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  1. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 Petrov Weak Galerkin Finite Element Method for Solving Coupled Burgers' Problem Maryam Mohammed Shnawa1, Hashim A Kashkool2 1Department of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah, Iraq, maryammohammedshnawa994@gmail.com 2Department of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah, Iraq, hashim.kashkool@uobasrah.edu.iq Abstract This paper presents the novel idea of the weak Galerkin finite element method (WG-FEM). Focuses on the usage of weak functions, weak gradients, and their approximations result in a new concept called discrete weak gradients which is predicted to play important roles in numerical methods for partial differential equations. We defined two spaces for the test function and the trial function in the Petrov weak Galerkin finite element method (PWG-FEM), we apply the Petrov weak Galerkin finite element method for solving coupled Burgers' equations in two- dimension by using conservation form of nonlinear terms. We proved the stability and the optimal order error in ?2- norm ??? ?1- norm in the case of semi-discrete Petrov weak Galerkin finite element method (SDPWG-FEM). Unlike traditional methods, the slicing in semi-discrete Petrov weak Galerkin finite element method with respect to variables of space only is applied while time remains continuous. When the diffusion coefficient ? is greater than h the mesh size, the solutions are free from oscillations, when ? << ℎ (where ? is the diffusion coefficient and ℎ ?? ???ℎ ???? ), the solution is in this way oscillating. We proposed to improve the solution and eliminate the oscillation by using the Petrov weak Galerkin finite element method to which the shape function (trail function and test function) belongs to two different spaces, where we got a solution accurate and free of oscillations. Furthermore, we obtained the numerical results to confirm the theoretical results obtained. Keywords: Petrov weak Galerkin finite element, Semi-discrete, Coupled Burgers′ equations, Stability, Optimal order error. 1. Introduction In this study, we consider the nonlinear time-dependent coupled Burgers^, problem in two dimensions [1]. 236

  2. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 ?? ?? − ?∆? + ???+ ???= ?(?,?,?), (?,?,?) ∈ ? × (0,?], (1) ?? ?? − ?∆? + ???+ ???= ?(?,?,?), (?,?,?) ∈ ? × (0,?], (2) with Dirichlet boundary conditions ?(? ,? ,?) = ? (?,?,?),(? ,? ,?) ∈ ?? ? (0,?], (3) ?(? ,? ,?) = ? (?,?,?),(? ,? ,?) ∈ ?? ? (0,?], (4) and initial conditions ?(?,?,0) = ?0(? ,?),(? ,?) ∈ ?, (5) ?(?,?,0) =?0(? ,?), (? ,?) ∈ ?. (6) Where ? is a bounded region in ?2with a Lipschitz continuous boundary ??, ? and ? are known functions, ?∆? are the diffusion terms, ?,?∈ ?2(? ,?) are given functions. Nonlinear partial differential equations (NPDEs) play a critical role in formulating continuum models and have widespread applications in various scientific disciplines, including physics, engineering, chemistry, finance, and more. Burgers' equations are a fundamental component of such equations and are frequently employed as a useful model in many intriguing applied mathematics situations, particularly in two dimensions. It effectively simulates various fluid flow-related issues such as shock flows, traffic flow, acoustic transmission in fog, airflow over the air, oil, gas dynamics [2], etc., in which either shocks or viscous dissipation is a significant influence. Furthermore, it is crucial for comprehending convection-diffusion events. In particular, for computational needs, Burgers’ equations can be utilized as a forerunner of solving fluid flow issues with the Naiver-Stokes equations. It serves as a good example it can be used as a model for any nonlinear wave propagation problem subject to dissipation [3]. This dissipation might be caused by viscosity, heat conduction, mass diffusion, thermal radiation, chemical reaction, or other factors depending on the situation being modeled. Are taken from the previous study, Hussein et al., [4] introduced two numerical schemes, the first continuous time WG-FEM and the second discontinuous time WG-FEM, as well as a WG- FEM that was proposed for solving the 2D coupled Burgers’ equations with a stabilization term and a special trilinear form nonlinear term.Kashkool [5] proposed the semi-discrete formulation of the Galerkin and Galerkin-Conservation finite element method for the two-dimensional coupled Burgers' problem employed the artificial diffusion approach to enhance the approximate solution when (? < ℎ) (where ? is the diffusion coefficient and ℎ ?? ???ℎ ???? ) and the analytical solution of the issue. Wang et al., [6] studied a class of time-fractional generalized Burgers’ equations using the WG method, they demonstrated the existence of numerical solutions, the stability of a completely discrete scheme, and an ideal estimation of order error in discrete systems using the energy approach ?2- norm.Hussein [7] presented a continuous and ?? ??,?? ?? are unsteady terms,???,??? are the nonlinear convection terms, ?∆?, 237

  3. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 discrete time WG-FEM for solving 2D time-fractional coupled Burgers’ equations, the optimum order error in ?2- norm was derived based on the definition of fractional derivatives, and the stability for continuous time WG-FEM is demonstrated. It is commonly known that directly applying the Galerkin finite element approach to singularly perturbed Burgers' equations may produce spurious oscillations in the approximate solution. Researchers have employed various approaches to tackle this issue, including the Petrov-Galerkin approach [8-10], the Discontinuous Galerkin method [11-13], and the Petrov Discontinuous Galerkin method [14-15]. The weak Galerkin finite element method was first introduced by Wang and Ye [16]. The weak form differential operators, such as ( gradient, divergence curl, Laplacian, etc., ) are approximated by discrete generalized distributions in the WG-FEM for partial differential equations. These weak differential operators shall serve as building blocks for WG-FEM to partial differential equations, such as the Stokes equations, the biharmonic equation, and the Maxwell equation. The development of numerical methods and their applications in solving mathematical models in engineering and physics were aided by various related studies, for example, Ali et al., [17] conducted a study comparing the accuracy and efficiency of finite difference and finite volume methods with numerical simulations for solving the Burger?, equations model, are taken from the previous study, Arshad et al., [18] investigated three numerical techniques for solving fractional-order electrical RLC circuit equations. Sultana et al., [19] proposed new efficient computations for solving higher-order fractional partial differential equations with symmetrical and dynamic analysis. Ali et al., [20] studied the continuous dependence and symmetric properties of double-diffusive convection in a Forchheimer model, while Meften and Ali [21] examined the continuous dependence for double-diffusive convection in a Brinkman model with variable viscosity. Hussein [22] provided the trilinear weak form for the nonlinear term and the discrete-time and continuous-time WG-FE scheme to the solution of the nonlinear 2D coupled Burgers’ problems. Based on the dual argument approach, they were able to determine the ideal order error in the ?2- norm for both discrete time and continuous time WG scheme. Zhao [23] devised a reliable WG discretization for the nonlinear time-dependent incompressible miscible displacement problem in porous media. They created a flexible, positive definite technique that does not impose additional stabilizing criteria thanks to its arbitrary polygonal mesh form. The opener to WG-FEM is the use of a discrete weak gradient operator, which is defined and calculated by solving inexpensive problems locally on each element. The classic Galerkin finite element method is naturally extended by the widely used WG-FEM, which has several benefits, as it has a highly flexible structural design. WG-FEM is well-suited for most partial differential equations, providing the necessary stability and accuracy in approximation. The importance of the study is to improve the numerical solution using Petrov weak Galerkin finite element method by choosing the shape function (trail function and test function) that belongs to two different spaces, unlike the shape function previously used by the weak Galerkin finite element method, where the shape function (trail function and test function) in two similar spaces, but in the proposed way, the shape function (trail function and test function) in two different spaces. This paper proposes the use of the Petrov weak Galerkin finite element method for solving the coupled Burgers' equations in two dimensions. The PWG-FEM is designed to eliminate 238

  4. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 inaccuracies and oscillations that may occur when using the WG-FEM with ℎ > ? (where ? is the diffusion coefficient and ℎ ?? ???ℎ ???? ). The paper is structured as follows: In section 2, the definition of PWG-FE space is introduced. In section 3, the Petrov weak variational form is defined. Section 4 presents the definition of semi-discrete PWG-FEM and some necessary lemmas for error estimation. The stability of the PWG-FEM is proved in section 5, while section 6 provides the error analysis of the semi-discrete PWG-FEM. In section 7, a numerical results is presented. Finally, section 8 contains a discussion and conclusion. 2. A Petrov Weak Galerkin Spaces: Let U, V be two trial spaces and ?,∅ be two test spaces defined as follows: ? = {? = {?0,??}:{?0,??} ∈?2(?) × ?2(??), ∀ ? ∈ ?ℎ}, (7) ? = {? = {?0,??}:{?0,??} ∈ ?2(?) × ?2(??),∀ ? ∈ ?ℎ}, (8) ? = {?:? = ?0 + ??.??? : ?∈?}, (9) ∅ = { ?:? =?0+??. ?? ? : ?∈? }. (10) We define PWG – FEspaces, there are two trial finite element spaces defined as follows ?ℎ= {? = {?0,??}:{?0,??} |?∈ ??(?)× ??(??),∀? ∈ ?ℎ}, ( 11) ?ℎ = {? ={?0,??}:{?0,??} |?∈??(?)× ??(??), ∀?∈ ?ℎ}. (12) Define two test spaces by, ?ℎ= { ?:? = ?0 + ??.??,?? : ?∈?ℎ}, (13) ∅ℎ= { ?:? =?0+ ??. ??,? ? : ?∈?ℎ }, (14) and 0= {? = { ?0, ??} ∈ ?ℎ: ?? |?? ∩ ?? = 0 }, (15) ?ℎ 0= {? = ?0+ ??. ??,??: ?∈ ?ℎ ? }, (16) ?ℎ and 0= {? = { ?0,??} ∈ ?ℎ: ?? |?? ∩ ?? = 0 }, (17) ?ℎ 0= {? = ?0+ ??. ??,?? : ?∈?ℎ 0 }, (18) ∅ℎ 239

  5. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 here ? indicates a constant stability parameter. It will be chosen as [24]: ?ℎ ?? ? < ℎ 1 4 , (small constant), ? = ; 0 < ? < 0 ?? ? ≥ ℎ and dim U, V = dim ?, ∅, respectively. Here ? indicate the convection coefficient and ? represent diffusion coefficient Th represent a collection of all triangulation on Ω ?2(?) indicates space of square-integrable functions ??(K) indicates the set of polynomials on K with a degree no more than ? ??(∂K) represent the set of polynomials on ∂K with a degree no more than j ∇ represent gradient operator K indicates a triangle element ∂K indicates the boundary for the polygonal domain 3. Petrov Weak Variational Form Multiply Eq. (1) and Eq. (2) by the test functions (?0+ ??.???) and ( ?0+ ?? ?? ? ) respectively and integrating by part, we get [10] (??,?0+ ??.???) + ?(??,??) + (???,?0+ ??.???) + (???,?0+ ??.???) = (?,?0+ ??.???), ∀? ∈ ? (19) (??,?0+ ??.???) + ?(??,?? ) + (??? ,?0+ ??.???) + (???,?0+ ??.???) = ( ?,?0+ ??.???), ∀? ∈ ? (20) and (? (?,?,0), ?0+ ??.???) = (?0, ?0+ ??.???), (? (?,?,0), ?0+ ??.?? ?) = ( ?0, ?0+ ??.?? ?). 240

  6. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 We can write the nonlinear terms ??? and ??? in conservation form and integrating by part, we get (???, ?0+ ??.???) = 1 2 ((?2)?, ?0+ ?????) = −1 2 (?2, ??), (???, ?0+ ??.?? ? ) = 1 2 (( ?2)?, ?0+ ??.?? ?) = −1 2(?2, ??). Substituting in Eq. (19) and Eq. (20) thePetrov weak variational form is find ? ∈ ? ??? ? ∈ ?, such that (?? , ?0+ ??.???) + (? ??,? ? ) −1 2 (?2, ??) + (???, ?0+ ??.???) = ( ?, ?0+ ??.???), ?(?,?,0)=?0(?,?) ∀ (?,? ) ∈ ? ∀ ?∈?, (21) (?? , ?0+ ??.?? ?) + ( ? ??,?? ) + (??? , ?0+ ??.?? ?) −1 2(?2, ??) = ( ? , ?0+ ??.?? ?), ?(? ,? ,0) = ?0( ? ,?) ∀ (?,?) ∈ ? ∀? ∈ ?, (22) where ?(? ,?) = (? ??,?? ) −1 2 (?2,??) + (???, ?0+ ??.???), ?(?,?)= (???,?? ) + (??? , ?0+ ??.?? ?) −1 2(?2, ??). And for constants ?1,?2> 0 the property (coercive) hold. i.e., [25] ?(? ,?)≥ ?1‖?? ?‖2∀? ∈? (23) ?(? ,?)≥ ?2‖?? ?‖2 ∀?∈? (24) 4. The Semi-Discrete PWG – FEM In this section, we analyze semi-discrete PWG – FEM for two – dimensional coupled Burgers′ equations and derive the error estimations in ?1– norm and ?2– norm respectively. Based on Petrov weak variational formulation Eq. (21 ) and Eq. (22). The semi-discrete PWG – FEM is find ?ℎ∈ ?ℎ and ?ℎ∈ ?ℎ such that 241

  7. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 2 (?ℎ?ℎ,??,?? (?ℎ,?,?0) + ( ?ℎ,?,??.??,??) + ? (??,??ℎ,??,??)−1 ??) + (?ℎ(??,? ?ℎ ),?0 )+ (?ℎ(??,? ?ℎ ),??.??,??) ?? ?? 0, (25) = ( ? ,?0) + (? ,??.??,??) ∀ w ∈ ?ℎ (?ℎ,? ,?0)+ ( ?ℎ,?,??.??,? ?) + ?(??,??ℎ ,??,??) +(?ℎ(??,? ?ℎ ??),?0) + (?ℎ(??,? ?ℎ 2 (?ℎ?ℎ,(??,?? ??),??.??,? ?) −1 ?? )) ?, (26) = ( ? ,?0) + ( ? ,??.??,? ?) ∀ p ∈?ℎ or, (?ℎ,?,?0) + ( ?ℎ,?,??.??,??) + ??? (?ℎ,?) 0, (27) = ( ? ,?0) + (? ,??.??,??) ∀ w ∈ ?ℎ (?ℎ,? ,?0)+ ( ?ℎ,?,??.??,? ?) + ??? (?ℎ,?) ?, (28) = ( ? ,?0) + ( ? ,??.??,? ?) ∀ p ∈?ℎ 2 (?ℎ?ℎ,??,?? where, ??? (?ℎ,?) = ? (??,??ℎ,??,??) −1 ??) + (?ℎ(??,? ?ℎ ),?0)+ (?ℎ(??,? ?ℎ ),??.??,??), ?? ?? ??? (?ℎ,?) = ?(??,??ℎ ,??,??) + (?ℎ(??,? ?ℎ ),?0) ?? +(?ℎ(??,? ?ℎ 2 (?ℎ?ℎ,(??,?? ??),??.??,? ?) −1 ?? )) here aPW (uh,w) indicate bilinear form of Petrov weak Galerkin method ?(?,?) and ?ℎ(x,y,0) = ?ℎ ?(?,?), ?ℎ(?,?,0) = ?ℎ ?, ?ℎ ? are a proper approximation of functions ?0 and ?0 respectively. where ?ℎ ?or ?ℎ? ∈ ?ℎ ?. Then 4.1 Lemma.[16] If ?∈?0 1(Ω) ∩ ??+1 (Ω), ?ℎ? ∈ ?ℎ ‖?ℎ? − ?‖≤ ? ℎ?‖?‖?0 ≤ ? ≤ ? + 1, (29) ‖?? ?ℎ? − ??‖ ≤ ? ℎ? ‖?‖1+?0 ≤ ? ≤ ? + 1. (30) ?ℎ? ???????? ?ℎ? ?2?????????? ?? ?1(?)?? ?? ??(?) × ??(??) 242

  8. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 4.2 Lemma. [26]for ? ∈ ?1+? with ? > 0, we have ‖? − Πℎ?‖≤? ℎ?‖?‖1+?, (31) ‖?? − ?Πℎ?‖ ≤ ? ℎ?‖?‖1+?. (32) ?1+? ???????? ??????? ????? ?? ????? ? + 1, Πℎ???????? ?ℎ? ?????????? ?? ?(???,?) 5. Stability for PWG− FEM In this section, we proved the stability of the PWG−FEM given in Eq. (27). 5.1 Theorem. For the numerical solution to Eq. (27) and Eq. (28), there exists a constant ? > 0 dependent on ?,??? ? such that ? 0 ? 2 ‖?ℎ(?)‖2+ ∫ ‖?ℎ‖?? ?? + ?∫ ‖?ℎ,?‖2 0 ?? ? ? ≤ ?∫ ‖?‖2 0 ?? + ?∫ ‖ ?ℎ‖2 ?? + ‖?ℎ(0)‖2, (33) 0 ? 0 ? 2 ‖?ℎ(?)‖2+ ∫ ‖?ℎ‖?? ?? + ?∫ ‖?ℎ,?‖2 0 ?? ? ? ≤ ?∫ ‖?‖2 0 ?? + ?∫ ‖ ?ℎ‖2 ?? + ‖?ℎ(0)‖2 . (34) 0 Proof. Put ? = ?ℎ, in Eq. (27), we get (?ℎ,?,?ℎ) + ( ?ℎ,?,??.??,??ℎ) +??? (?ℎ,?ℎ) = (? ,?ℎ ) + (? ,??.??,??ℎ). (35) Where, 2 (?ℎ?ℎ,??,? ?ℎ ??? (?ℎ,?ℎ) = ? (??,??ℎ,??,??ℎ) −1 ) ?? + (?ℎ(??,? ?ℎ ),?ℎ) + (?ℎ(??,? ?ℎ ),??.??,??ℎ). ?? ?? Here??? (?ℎ,?ℎ) ?ndicate bilinear form of Petrov weak Galerkin method we take the first terms on the left-hand side of Eq. (35) and we get 243

  9. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 (?ℎ,?,?ℎ) = 1 2? ??‖?ℎ(?)‖2, (36) by Cauchy – Schwartz- inequality and Young's-inequality, we have ( ?ℎ,?,??.??,??ℎ) = ‖?ℎ,?‖‖ ??.??,??ℎ‖≤ 1 2‖?ℎ,?‖2+1 2‖ ??.??,??ℎ‖2 ≤ ?(‖?ℎ,?‖2+ ‖ ??.??,??ℎ‖2). (37) Using property in Eq. (23), we get ???(?ℎ,?ℎ) ≥ 1 2. (38) 2‖?ℎ‖?? Where, = ? ∥ ??,??ℎ∥2+ ?1∥ ?ℎ∥2∥ ?ℎ,?∥ +?1 2 ‖?ℎ‖?? 2∥ ?ℎ∥∥ ?ℎ,?∥∥ ?ℎ∥ +? ∥ ?ℎ∥∥ ?ℎ,?∥ ∥ ? ⋅ ??,??ℎ∥. Cauchy-Schwartz-inequality and Young'-inequality are applied to the first term on the right-hand side of Eq. (35) and the result is (? ,?ℎ ) ≤ ?(‖?‖2+ ‖ ?ℎ‖2). (39) By applying Cauchy-Schwartz- inequality and Young's -inequality to the second term on the right-hand side of Eq. (35), we obtain (? ,??.??,??ℎ) ≤ ?(‖?‖2+ ‖ ??.??,??ℎ‖2). (40) Substituting Eq. (36) - Eq. (40) in Eq. (35), we get 1 2? ??‖?ℎ(?)‖2+ ?(‖?ℎ,?‖2+ ‖ ??.??,??ℎ‖2) +1 2 2‖?ℎ‖?? ≤ ?(‖?‖2+ ‖ ?ℎ‖2) + ?(‖?‖2+ ‖ ??.??,??ℎ‖2). (41) Integration of both sides from (0 to t), we obtain ? 0 ? 0 2 ‖?ℎ(?)‖2− ‖?ℎ(0)‖2+ ∫ ‖?ℎ‖?? ?? + 2? ∫ ‖?ℎ,?‖2 ?? ? ? ≤ 2∫ (? + ?)‖?‖2 0 ?? + 2? ∫ ‖ ?ℎ‖2 ?? . (42) 0 ? 0 ? 2 ‖?ℎ(?)‖2+ ∫ ‖?ℎ‖?? ?? + ?∫ ‖?ℎ,?‖2 0 ?? 244

  10. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 ? ? ≤ ?∫ ‖?‖2 0 ?? + ?∫ ‖ ?ℎ‖2 ?? + ‖?ℎ(0)‖2 , (43) 0 where ? = 2(? + ?), ? = 2?, and ? = 2? . For ?,?,? > 0 represent constants Similarly, for proof Eq. (34). 6. The Error Analysis of PWG – FEM The goal of this section to is prove the error estimates for Semi-discrete PWG – FEM, in the ?1−norm and ?2−norm respectively. 6.1 Lemma. (?2– Error in SDPWG)Let ?(?,?,?),?(?,?,?) be the exact solutions and ?ℎ(?,?,?),?ℎ (?,?,?) be approximation solutions of Eq. (21), Eq. (22) and Eq. (25), Eq. (26) respectively. Denote by ? = ?ℎ? − ?ℎ. Then there exists constant ?, such that 2 ?? ≤ ‖??(0)‖2 + ? ∫ ‖??,??? ? ? ? ‖??‖2+ ∫ ‖???‖2 ?? ??‖ 0 ? ? 2 + ?ℎ2?∫ ‖?(?)‖1+? ? ?? + ? ∫ ‖??‖2 ??, (44) 0 2 ?? ≤ ‖??(0)‖2 + ? ∫ ‖??,??? ? ? ? ‖??‖2+ ∫ ‖???‖2 ?? ??‖ 0 ? ? 2 + ?ℎ2?∫ ‖?(?)‖ 1+? ? ?? + ? ∫ ‖??‖2 ??. (45) 0 Proof. Subtracting Eq. (25) from Eq. (21) and applying the fact (?ℎ??,?0) = (??,?0) and (Πℎ?ℎ,?0) = (? ,?0), we get ?ℎ? ???????? ?ℎ? ?2?????????? ?? ?1(?)?? ?? ??(?) × ??(??) Πℎ???????? ?ℎ? ?????????? ?? ?(???,?) ? ????????? ????????? ??????????? ??? ? indicates convection coefficient (?ℎ??− ?ℎ,?,?0) + (?ℎ??− ?ℎ,?, ??.??,??) 2( Πℎ ?2 ,??,?? + ? (Πℎ??,??,??)− ?(??,??ℎ,??,??) =1 ?? ) 245

  11. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 2(?ℎ?ℎ ,??,?? ??)+ (?ℎ(??,? ?ℎ − 1 ),?0) − (???,?0) ?? + (?ℎ(??,? ?ℎ ),??.??,??) − (???,??.??,??). (46) ?? The result of adding and subtracting the term ?(??,??ℎ?,??,??), is ((?ℎ? − ?ℎ)? ,?0) + ((?ℎ? − ?ℎ)?,??.??,? ?) 2(Πℎ?2,??,?? 2 (?ℎ?ℎ ,??,?? + ? (??,?(?ℎ? − ?ℎ),??,??)= 1 ??) −1 ??) + (?ℎ (??,? ?ℎ ),?0) − (???,?0)+ (?ℎ (??,? ?ℎ ),??.??,??) ?? ?? − (???,??.??,??)+ ?(??,??ℎ? ,??,??) − ?(Πℎ?? ,??,??).(47) 0u−?ℎ 0, ?ℎ ?? − ?ℎ ? } = { ?0 ? }, and ? , ?? Let ?? = ?ℎ? −?ℎ= {?ℎ 0? −?ℎ 0 , ?ℎ ?? −?ℎ ? } = { ?0 ?}. ? , ?? ??= ?ℎ? − ?ℎ = { ?ℎ Put ? = ?? in Eq. (47), we get 2(Πℎ?2,??,? ?? (??? ,??) + (???,??.??,? ??) + ?(??,? ??,??,? ??) = 1 ) ?? 2 (?ℎ?ℎ , ??,??? )+ (?ℎ(??,? ?ℎ ),??) − (???,??)+ (?ℎ( ??,? ?ℎ −1 ),??.??,???) ?? ?? ?? −(???,??.??,???) + ? (??,??ℎ? ,??,???) − ?(Πℎ?? ,??,???). (48) By Cauchy – Schwartz-inequality and Young's-inequality, we have 1 2? ??‖??‖2 + ?‖??,???‖2+ 1 2‖???‖2 + 1 4 ?=1 2?‖?.??,? ??‖2=∑ , (49) ?? where, 2(Πℎ?2,??,? ?? 2 (?ℎ?ℎ ,??,??? ?1=1 ) −1 ??) ?? ?2= (?ℎ(??,? ?ℎ ),??) − (???,??) ?? ?3= (?ℎ(??,? ?ℎ ),??.??,???) − (???,??.??,???) ?? ?4= ?(??,??ℎ? ,??,???) − ?(Πℎ??,??,???). 246

  12. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 To estimate ?1, we write it as following ??,??? ??) + (?ℎ(?ℎ? −?ℎ),??,??? 1 2 ((Πℎ ?2 – ?2 , ??) + ?1= (?ℎ?(?ℎ? −?ℎ),??,??? ??) + (?2− (?ℎ?)2 ,??,??? )), ?? by Cauchy – Schwartz-inequality and Young's-inequality, we obtain |?1| ≤ 1 4‖Πℎ?2− ?2‖2+ 1 2‖?ℎ‖2‖(??)‖2+1 2‖(??)‖2 2‖?‖∞ 2 2‖ (? − ?ℎ?) ‖2+‖??,??? +1 2‖?‖∞ , ??‖ by Eq. (29) and Eq. (31), we have 2+ ‖?ℎ‖2)‖(??)‖2+ ‖??,??? + 1 2 |?1| ≤ ?ℎ2?‖?‖ 1+? ‖2. (50) 2 (‖?‖∞ ?? To estimate ?2, we add and subtract (?ℎ??ℎ??,??), we get ?2 = (?(?ℎ ?? – ?? ),??) –(?ℎ??ℎ?? –?ℎ??,? ?ℎ ,?? ), ?? again by Cauchy – Schwartz-inequality and Young's-inequality, we obtain 2‖(?ℎ??− ??)‖2+ ‖?ℎ??ℎ??–?ℎ??,??ℎ ?? ‖2+1 2 ‖??‖ 2, |?2| ≤ ‖?‖∞ by Eq. (29), we have + ‖?ℎ??ℎ??–?ℎ??,??ℎ ?? ‖2+1 2 |?2| ≤ ?ℎ2?‖?‖ 1+? 2 ‖??‖ 2. (51) (?ℎ??ℎ?? ,??.??,? ??), is added and subtracted to estimate ?3, giving us ??,? ?ℎ ?? ?3= (?(?ℎ ?? – ?? ),??.??,? ??)–(?ℎ??ℎ?? –?ℎ ,??.??,? ??), again by Cauchy – Schwartz-inequality, Young's-inequality and Eq. (29), we obtain 2‖(?ℎ??− ??)‖2 + ‖?ℎ? ?ℎ??–?ℎ ??,??ℎ ‖2+1 2 ?‖?.??,???‖2 |?3| ≤ ‖?‖∞ ?? + ‖?ℎ??ℎ??–?ℎ??,??ℎ ?? ‖2+1 2 |?3| ≤ ?ℎ2?‖?‖ 1+? 2?‖?.??,???‖2. (52) To calculate ?4, we must add and subtract ?(??,??,???), we get 247

  13. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 ?4= ?(??,??ℎ? − ?? ,??,? ??) + ? (?? − Πℎ??,??,???), by Eq. (30) and Eq. (32), as well as Cauchy-Schwartz-inequality and Young's-inequality, we have |?4| ≤? 2‖( ??,??ℎ? − ??)‖2+? 2 ‖?? − Πℎ??‖2+ ? ‖??,???‖2 2 ≤ ?ℎ2?‖?‖ 1+? + ? ‖??,???‖2. (53) Substituting Eq. (50) - Eq. (53) in Eq. (49) by noting that ‖?ℎ? ?ℎ?? –?ℎ??,??ℎ negative term, we get ?? ‖2is a non- 1 2 ? ??‖??‖2+ ?‖??,???‖2+1 2‖???‖2+ 1 2 2?‖?.??,? ??‖2≤ ?ℎ2?‖?‖ 1+? 2 + ‖??,??? + 1 2+ ‖?ℎ‖2)‖(??)‖2 + ? ‖??,???‖2+1 2?‖?.??,???‖2. 2 (1 + ‖?‖∞ ‖ ?? (54) Using the Grönwall lemma and integrating concerning ?, we get to obtain the following estimate. 2 ?? ≤ ‖??(0)‖2 + ? ∫ ‖??,? ?? ? ? ? ‖??‖2+ ∫ ‖???‖2 ?? ‖ 0 ?? ? ? 2 + ?ℎ2?∫ ‖?(?)‖1+? ?? + ? ∫ ‖??‖2 ??. (55) ? 0 Similarly, to prove Eq. (45). 6.3 Lemma. (?1– Error in SDPWG) Let ?(?,?,?),?(?,?,?) be the exact solutions and ?ℎ(?,?,?),?ℎ (?,?,?) be approximation solutions of Eq. (21), Eq. (22) and Eq. (25), Eq. (26) respectively. Then there exists a constant ?, such that ‖(??)‖2+ ?‖??,???‖2≤ ?‖??,???(0)‖2+ ‖(??(0))‖2 ? ? 2 2 2 + ?ℎ2?(∫ ‖?‖?+1 ) + ?? + ∫ ‖??‖?+1 ? ?? + ‖?‖ 1+? ? 2 +‖??,? (??) ? + ∫ ?(?) 0 ?? , (56) ‖ ?? ‖(??)‖2+ ?‖??,???‖2≤ ?‖??,???(0)‖2+ ‖(??(0))‖2 248

  14. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 ? ? 2 2 2 + ?ℎ2?(∫ ‖?‖?+1 ) + ?? + ∫ ‖??‖?+1 ? ?? + ‖?‖ 1+? ? 2 +‖??,? (??) ? + ∫ ?(?) 0 ?? . (57) ‖ ?? Proof. Put?? = (??)? in Eq. (49), we obtain ‖(??)?‖2+ ? 2? ??‖??,???‖2+1 2‖???‖2+1 4 ?=1 2 ?‖?.??,? ???‖2= ∑ , (58) ?? where, 2(Πℎ?2,??,? ??? 2 (?ℎ?ℎ ,??,? ??? ?1=1 ) −1 ) ?? ?? ?2= (?ℎ(??,? ?ℎ ),???) − (???,???) ?? ?3= (?ℎ(??,? ?ℎ ),??.??,? ???) − (???,??.??,? ???) ?? ?4= ?(??,??ℎ? ,??,? ???) − ?(Πℎ?? ,??,? ???). To estimate ?1, we write it as follows: ??,? ??? ?? 2),??,? ??? 1 2(Πℎ ?2 – ?2 , 2 ?=1 ) + ((?ℎ??ℎ? − ?ℎ . (59) ) = ∑ ?1= ?1? ?? Using Eq. (31), the Cauchy-Schwartz-inequality, and Young's inequality we can estimate ?1?,? = 1,2, and obtain ??(Πℎ ?2 − ?2,??,?(??) ??(Πℎ ?2 − ?2 ), ??,?(??) ?11=1 2? )−1 2(? ) ?? ?? 2 ? ??(Πℎ ?2 − ?2,??,?(??) 4 ‖??,? (??) 1 ) +1 ? ??(Πℎ ?2 – ?2)‖2+ 1 4‖ , |?11| ≤ ‖ 2 ?? ?? 2 ? ??(Πℎ ?2− ?2,??,?(??) 4 ‖??,? (??) 1 +1 2 ) + ?ℎ2? ‖??‖ 1+? . (60) ≤ ‖ 2 ?? ?? 2− (?ℎ ? )2),??,? (??) 2− (?ℎ ? )2), ??,? (??) ?12=1 ? ?? ( − (?ℎ ) +1 2(? ) ?? (?ℎ 2 ?? ?? 2 2− (?ℎ ? )2),??,?(??) 4 ‖ ??,? (??) ≤ 1 ? ?? ( −(?ℎ ) +1 ? ?? (?ℎ 2 –(?ℎ?)2)‖2+ 1 . (61) 4‖ ‖ 2 ?? ?? Substituting Eq. (60) and Eq. (61) in Eq. (59), we get 249

  15. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 2 2 ‖??,? (??) +1 ? ?? (?ℎ 2 –(?ℎ?)2)‖2 +1 2 |?1| ≤ ?ℎ2?‖??‖ 1+? 4‖ ‖ ?? ? ??(Πℎ ?2 − ?2,??,?(??) 2− (?ℎ ? )2),??,?(??) + 1 ) +1 ? ?? (−( ?ℎ ). (62) 2 ?? 2 ?? To estimate ?2, we write ?? as follows: ?2= ( ?(?ℎ ?? – ?? ),(??)?) –(?ℎ??ℎ ?? –?ℎ??,? ?ℎ ,(??)? ) ?? Using Cauchy – Schwartz-inequality, Young's inequality, and Eq. (29), we obtain ??,? ?ℎ ?? |?2| ≤ 1 2‖?ℎ ??− ??‖2+1 ‖2+ ‖(??)?‖2 2‖?‖∞ 2‖?ℎ ? ?ℎ ??− ?ℎ ??,? ?ℎ ?? +1 2 ≤ ?ℎ2?‖?‖ 1+? ‖2+ ‖(??)?‖2 . (63) 2‖?ℎ ? ?ℎ ??− ?ℎ To estimate ?3, we write ?? as follows: ?3= (?(?ℎ ?? – ??),??.??,? ???) –(?ℎ??ℎ ?? –?ℎ??,? ?ℎ ,??.??,? ??? ) ?? Using Cauchy – Schwartz-inequality, Young's inequality, and Eq. (29), we obtain ??,? ?ℎ ?? ‖2 + 1 2 ‖??.??,? ???‖2 2‖?ℎ ??− ??‖2+ ‖?ℎ ? ?ℎ ??− ?ℎ |?3| ≤ ‖?‖∞ ??,? ?ℎ ?? ‖2 +1 2 2 ‖??.??,? ???‖2 . (64) |?3| ≤ ?ℎ2?‖?‖ 1+? + ‖?ℎ ? ?ℎ ??− ?ℎ To estimate ?4, we write it as follows: 2 ?=1 ?4= ?(??,??ℎ ? − ?? ,??,?(???)) + ? (?? − Πℎ ?? ,??,?(???)) = ∑ . (65) ?4? Using Eq. (30) and Eq. (32), we can estimate ?4?, ? = 1,2, using the Cauchy-Schwartz- inequality, and Young-inequality ?41= ?? ??(??,??ℎ ? − ?? ,??,? (??)) − ? (? ??(??,??ℎ ? − ??),??,? (??)) ? ??(??,??ℎ ? − ?? ,??,? ??) + ? ‖ ? ??(??,??ℎ ? − ??)‖2+ ? 4 ‖??,?( ??)‖2 |?41| ≤ ? ? ??(??,??ℎ ? − ?? ,??,? ??) +?ℎ2? ‖??‖?+1 + ? 2 4 ‖??,?( ??)‖2. (66) | ?41| ≤? 250

  16. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 ? ??( ?? − Πℎ ??,??,? ??) − ? (? ??(?? − Πℎ ??),??,? ??) ?42= ? 2 2 ? ??(?? − Πℎ ??,??,? ??) + ?‖? +? 4‖??,? ( ?? )‖ | ?42| ≤ ? ??(?? − Πℎ ??)‖ ? ??(?? − Πℎ ?? ,??,? ??) +?ℎ2?‖??‖?+1 +? 2 4‖??,? ( ?? )‖2. (67) | ?42| ≤ ? Substituting Eq. (66) and Eq. (67) in Eq. (65), we get ? ??(??,??ℎ ? − ?? ,??,? ??) + ? ? ??(?? − Πℎ ?? ,??,? ??) | ?4| ≤ ? + ? 2 +?ℎ2?‖??‖?+1 2 ‖??,? ( ?? )‖2. (68) Substituting Eq. (62), Eq. (63), Eq. (64), and Eq. (68) in Eq. (58) with noting that ‖?ℎ? ?ℎ?? –?ℎ??,? ?ℎ ? ?? (?ℎ 2 –(?ℎ?)2)‖2are non-negative terms, we get ‖2???‖ ?? ‖(??)?‖2+ ? 2? ??‖??,???‖2+1 2‖???‖2+1 2 2 ?‖?.??,? ???‖2≤ ?ℎ2?‖??‖ 1+? 2 2 ‖??,? (??) ??(Π ℎ ?2 − ?2,??,?(??) +1 + ‖(??)?‖2 + 1 2? 2 + ?ℎ2?‖?‖ 1+? ) ‖ ?? ?? 2− (?ℎ ? )2),??,?(??) + 1 ? ?? (−( ?ℎ ) + ? 2 ‖?.??,? ???‖2 + ? 2 ‖??,? ( ?? )‖2 2 ?? + ?? ??(??,??ℎ ? − ?? ,??,? ??) + ?? ??(?? − Πℎ ?? ,??,? ??). (69) 2 Put ?(?) = ‖??,? (??) + ? ‖??,? ( ?? )‖2 ‖ ?? Eq. (69) can be expressed simply as follows. ? ?? ( 1 2 ‖(??)‖2+? 2 2 2‖??,???‖2) ≤ ?ℎ2?‖??‖ 1+? + ?ℎ2?‖?‖ 1+? ? ??(Πℎ ?2 − ?2,??,?(??) 2− (?ℎ ? )2),??,?(??) + 1 ) + 1 ? ?? (−( ?ℎ ) 2 ?? 2 ?? +?? ??(??,??ℎ ? − ?? ,??,? ??) + ?? ??(?? − Πℎ ?? ,??,? ??) + 1 2?(?). (70) Integration with respect to ?, we obtain 1 2‖(??)‖2+ ? 2‖??,???‖2≤ ? 2‖??,???(0)‖2+1 2 ‖(??(0)‖2+ 251

  17. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 ? ? ? ?? + 1 2 2 ?ℎ2?∫ ‖?‖?+1 ? ?? +?ℎ2?∫ ‖??‖?+1 ?? + 2∫ ?(?) 0 ? 2(Πℎ ?2 − ?2,??,?(??) 2− (?ℎ ? )2),??,?(??) 1 ) +1 ) 2(−( ?ℎ ?? ?? + ? (??,??ℎ ? − ?? ,??,? ??) + ?(?? − Πℎ ?? ,??,? ??). (71) Cauchy-Schwartz -inequality and Young's inequality are used, and the results are: 1 2‖(??)‖2+ ? 2‖??,???‖2≤ ? 2‖??,???(0)‖2+ 1 2‖(??(0)‖2 ? ? 2 2 + ?ℎ2?∫ ‖?‖?+1 ? ?? + ?ℎ2?∫ ‖??‖?+1 ?? ? ? +1 ?? + 2?‖(??,? ?ℎ ? − ??) ‖2+ ? 8‖??,?(??)‖2 2∫ ?(?) 0 2 4‖??,? (??) + 2 ? ‖(?? − Πℎ ??) ‖2 + ? 8‖??,?(??)‖2 + 1 ‖ ?? 2 4‖??,? (??) + 1 4 ‖Πℎ?2− ?2‖2 +1 2− (?ℎ ?)2‖2+1 . (72) 4 ‖?ℎ ‖ ?? 2− (?ℎ ?)2‖2 is the non-negative Using Eq. (30), Eq. (31), and Eq. (32), with noting that ‖?ℎ term, we can obtain ‖(??)‖2+ ?‖??,???‖2≤ ?‖??,???(0)‖2+ ‖(??(0))‖2 2 ) + ‖??,? (??) ? ? ? 2 2 2 + ? ℎ2?(∫ ‖?‖?+1 + ∫ ?(?) 0 ??. ?? + ∫ ‖??‖?+1 ? ?? + ‖?‖ 1+? (73) ‖ ? ?? Similarly, for proof Eq. (57). 7. Numerical Results and Discussion We employ a uniform triangular mesh ?ℎtogether with a discrete weak space ?ℎ(??(?), ??(??)), where,? = 0,1. The Raviart-Thomas element ???, a space of discontinuous weak gradient, was utilized in [16] for the numerical experiments of WG-FEM and PWG-FEM for second-order elliptic problems. It consists of piecewise polynomials of order k on the triangles and edges, with ??+1(?) as the Raviart-Thomas element. We use the following norms to provide the numerical results of the inaccuracy between the ?2projection ?ℎ? of the precise solution and the numerical results ?ℎ: ?1 - semi-norm, 252

  18. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 1 2 ∥ ???ℎ∥= ( ∑ ∫   |???ℎ|2 .   ??) ? ?∈?ℎ ?2 - norm based on elements, 1 2   |?0|2??) . ∥ ?0∥= ( ∑ ∫    ? ?∈?ℎ ?2 - norm based on the edge, 1 2 ∥ ??∥= (∑ ℎ?∫ |??|2 .     ??) ?ℎ ?∈?ℎ Where, ℎ? is the diameter of edge ? and ?ℎ is the set of all edges in ?ℎ. We compute the error ? − ?ℎ of ?2-norm and ?1-norm of the PWG-FEM in the caseof SDPWG-FEMby using Matlab R2014a software. To demonstrate the WG-FEM and PWG-FEM for the time-dependent coupled Burgers' Eq. (1) and Eq. (2) throughout the time range [0,?] = [0,1] we give the test issue. Coupled Burgers' equation exact solutions [27] are: ?(?,?,?) =(? + ? − 2??) (1 − 2?2) ,?(?,?,?) =(? − ? − 2??) . (1 − 2?2) We selected the approximate space (?1,?1,??1) with the following normalized basis for ??1: {(1 ?),(?2 0),(? 0),(? ??),(?? 0),(0 1),(0 ?),(0 ?2)}. The results of test time step ? = 0.01 and diffusion coefficient ? = 0.0001 are displayed in Tables 1, 2, and Figure 1, and the convergence rate is in Figure 3, for the WG -FEM. When stability parameter ? = 0, and Tables 3,4, and Figure 2, and the convergence rate is in Figure 4, for the PWG -FEM when stability parameter ? = indicate the sequence of convergence about time step size ? and mesh size ℎ. Table 1, Table 2, Table 3, and Table 4 shows the Convergence rate for ? ??? ? with (?1,?1,??1) elements in case(diffusion coefficient ? = 0.0001) for weak Galerkin finite element method and Petrov weak Galerkin finite element method. The error in Table 3 and Table 4 for the Petrov weak Galerkin finite element method is much less than the error in Table 1 and Table 2 for the weak Galerkin finite element method. Our findings demonstrate that Table 3 and Table 4 of PWG-FEM are significantly more accurate than Table 1 and Table 2 of ℎ 32, convection coefficient ? = [1,1], to 253

  19. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 WG-FEM. We notice that Table 3 and Table 4 are free from oscillations when ? << ℎ. When comparing Table 1, Table 2, Table 3, and Table 4 for unsteady-states WG-FEM and PWG-FEM a significant improvement and regularity were observed in the numerical results of the PWG- FEM compared to the numerical results for WG-FEM. Table 1?2 and ?1error for ?in the caseof WG-FEM ℎ ?1−error Order Order ?2−error 6.1042e-01 8.7218e-02 1 2 1 4 1 8 1 16 1 32 Table 2?2 and ?1error for ?in the caseof WG-FEM 2.3889e-01 1.3535 1.6219e-02 2.4270 1.1148e-01 1.0996 3.7707e-03 2.1047 5.4706e-02 1.0270 9.2590e-04 2.0259 2.7220e-02 1.0070 2.3043e-04 2.0065 Order Order ?1−error ℎ ?2−error 2.6529e-01 3.4014e-02 1 2 1 4 1 8 1 16 1 32 9.5527e-02 1.4736 5.5810e-03 2.6075 4.2445e-02 1.1703 1.2490e-03 2.1597 2.0483e-02 1.0512 3.0392e-04 2.0390 1.0145e-02 1.0136 7.5472e-05 2.0097 Table 3?2 and ?1 error for ?in the caseof PWG-FEM ℎ ?1−error Order Order ?2−error 3.0521e-01 4.3609e-02 1 2 1 4 1 8 1 16 1 32 1.1724e-01 1.3802 9.1095e-03 2.4270 5.5740e-02 1.0727 2.2569e-03 2.0130 1.8235e-02 1.3267 4.6295e-04 2.0259 7.6100e-03 1.2607 1.0521e-04 2.1375 254

  20. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 Table 4?2 and ?1error for ?in the caseof PWG-FEM Order Order ℎ ?1−error ?2−error 1.3264e-01 1.7007e-02 1 2 1 4 1 8 1 16 1 32 5.7763e-02 1.1993 1.5603e-03 1.9604 3.0712e-02 0.9113 4.1633e-04 1.9060 1.1941e-02 1.3627 1.0030e-04 2.0533 5.0725e-03 1.2352 2.2157e-05 2.1785 Figure 1 and Figure 2 show numerical and exact solutions for u and v in case (end time T=1, the time step τ = 0.01 and diffusion coefficient ϵ=0.0001) for weak Galerkin finite element method and Petrov weak Galerkin finite element method respectively. Note that the unsteady-state WG- FEM the exact solution and numerical results are not consistent in Figure 1, while the unsteady- state PWG-FEM, the exact solution, and numerical results are consistent in Figure 2. This indicates that Figure 2 of the Petrov weak Galerkin finite element method is free of oscillations, in contrast to Figure 1 of the weak Galerkin finite element method that contains oscillations. Our findings demonstrate that Figure 2 of PWG-FEM is significantly more accurate than Figure 1 for WG-FEM. A substantial improvement and regularity were seen in the numerical results in Figure 2 of PWG-FEM when compared to the numerical results in Figure 1 of WG-FEM. The oscillation and inaccuracy that occur when using the weak Galerkin finite element method have been eliminated by using Petrov weak Galerkin finite element method. 255

  21. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 Fig. 1. Numerical and exact solutions for u and v in case(? = 1,? = 0.01,? = 0.0001) for the WG -FEM Fig. 2. Numerical and exact solutions for u and v in case(? = 1,? = 0.01,? = 0.0001) for the PWG -FEM 256

  22. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 Figure 3 and Figure 4 show the Convergence rate in the weak Galerkin finite element method and Petrov weak Galerkin finite element method respectively, for diffusion coefficient ? = 0.0001 in ?2− norm. Through the practical results between the error and its order, we note that there is an agreement between the theoretical proof and the numerical results by drawing the relationship between the error and its order. It is clear through the use of practical results that the error order is approaching the value 2 of the weak Galerkin finite element method and Petrov weak Galerkin finite element method. The numerical result showed that the PWG- FEM ?2 and ?1 convergence orders are better than the WG-FEM order. Fig. 3. Convergence rate in the WG-FEM for ? = 0.0001 in ?2− norm Fig. 4. Convergence rate in the PWG-FEM for ? = 0.0001 in ?2− norm 257

  23. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 8. Conclusion In this paper, we consider the PWG-FEM for solving coupled Burgers' equations in two dimensions. We proved the stability and the optimal order error in ?2- norm ??? ?1- norm in the case of semi-discrete Petrov weak Galerkin finite element method. Our findings demonstrate that the PWG-FEM is significantly more accurate than the WG-FEM, we also provide the convergence rates for PWG-FEM and WG-FEM. The numerical result showed that the PWG- FEM ?2 and ?1 convergence orders are better than the WG-FEM order. We demonstrated consistency between the exact solutions and numerical outcomes for unsteady-state in PWG- FEM. When comparing the unsteady-states PWG-FEM and WG-FEM a significant improvement and regularity were observed in the numerical results of the PWG-FEM compared to the numerical results of the WG-FEM. The Stability of the PWG-EFM is dependent on the stability parameter ?. References [1] Fletcher, Clive A J. “Generating Exact Solutions of the Two-Dimensional Burgers’ Equations.” International Journal for Numerical Methods in Fluids 3, (1983): 213–16. https://doi.org/10.1016/j.ast.2012.02.006 [2] Burgers, Johannes Martinus. “A Mathematical Model Illustrating the Theory of Turbulence.” Advances in Applied Mechanics 1, (1948): 171–99. https://doi.org/10.1016/j.ast.2012.02.006 [3] Cole, Julian D. “On a Quasi-Linear Parabolic Equation Occurring in Aerodynamics.” Quarterly of Applied Mathematics 9, No.3, (1951): 225–36. https://doi.org/10.1090/qam/42889 [4] Hussein, Ahmed Jabbar, and Hashim A Kashkool. “A Weak Galerkin Finite Element Method for Two-Dimensional Coupled Burgers’ Equation by Using Polynomials of Order (k, k–1, k–1).” Journal of Interdisciplinary Mathematics 23, No. 4, (2020): 777–90. https://doi.org/10.1080/09720502.2019.1706844 [5] Kashkool, Hashim A. “The Semi Discrete Formulation of Galerkin and Galerkin- Conservation Finite Element Methods for Two Dimensional Coupled Burgers’ Problem.” Basrah Journal of Science 33, (2015): (1A). [6] Wang, Haifeng, Da Xu, Jun Zhou, and Jing Guo. “Weak Galerkin Finite Element Method for a Class of Time Fractional Generalized Burgers’ Equation.” Numerical Methods for Partial Differential Equations 37, No.1, (2021): 732–49. https://doi.org/10.1002/num.22549 [7] Hussein, Ahmed Jabbar. “A Weak Galerkin Finite Element Method for Solving Time- Fractional Coupled Burgers’ Equations in Two Dimensions.” Applied Numerical Mathematics 156,( 2020): 265–75. https://doi.org/10.1016/j.apnum.2020.04.016 [8] Demkowicz, L, and J T Oden. “An Adaptive Characteristic Petrov-Galerkin Finite Element Method for Convection-Dominated Linear and Nonlinear Parabolic Problems in One Space Variable.” Journal of Computational Physics 67, No. 1,( 1986): 188–213. https://doi.org/10.1016/0021-9991(86)90121-X 258

  24. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 [9] Westerink, J J, and D Shea. “Consistent Higher Degree Petrov–Galerkin Methods for the Solution of the Transient Convection–Diffusion Equation.” International Journal for Numerical Methods in Engineering 28, No.5, (1989): 1077–1101. https://doi.org/10.1002/nme.1620280507 [10] Keshaish, Dhiaa A, and Hashim A Kashkool. “Petrov-Galerkin Finite Element Method for Convection-Diffusion-Reaction Problem.” Journal of Basrah Researches (Sciences) 46, No.2,( 2020):136-152. [11] Saadoon, Jawad Jaaywel, and Hashim Abdul-Khaliq Kashkool. “The Error Analysis of Linearized Discontinuous Galerkin Finite Element Method for Incompressible Miscible Displacement in Porous Media.” Journal of Interdisciplinary Mathematics 22, No. 8, (2019): 1471–84. https://doi.org/10.1080/09720502.2019.1706845 [12] Kashkool, Hashim Abdul-Khaliq, and Jawad Jaaywel Saadoon. 202 “Hp-Discontinuous Galerkin Finite Element Method for Incompressible Miscible Displacement in Porous Media.” In Journal of Physics: Conference Series, 1530, (2020): 1-18. https://doi.org/10.1088/1742-6596/1530/1/012001 [13] Kashkool, Hashim A, and Mohammed W AbdulRidha. “The Error Analysis for the Discontinuous Galerkin Finite Element Method of the Convection-Diusion Problem.” Journal of Basrah Researches (Sciences) 45, no. 2A,( 2019): 88–107. [14] AbdulRidha, Mohammed Waleed, and Hashim A Kashkool. “Space-Time Petrov- Discontinuous Galerkin Finite Element Method for Solving Linear Convection-Diffusion Problems.” In Journal of Physics: Conference Series, 2322, No. 1, (2022): 1-14. https://doi.org/10.1088/1742-6596/2322/1/012007 [15] AbdulRidha, Mohammed Waleed, Hashim A Kashkool and Ali Hasan Ali."Petrov- Discontinuous Galerkin Finite Element Method for Solving Diffusion-Convection Problems." Ital. J. Pure Appl. Math. in press(2023). [16] Wang, Junping, and Xiu Ye. “A Weak Galerkin Finite Element Method for Second-Order Elliptic Problems.” Journal of Computational and Applied Mathematics ,241, No.1, (2013): 103-115. https://doi.org/10.1016/j.cam.2012.10.003 [17] Ali, Ali Hasan, Ahmed Shawki Jaber, Mustafa T Yaseen, Mohammed Rasheed, Omer Bazighifan, and Taher A Nofal. “A Comparison of Finite Difference and Finite Volume Methods with Numerical Simulations: Burgers Equation Model.” Complexity 2022, (2022). https://doi.org/10.1155/2022/9367638 [18] Arshad, Uroosa, Mariam Sultana, Ali Hasan Ali, Omar Bazighifan, Areej A Al-moneef, and Kamsing Nonlaopon. “Numerical Solutions of Fractional-Order Electrical RLC Circuit Equations via Three Numerical Techniques.” Mathematics 10, No. 17, (2022): 3071. https://doi.org/10.3390/math10173071 [19] Sultana, Mariam, Uroosa Arshad, Ali Hasan Ali, Omar Bazighifan, Areej A Al-Moneef, and Kamsing Nonlaopon. “New Efficient Computations with Symmetrical and Dynamic Analysis for Solving Higher-Order Fractional Partial Differential Equations.” Symmetry 14, No. 8, (2022): 1653. https://doi.org/10.3390/sym14081653 [20] Ali, Ali Hasan, Ghazi Abed Meften, Omar Bazighifan, Mehak Iqbal, Sergio Elaskar, and Jan Awrejcewicz. “A Study of Continuous Dependence and Symmetric Properties of Double Diffusive Convection: Forchheimer Model.” Symmetry 14, No.4, (2022): 682. https://doi.org/10.3390/sym14040682 259

  25. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 [21] Meften, Ghazi Abed, and Ali Hasan Ali. “Continuous Dependence for Double Diffusive Convection in a Brinkman Model with Variable Viscosity.” Acta Universitatis Sapientiae, Mathematica 14, No. 1, (2022): 125–46. https://doi.org/10.2478/ausm-2022-0009 [22] Hussein, A J, and H A Kashkool. “L2-OPTIMAL ORDER ERROR FOR TWO- DIMENSIONAL COUPLED BURGERS’EQUATIONS BY WEAK GALERKIN FINITE ELEMENT METHOD.” TWMS Journal of Applied and Engineering Mathematics 12, No. 1, (2022): 34. [23] Zhao, Jijing, Fuzheng Gao, and Hongxing Rui. “The Weak Galerkin Method for the Miscible Displacement of Incompressible Fluids in Porous Media on Polygonal Mesh.” Applied Numerical Mathematics 185, (2023): 530–48. https://doi.org/10.1016/j.apnum.2022.12.012 [24] Perella, Andrew James. “A Class of Petrov-Galerkin Finite Element Methods for the Numerical Solution of the Stationary Convection-Diffusion Equation.” Durham University, (1996). [25] Kashkool, Hashim A, and Ahmed J Hussein.“Error Estimate for Two-Dimensional Coupled Burgers’ Equations By Weak Galerkin Finite Element Method.” In Journal of Physics: Conference Series, 1530, (2020):12065. https://doi.org/10.1088/1742-6596/1530/1/012065 [26] Thomée, Vidar. 2007. Galerkin Finite Element Methods for Parabolic Problems. Vol. 25. Springer Science & Business Media. [27] Zhu, Hongqing, Huazhong Shu, and Meiyu Ding. “Numerical Solutions of Two- Dimensional Burgers’ Equations by Discrete Adomian Decomposition Method.” Computers & Mathematics with Applications 60, No.3, (2010): 840–48. https://doi.org/10.1016/j.camwa.2010.05.031 260

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