocation method[24], a non-polynomial spline based method[22], an explicit numerical scheme[13], Petrov-Galerkin method[33] and explicit exponential finite difference schemes have been used to obtain numerical solution of the modified Burgers equation by several authors[16]. Many direct methods of nonlinear evolutions equations have been developed to find solutions, such as tanh-function method[25], functional variable method[4, 5, 7, 9, 10], Hirota method[23], Backlund transform method[32], exp-function method[28], G/G′ ex- pansion method[6, 8] and extended tanh-method[19] are used for searching the exact solutions[2, 3, 11, 20, 26, 27]. In[17], the arteries were considered as thin-wall prestressed elastic tubes of variable radius, and the long-wavelength approximation was used. The propagation of weakly nonlinear waves in such an elastic tube filled with a liquid was investigated using the modified Korteweg-de Vries equation with a variable coefficient ut + 6u2ux − uxxx = h(t)ux, where t is the scale coordinate along the vessel axis after a static deformation (this coor- dinate characterizes the axi symmetric stenosis on the surface of the arterial wall), x is a variable depending on time and the coordinate along the vessel axis, h(t) is the shape of the stenosis, and the function u(x, t) characterizes the average axial velocity of the liquid. The modified KdV-Burgers equation with variable coefficients is defined as ut + uxxx + 3αu2ux + βuxx = 0, where α and β are constant coefficients, and they incorporate the effects of nonlinearity (αu2ux) and dissipation (βuxx) into the equation; β is the coefficient of the kinematic viscosity of a fluid (β < 0). When the dispersion term uxxx = 0, then this equation was was formulated from the modified Burgers equation[30]. When β = 0, this equation is just the so called mKdV equation, which originates from nonlin ear optics[1] and the propagation of long internal waves in a fluid when the coefficient of the ordinary nonlinear term in the KdV equation. The higher order nonlinear term u2ux dominates over higher or dispersive terms[21]. In this article, we consider the modified Burgers equation with additional time-dependent variable coefficient ut + h1(t)u 2ux − h2(t)uxx + ω(t)ux = 0, (1) B. Babajanov, F. Abdikarimov, S. Bazarbaeva / Eur. J. Pure Appl. Math, 17 (2) (2024), 945-955 947 where u(x, t) is an unknown function, x ∈ R, t ≥ 0, h1(t) ̸= 0, h2(t) ̸= 0, ω(t) ̸= 0 are given continuous differentiable functions and h2(t) > 0 is a variable representing the kinematics viscosity of the fluid. The equation (1) arises in many physical problems including the motions of waves in nonlinear optics, plasma or fluids, water waves, ion-acoustic waves in a collision less plasma. The first element ut designates the evolution term and the second one shows the term of dispersion. The main aim of this paper is to find the exact soliton solutions of the equation (1) via functional variable method. The main advantage of the proposed method over other methods is that it provides more new exact traveling wave solutions. All solutions of this equation have been examined and three dimensional graphics of the obtained solutions have been drawn by using the Matlab program. The exact solutions have its great importance to reveal the internal mechanism of the physical phenomena. 2. Description of the method The basic idea of the functional variable method proposed in[18]. Let us consider the nonlinear differential equation with independent variables x, y, z, t and a dependent variable u P (u, ut, ux, uy, uz, uxy, uyz, uxz, ...) = 0, (2) where P is a polynomial in u(t, x, y, z, ...) and its partial derivatives. The equation (2) is a nonlinear partial differential equation that is not integrable, in general. Sometime it is difficult to find a complete set of solutions. Step 1. The following transformation is used for the new wave variable as ξ = p∑ i=0 αiχi + δ, (3) where χi are distinct variables, when p = 1, ξ = α0χ0 +α1χ1 + δ. If the quantities α0, α1 are constants, then, they are called the wave pulsation and χ0, χ1 are the variables t and x, respectively. We can introduce the following transformation for a travelling wave solution of equation (2) u(χ0, χ1, ...) = u(ξ), (4) and the chain rule ∂u ∂χi = αi du dξ , ∂2u ∂χi∂χj = αiαj d2u dξ2 , .... (5) Using equation (3) and equation (5), the nonlinear partial differential equation (2) can be transformed into an ordinary differential equation of the form Q(u, u′, u′′, u′′′, ...) = 0, (6) where Q is a polynomial in u(ξ) and its total derivatives, while u′ = du dξ . B. Babajanov, F. Abdikarimov, S. Bazarbaeva / Eur. J. Pure Appl. Math, 17 (2) (2024), 945-955 948 Step 2. We make a transformation in which the unknown function u is considered as a functional variable in the form u′ = F (u), (7) then, the solution can be found by the relation∫ du F (u) = ξ + C, (8) here C is a constant of integration which is set equal to zero for convenience. Some successive differentiations of u in terms of F are given as u′′ = dF (u) du du dξ = dF (u) du F (u) = 1 2 d(F 2(u)) du , u′′′ = 1 2 d2(F 2(u)) du2 √ F 2(u), u(IV ) = 1 2 [ d3(F 2(u)) du3 F 2(u) + d2(F 2(u)) du2 d(F 2(u)) du ] , ........................................................................ (9) Step 3. The ordinary differential equation (6) can be reduced in terms of u, F and its derivatives upon using the expressions of equation (7) and (9) into equation (6) gives R(u, dF (u) du , d2F (u) du2 , d3F (u) du3 , ...) = 0. (10) After integration, equation (10) provides the expression of F (u) and this, together with equation (7), give appropriate solutions to the being considered problem. 3. Algorithm for finding solutions We use the following algorithm to calculate the exact solution of the equation (1) by the functional variable method. Using the wave variable u(x, t) = u(t, ξ), ξ = a(t) + b(t)x, (11) that will convert equation (1) to following form u′t + (at(t) + bt(t))u ′ ξ + h1(t)b(t)u 2u′ξ − h2(t)b 2(t)u′′ξ + ω(t)b(t)u′ξ = 0, (12) where a(t) and b(t) are an unknown time-dependent functions, we will determine these functions later. Let a(t), b(t), h1(t), h2(t) and ω(t) are constant functions. We use the following transformation ξ = a+ bx. (13) We put a(t), b(t), h1(t), h2(t) and ω(t) into (11) and (12), integration constants are considered zero. It is easy to show that after transformation, the equation (12) can be transformed into an ordinary differential equation of the form h1bu 2u′ξ − h2b 2u′′ξ + ωbu′ξ = 0. (14) B. Babajanov, F. Abdikarimov, S. Bazarbaeva / Eur. J. Pure Appl. Math, 17 (2) (2024), 945-955 949 Integrating once equation (14), we have h1 3 u3 − h2bu ′ ξ + ωu = 0. (15) It is easy to deduce from equation (15) an expression for the function u′ξ u′ξ = ru+ nu3, (16) where r = ω h2b , n = h1 3h2b . We search the solution of equation (1) in the form: u(t, ξ) = m∑ k=0 qk(t)Φ k(ξ) = q0(t) + q1(t)Φ(ξ) + ...+ qm(t)Φ(ξ)m, (17) where Φ satisfies equation (16) as Φ′ = λΦ+ µΦ3, (18) where λ and µ are free parameters and m is an undetermined integer and qk(t) are coeffi- cients to be determined later. One of the most useful techniques for obtaining the parameter m in (17) is the ho- mogeneous balance method. Substituting (17) into equation (12) and by making balance between the linear term u′′ and the nonlinear term uu′ to determine the value of m, and by simple calculation we have got that 3m+2 = m+4, this in turn gives m = 1, and the solution (17) takes the form u(t, ξ) = 1∑ k=0 qk(t)Φ k(ξ) = q0(t) + q1(t)Φ(ξ). (19) Now, we substitute (19) into (12) along with (18) and set each coefficient of Φk (Φ′)p (k = 0, 1, 2 and p = 0, 1 ) to zero to obtain a set of algebraic equations for q0(t), q1(t), a(t) and b(t):  bt(t) = 0, q0t(t) + h1(t)q 2 0(t) = 0, q1t(t) + 2q0(t)q1(t)h1(t)b(t) = 0, at(t)− λh2(t)b 2(t) + ω(t)b(t) = 0 h1(t)q 2 1(t)− 3µh2(t)b(t) = 0. (20) Solving the system of algebraic equations, we can obtain a(t), b(t), q0(t) and q1(t). For this, we consider the following 2 cases in the system of equations (20). Let q0(t) = 0, then the system of algebraic equations (20) has the following solution a(t) = ∫ t 0 ( λS2 2h2(τ)− S2ω(τ) ) dτ + S1, b(t) = const = S2, q0(t) = 0, q1(t) = const = S3. (21) B. Babajanov, F. Abdikarimov, S. Bazarbaeva / Eur. J. Pure Appl. Math, 17 (2) (2024), 945-955 950 h2(t) = kh1(t), k = const, (22) where S1, S2 and S3 are the integration constants and are identified from initial data of the pulse. Notice that h1(t) and h2(t) serve as constraint relations between the coefficient functions and which indicate that (22) must be satisfied to assure the existence and the formation process of soliton structures. Taking account of (11), (18), (19) and (21), we get the exact solutions for equation (1) u1(x, t) = S3 √√√√ e2( ∫ t 0 (λS 2 2h2(τ)−S2ω(τ))dτ+S1+S2x) 1− e2( ∫ t 0 (λS 2 2h2(τ)−S2ω(τ))dτ+S1+S2x) . (23) Let q0(t) ̸= 0, then the system of algebraic equations (20) has the following solution a(t) = ∫ t 0 ( λC2 2h2(τ)− C2ω(τ) ) dτ + C1, b(t) = const = C2, q0(t) = 1∫ t 0 h1(τ)dτ+C3 , q1(t) = C4 ( ∫ t 0 h1(τ)dτ+C3) 2C2 . (24) h2(t) = C2 4 3µC2 h1(t)(∫ t 0 h1(τ)dτ + C3 )4C2 , (25) where C1, C2, C3 and C4 are the integration constants and are identified from initial data of the pulse. Notice that h1(t) and h2(t) serve as constraint relations between the coefficient functions and which indicate that (25) must be satisfied to assure the existence and the formation process of soliton structures. Taking account of (11), (18), (19) and (24), we get the exact solutions for equation (1) u2(x, t) = 1∫ t 0 h1(τ)dτ + C3 + C4(∫ t 0 h1(τ)dτ + C3 )2C2 √√√√ e2( ∫ t 0 (λC 2 2h2(τ)−C2ω(τ))dτ+C1+C2x) 1− e2( ∫ t 0 (λC 2 2h2(τ)−C2ω(τ))dτ+C1+C2x) . (26) 4. Examples Solitary wave solutions represent an important type of solutions for nonlinear partial differential equations as many nonlinear partial differential equations have been found to have a variety of solitary wave solutions. The solitary wave solutions were obtained in this article and could be helpful in analyzing long wave propagation on the surface of a fluid layer, iron sound waves in plasma, and vibrations in a nonlinear string. Also, solitary wave in the concept of mathematical physics is defined as a self-reinforcing wave package that retains its shape. It propagates at a constant amplitude and velocity. B. Babajanov, F. Abdikarimov, S. Bazarbaeva / Eur. J. Pure Appl. Math, 17 (2) (2024), 945-955 951 We illustrate the application of algorithm to solving the equation (1). Exact soliton solution of the equation (1) can be defined explicitly for exact values of h1(t) = t, h2(t) = t, ω(t) = t, λ = 1, µ = 1. According to (21), we obtain q0(t), q1(t), a(t) and b(t): q0(t) = 0, q1(t) = 3, a(t) = 3t2, b(t) = 3. (27) In this case, the soliton solution of the equation (1) has the form u1(x, t) = √ 9e6(t2+x) 1− e6(t2+x) . (28) This solution of the equation (1) have been checked and using mathematical software Matlab and three-dimensional graphics of the obtained solutions have been shown. Solitary wave solutions represent an important type of solutions for nonlinear partial differential equations as many nonlinear partial differential equations have been found to have a variety of solitary wave solutions. Figure 1: Soliton wave solution of the equation (1) for h1(t) = t, h2(t) = t, ω(t) = t, λ = 1, µ = 1. We illustrate the application of algorithm to solving the equation (1). Exact soliton solution of the equation (1) can be defined explicitly for exact values of h1(t) = 2t, h2(t) = t t2+1 , ω(t) = −8t, λ = 32, µ = 8 3 . According to (24), we obtain q0(t), q1(t), a(t) and b(t): q0(t) = 1 t2 + 1 , q1(t) = 1√ t2 + 1 , a(t) = ln(t2 + 1) + t2, b(t) = 1 4 . (29) In this case, the soliton solution of the equation (1) has the form u2(x, t) = 1 t2 + 1 + √ t2 + 1 √√√√ e2(t 2+ 1 4 x) 1− (t2 + 1)2e2(t 2+ 1 4 x) . (30) This solution of the equation (1) have been checked and using mathematical software Matlab and three-dimensional graphics of the obtained solutions have been shown. Solitary wave solutions represent an important type of solutions for nonlinear partial differential equations as many nonlinear partial differential equations have been found to have a variety of solitary wave solutions. B. Babajanov, F. Abdikarimov, S. Bazarbaeva / Eur. J. Pure Appl. Math, 17 (2) (2024), 945-955 952 Figure 2: Soliton wave solution of the equation (1) for h1(t) = 2t, h2(t) = t t2+1 , ω(t) = −8t, λ = 32, µ = 8 3 . 5. Conclusion This paper discusses several traveling wave solutions of the modified Burgers equation with additional time-dependent variable coefficient by the functional variable method. The main advantage of the proposed method over other methods is that it provides more new exact traveling wave solutions. We have found soliton solutions of this equation and three dimensional graphics of the obtained solutions have been drawn by using the Matlab program. After visualizing the graphs of the soliton solutions wave solutions, the use of distinct values of random parameters is demonstrated to better understand their physical features. It is known that the parameters included in the solutions have a deep connection with the amplitudes and velocities. In this regard, we can explore some of the nonlinear phenomena that take place in physics, applied mathematics and technology. We conclude that the exact solutions have its great importance to reveal the internal mechanism of the physical phenomena. Conflict of Interest The author declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Author contributions Bazar Babajanov and Fakhriddin Abdikarimov conceived of the presented idea. Bazar Babajanov developed the theory. Fakhriddin Abdikarimov performed the computations. Fakhriddin Abdikarimov and Sarbinaz Bazarbaeva verified the methods. All authors dis- cussed the results and contributed to the final manuscript and contributed to the article and approved the submitted version. REFERENCES 953 References [1] G. P. Agrawal. Nonlinear Fiber Optics. Academic Press, San Diego, California, 1989. [2] O. Abu Arqub. Computational algorithm for solving singular fredholm time-fractional partial integrodifferential equations with error estimates. Journal of Applied Mathe- matics and Computing, 59:227–243, 2019. [3] O. Abu Arqub and H. Rashaideh. The rkhs method for numerical treatment for integrodifferential algebraic systems of temporal two-point bvps. Neural Computing and Applications, 30:2595–2606, 2018. [4] B. Babajanov and F. Abdikarimov. The application of the functional variable method for solving the loaded non-linear evaluation equations. Frontiers in Applied Mathe- matics and Statistics, 8:912674, 2022. [5] B. Babajanov and F. Abdikarimov. Exact solutions of the nonlinear loaded benjamin- ono equation. WSEAS Transactions on Mathematics, 21:666–670, 2022. [6] B. Babajanov and F. Abdikarimov. Expansion method for the loaded modi- fied zakharov-kuznetsov equation. Advanced Mathematical Models & Applications, 7(2):168–177, 2022. [7] B. Babajanov and F. Abdikarimov. Solitary and periodic wave solutions of the loaded modified benjamin-bona-mahony equation via the functional variable method. Re- searches in Mathematics, 30(1):10–20, 2022. [8] B. Babajanov and F. Abdikarimov. Soliton solutions of the loaded modified calogero- degasperis equation. International Journal of Applied Mathematics, 35(3):381–392, 2022. [9] B. Babajanov and F. Abdikarimov. New exact soliton and periodic wave solutions of the nonlinear fractional evolution equations with additional term. Partial Differential Equations in Applied Mathematics, 8:100567, 2023. [10] B. Babajanov and F. Abdikarimov. Solitary and periodic wave solutions of the loaded boussinesq and the loaded modified boussinesq equation. Journal of Mathematics and Computer Science, 30(1):67–74, 2023. [11] B. Babajanov and F. Abdikarimov. Soliton and periodic wave solutions of the non- linear loaded (3+1)-dimensional version of the benjamin-ono equation by functional variable method. Journal of Nonlinear Modeling and Analysis, 5(4):782–789, 2023. [12] H. Bateman. Some recent researches on the motion of fluids. Monthly Weather Review, 43:163–170, 1915. REFERENCES 954 [13] A. G. Bratsos and L. A . Petrakis. An explicit numerical scheme for the modified burgers equation. International Journal for Numerical Methods in Biomedical En- gneering, 27(2):232–237, 2011. [14] A.G. Bratsos. A fourth-order numerical scheme for solving the modified burgers equation. Computers & Mathematics with Applications, 60(5):1393–1400, 2010. [15] J. M. Burgers. A mathematical model illustrating the theory of turbulence. Advances in Applied Mechanics, 1:171–199, 1948. [16] G. Celikten and E. N. Aksan. Explicit exponential finite difference methods for the numerical solution of modified burgers equation. Eastern Anatolian Journal of Science, 3(1):45–50, 2017. [17] H. Demiray. Variable coefficient modified kdv equation in fluid-filled elastic tubes with stenosis: solitary waves. Chaos, Solitons & Fractals, 42:358–364, 2009. [18] W. Djoudi and A. Zerarka. Exact solutions for the kdv-mkdv equation with time- dependent coefficients using the modified functional variable method. Cogent Math- ematics, 3(1):1–9, 2016. [19] S. A. El-Wakil and M. A. Abdou. New exact travelling wave solutions using modified extended tanh-function method. Chaos, Solitons & Fractals, 31(4):840–852, 2007. [20] M. Farhan, Z. Omar, F. Mebarek-Oudina, J. Raza, Z. Shah, R. V Choudhari, and O. D. Makinde. Implementation of the one-step one-hybrid block method on the nonlinear equation of a circular sector oscillator. Computational Mathematics and Modeling, 31:116–132, 2020. [21] J. A. Gear and R. Grimshaw. A second-order theory for solitary waves in shallow fluids. Physics of Fluid, 26(14):14–29, 1983. [22] A. Griewank and T. S. El-Danaf. Efficient accurate numerical treatment of the mod- ified burgers equation. Applicable Analysis, 88(1):75–87, 2009. [23] R. Hirota. Exact solution of the kdv equation for multiple collisions of solutions. Physical Review Letters, 27:1192–1194, 1971. [24] D. Irk. Sextic b-spline collocation method for the modified burgers equation. Mat- mematics and Computers in Simulation, 38(9):1599–1620, 2009. [25] W. Malfliet. Solitary wave solutions of nonlinear wave equations. American Journal of Physics, 60(7):650–654, 1992. [26] Sh. Momani, O. Abu Arqub, and B. Maayah. Piecewise optimal fractional reproducing kernel solution and convergence analysis for the atangana-baleanu-caputo model of the lienard’s equation. Fractals, 28(8):2040007, 2020. REFERENCES 955 [27] Sh. Momani, B. Maayah, and O. Abu Arqub. The reproducing kernel algorithm for numerical solution of van der pol damping model in view of the atangana-baleanu fractional approach. Fractals, 28(8):2040010, 2020. [28] H. Naher, F. A. Abdullah, and M. A. Akbar. The exp-function method for new exact solutions of the nonlinear partial differential equations. International Journal of Physical Sciences, 6(29):6706–6716, 2011. [29] M. A. Ramadan and T. S. El-Danaf. Numerical treatment for the modified burgers equation. Matmematics and Computers in Simulation, 70(2):90–98, 2005. [30] M. A. Ramadan and T. S. El-Danaf. Numerical treatment for the modified burgers equation. Mathematics and Computers in Simulation, 70(2):90–98, 2005. [31] M. A. Ramadan, T. S. El-Danaf, and F. E. I. ABD Alaal. A numerical solution of the burgers equation using septic b-splines. Chaos, Solitons & Fractals, 26(3):795–804, 2005. [32] C. Rogers and W. F. Shadwick. Backlund transformations and their applications. Mathematics in Science and Engineering, 161:334, 1982. [33] T. Roshan and K. S. Bhamra. Numerical solutions of the modified burgers equation by petrov-galerkin method. Applied Mathematics and Computation, 218(7):3673–3679, 2011.