Solution of generalized Riemann problem for hyperbolic p− system with damping

Journal Pre-proof Solution of generalized Riemann problem for hyperbolic 𝑝−system with damping Rahul Kumar Chaturvedi, Pooja Gupta, L.P. Singh

PII: DOI: Reference:

S0020-7462(19)30301-4 https://doi.org/10.1016/j.ijnonlinmec.2019.07.014 NLM 3229

To appear in:

International Journal of Non-Linear Mechanics

Received date : 2 May 2019 Revised date : 24 July 2019 Accepted date : 25 July 2019 Please cite this article as: R.K. Chaturvedi, P. Gupta and L.P. Singh, Solution of generalized Riemann problem for hyperbolic 𝑝−system with damping, International Journal of Non-Linear Mechanics (2019), doi: https://doi.org/10.1016/j.ijnonlinmec.2019.07.014. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Journal Pre-proof

Solution of generalized Riemann problem for hyperbolic 𝑝−system with damping

Rahul Kumar Chaturvedia,∗, , Pooja Guptaa and L P Singha a Department

India.

of Mathematical Sciences, Indian Institute of Technology, Banaras Hindu University, Varanasi 221005,

ABSTRACT

Keywords: Hyperbolic system Generalized Riemann problem Exact solutions Differential constraint method

The aim of this paper is to determine the exact solution of the generalized Riemann problem for the 2 × 2 hyperbolic 𝑝−system with linear damping. Differential constraint method is used to find the solution of the system of governing equations. Also, consistency conditions and constraint equations for the hyperbolic system are obtained. Further, we solve the generalized Riemann problem of non-homogeneous hyperbolic 𝑝−system which involves discontinuous initial data.

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ARTICLE INFO

1. Introduction

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In the present paper, we consider the model, hyperbolic 𝑝−system with linear damping and determine the exact solution of generalized Riemann problem using differential constraint method. The governing equations describing the hyperbolic 𝑝−system with linear damping of a compressible flow, in the Lagrangian coordinate is given by [1, 2] { 𝜌𝑡 − 𝑣𝑥 = 0, (1) 𝑣𝑡 + 𝑝(𝜌)𝑥 = −𝛼𝑣,

Jo

where (𝑥, 𝑡) ∈ 𝑅 × 𝑅+ . 𝑥 and 𝑡, respectively, represent space and time coordinates. 𝑣(𝑥, 𝑡) and 𝜌(𝑥, 𝑡) > 0 are the velocity and the specific volume respectively. The function 𝑝(𝜌) (pressure) is the smooth function such that 𝑝(𝜌) > 0 and 𝑝′ (𝜌) < 0. The term −𝛼𝑣 which appears in the momentum equation represents a linear damping with 𝛼 > 0. In the present paper, we consider a particular case that the pressure 𝑝 is given by a 𝛾-law with 𝛾=1 as [3, 4] 𝑝(𝜌) = 𝜌1 . The system (1) is considered by several authors in past decades and various results have been examined. The authors Mei [1, 5], Nishihara [2] etc. have discussed the behavior of solution of system (1). In the present paper we use the method of differential constraints to obtain the exact solution of generalized Riemann problem for the hyperbolic system (1). In past decades several types of mathematical methods have been used to obtain the exact or approximate analytical solution of the system of PDEs. For example, perturbation method, similarity method, differential constraints method etc.. Among these methods the differential constraints method is of great interest because ∗ Corresponding

author [email protected] (R.K. Chaturvedi) ORCID (s):

RK Chaturvedi et al.: Preprint submitted to Elsevier

Page 1 of 8

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of its systematic procedure to find the exact solution of hyperbolic system of PDEs. This method is applicable to the system of PDEs and it was first used by Janenko [6] in the gasdynamics regime. Curro et al. [7, 8, 9] have discussed the exact solutions of non-homogeneous and homogeneous 2×2 hyperbolic system using differential constraints method. The exact solution describing soliton-like interaction in a non dispersive medium is obtained by Seymour and Varley [10]. Further, C. Curro et al. [11, 12] have studied the exact description of simple wave for multicomponent hyperbolic system. Within this theoretical framework, differential constraint method is used for solving generalized Riemann problem for hyperbolic 𝑝−system with linear damping. In recent years, Curro et al. [13, 14, 15], Manganaro [16], Sahoo et al. [17] have discussed the solution of Riemann problem as well as generalized Riemann problem for hyperbolic balance law. Chaiyasena et al. [18] have studied the generalized Riemann problem and their adjoinment through a shock wave. Shekhar et al. [19], Singh et al. [20], Kuila et al. [21], Gupta et al. [22] have studied the classical Riemann problem for several relevant cases. The study of generalized Riemann problem (GRP) involves complexities and requires more attentions. Since the analytical expressions for the exact or approximate solution of GRP, in the case of quasilinear hyperbolic system, are usually not available. In order to find the solution of GRP, authors have used this method for several models like 𝑝−system with relaxation ([23]), rate-type material ([17]), fast diffusion equations ([24]), traffic flow model ([13, 14]), ET6 model ([25]), non-linear transmission lines ([15]), non-linear diffusion-convention equations ([26]) etc. The rest of the paper is organized as: section 2 contains the procedure of the method to find the exact solution of system (1). section 3 contains the brief analysis of model taken in this paper. Section 4 sketches the phenomenon of GRP and the exact solution of GRP is obtained. In last section 5, we discuss the results and conclusions of the problem. 2. Differential constraint method

In this section for our convenience, we discuss the important steps of differential constraint method, described by Curro et al. [23], which will be helpful to obtain the solution of system (1). We consider the one-dimensional hyperbolic system of quasilinear PDEs 𝑈𝑡 + 𝑀(𝑈 )𝑈𝑥 = 𝑁(𝑈 ),

(2)

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where 𝑈 ∈ 𝑅𝑛 , 𝑀(𝑈 ) and 𝑁(𝑈 ) are column vectors representing field variables, the coefficient matrix and column vector representing source terms respectively. Since the system (2) is hyperbolic in nature, therefore the matrix 𝑀𝑛×𝑛 has 𝑛 real eigenvalues 𝜆(𝑖) , 𝑖 = 1, 2, ....𝑛., and corresponding to each eigenvalues 𝜆(𝑖) we have 𝑛 linearly independent left eigenvectors 𝐿(𝑖) and 𝑛 right eigenvectors 𝑅(𝑖) . Let us consider that the system (2) is strictly hyperbolic system i.e. 𝜆(𝑖) ≠ 𝜆(𝑘) , ∀𝑖, 𝑘 = 1, 2, ...𝑛, 𝑖 ≠ 𝑘. Let us assume that { 0 if 𝑖 ≠ 𝑘, (𝑖) (𝑘) 𝐿 .𝑅 = (3) 1 if 𝑖 = 𝑘. We append the set of 𝑛 − 1 differential constraints of first order as [15] 𝐿(𝑖) ⋅ 𝑈𝑥 = 𝑄(𝑖) (𝑥, 𝑡, 𝑈 ),

𝑖 = 1, 2, ....., 𝑛 − 1,

RK Chaturvedi et al.: Preprint submitted to Elsevier

(4) Page 2 of 8

Journal Pre-proof where the function 𝑄(𝑖) is an unspecified function which will be determined during the reduction process. The consistency conditions for the arbitrary function 𝑄(𝑖) can be determined from (2) and (4), given as +𝜆 +

(𝑖)

𝑄(𝑖) 𝑥

∑∑ 𝑛−1 𝑛−1

+

𝑛−1 ∑

( ) 𝑄(𝑘) 𝐿(𝑖) (∇𝑅(𝑘) 𝑁 − ∇𝑁𝑅(𝑘) ) + 𝑄(𝑖) ∇𝜆(𝑖) 𝑅(𝑘)

𝑘=1

(

𝑄(𝑗) 𝑄(𝑘) (𝜆(𝑗) − 𝜆(𝑖) )𝐿(𝑖) ∇𝑅(𝑗) 𝑅(𝑘) + ∇𝑄(𝑖)

𝑁−

(𝑖)

(5)

) 𝑄(𝑗) (𝜆(𝑗) − 𝜆(𝑖) )𝑅(𝑗)

= 0,

𝑗=1

(𝑛)

(𝑖)

(𝜆 − 𝜆 )∇𝑄 𝑅

(𝑛)

+

𝑛−1 ∑

pro

𝑗=1 𝑘=1

𝑛−1 ∑

of

𝑄𝑡(𝑖)

𝑄(𝑘) (𝜆(𝑘) − 𝜆(𝑛) )𝐿(𝑖) (∇𝑅(𝑘) 𝑅(𝑛) − ∇𝑅(𝑛) 𝑅(𝑘) )

𝑘=1

(6)

+ 𝐿(𝑖) (𝑅(𝑛) 𝑁 − ∇𝑁𝑅(𝑛) ) + 𝑄(𝑖) ∇𝜆(𝑖) 𝑅(𝑛) = 0,

(𝑛)

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where ∇ = 𝜕∕𝜕𝑈 , 𝑖 = 1, 2, ...𝑛 − 1. Once the function 𝑄(𝑖) satisfies the consistency conditions (5) and (6), the systems (2) and (4) become compatible. Then we can obtain a class of exact solution of the given system of governing PDEs by direct integration of overdetermined system (2) and (4). The overdetermined constraint equations and consistency conditions help to rewrite the original system in to a partial decoupled form so that we can solve it and obtain exact solution which satisfies the original system and constraint equations. The reduced system is written in the following form 𝑈𝑡 + 𝜆 𝑈𝑥 = 𝑁 +

𝑛−1 ∑

( ) 𝑄(𝑖) 𝜆(𝑛) − 𝜆(𝑖) 𝑅(𝑖) .

(7)

𝑖=1

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Once 𝑄(𝑖) satisfies the consistency conditions, the solution of the systems (2), (4) can be obtained by direct integration of the equations (7) along with the constraint equations (4). Furthermore since the constraint equations (4) are involutive, they will characterize the set of initial conditions compatible with the class of solution under interest. Also, it is noticeable from (2) and (4) that when the source term 𝑁(𝑈 ) = 0 and 𝑄(𝑖) = 0, system (7) results in a classical wave solution to a hyperbolic system. Therefore, the obtained solution of (2) characterized by (7) generalizes the simple wave of the homogeneous case taking into account source-like effects in hyperbolic wave process. For more details one may refer [8, 13, 15, 23]. 3. Exact solution

In this section, we use the above mentioned method to obtain the exact solution of the hyperbolic system (1) describing 𝑝−system with linear damping. The system (1) can be written in the form of (2), where [ ] [ ] [ ] 𝜌 0 −1 0 𝑈= , 𝑀(𝑈 ) = ′ , 𝑁(𝑈 ) = . (8) 𝑣 𝑝 (𝜌) 0 −𝛼𝑣 RK Chaturvedi et al.: Preprint submitted to Elsevier

Page 3 of 8

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where 𝜆0 = ±1. and corresponding eigenvectors are

[

1 √ 2 −𝑝′ (𝜌) 1 2

]

, 𝑅(2) =

pro

𝐿(1)

] [ √ ] [√ (2) ′ ′ = −𝑝 (𝜌) 1 , 𝐿 = − −𝑝 (𝜌) 1 , 𝑅(1) =

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Consider 𝜆(𝑖) , 𝑖 = 1, 2 are eigenvalues of the coefficient matrix 𝑀2×2 , 𝑅(𝑖) and 𝐿(𝑖) are the right and left eigenvectors of the corresponding eigenvalues, 𝜆(𝑖) , 𝑖 = 1, 2, defined as √ √ 𝜆(2) = −𝑝′ (𝜌) 𝜆(1) = − −𝑝′ (𝜌), which can be written as √ (𝑖) 𝜆 = 𝜆0 −𝑝′ (𝜌), (9)

[

−

1 √ 2 −𝑝′ (𝜌) 1 2

]

, (10)

respectively. From equation (4), the required differential constraint associated to the system (1) can be written in the form 𝑣𝑥 + 𝜆(𝜌)𝜌𝑥 = 𝑄(𝑥, 𝑡, 𝜌, 𝑣).

(11)

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Substituting equations (8), (9), (10) and (11) in the consistency conditions (5) and (6), we obtain the following consistency equations

𝑄𝜌 + 𝜆𝑄𝑣 = −

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𝑄𝑡 + 𝜆𝑄𝑥 − 𝛼𝑣𝑄𝑣 + 𝑄(𝜆𝑄𝑣 + 𝑄𝜌 ) = −𝛼𝑄, 1 (𝜆𝛼 + 𝜆′ 𝑄), 2𝜆

(12) (13)

where the prime denotes the differentiation. Once 𝑄(𝑥, 𝑡, 𝜌, 𝑣) is obtained by (12) and (13), from (7) we obtain the solution of the system (1) and (11) by integrating the following set of constraints along with (11) { 𝜌𝑡 + 𝜆𝜌𝑥 = 𝑄, (14) 𝑣𝑡 + 𝜆𝑣𝑥 = −(𝛼𝑣 + 𝜆𝑄).

Prior to determine the exact solution of the system (14), we determine the arbitrary function 𝑄(𝑥, 𝑡, 𝜌, 𝑣) from the consistency equations (12) and (13). On integrating (13), we obtain the following general solution of (13)

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𝑄 = −𝛼𝜌 + 𝜌1∕2 𝜙(𝑥, 𝑡, 𝑚),

(15)

where 𝑚 = 𝑣 − 𝜆0 log 𝜌 and 𝜙(𝑥, 𝑡, 𝑚) is an arbitrary function. Next the substitution of (15) into (12) yields 𝛼 1 𝜙𝑡 + 𝜆𝜙𝑥 − 𝛼𝑣𝜙𝑚 − 𝜌−1∕2 𝜙2 − 𝜙 = 0. 2 2

(16)

The last step of the process is to determine 𝜙(𝑥, 𝑡, 𝑚) which satisfies (16) for all 𝜌. It is noticeable that 𝜙 = 0 must satisfy the above equation (16) for all 𝜌 which implies that 𝜙 = 0 must be a solution RK Chaturvedi et al.: Preprint submitted to Elsevier

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of

of equation (16). On the insertion in (15), we find that 𝑄 = −𝛼𝜌. Now we look for a special solution of the compatibility conditions (12) and (13) under the form 𝑄 = 𝑄(𝜌). On substituting it in (12) and (13), and solving the resulting equations we obtain that 𝑄 = −𝛼𝜌 is the solution of the consistency conditions (12) and (13) as well as 𝑝(𝜌) = 𝑐1 ∕𝜌 + 𝑐2 , where 𝑐1 and 𝑐2 are arbitrary constants. Substituting this to the system (14) specialize to { 𝜌𝑡 + 𝜆𝜌𝑥 = −𝛼𝜌, (17) 𝑣𝑡 + 𝜆𝑣𝑥 = −𝛼𝑣 + 𝜆𝛼𝜌. along with (11)

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𝑣𝑥 + 𝜆(𝜌)𝜌𝑥 = −𝛼𝜌.

(18)

⎧ −𝛼𝑡 ⎪𝜌(𝑧, 𝑡) = 𝜌0 (𝑧)𝑒 , ⎪𝑣(𝑧, 𝑡) = 𝑣0 (𝑧)𝑒−𝛼𝑡 + 𝜆0 (1 − 𝑒−𝛼𝑡 ), ⎨ ⎪𝑧 = 𝑥 + 𝜆0 (1 − 𝑒𝛼𝑡 ) . ⎪ 𝛼𝜌0 ⎩

re-

Now, we integrate the system (18) with initial data 𝜌(𝑥, 0) = 𝜌0 (𝑥) and 𝑣(𝑥, 0) = 𝑣0 (𝑥) gives the following solution of the system (1)

(19)

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Within the context of differential constraint method, we have obtained the solution (19) of the system (1) for arbitrary initial data 𝜌0 (𝑥) and 𝑣0 (𝑥). Now we are interested to introduce the initial conditions in the next section which are known as GRP. In next section we solve the GRP to determine the exact solution for the model (1). 4. Generalized Riemann problem

The purpose of this section is to discuss how the differential constraint method helps to solve the generalized Riemann problem of interest in 2 × 2 hyperbolic 𝑝−system with damping. The generalized Riemann problem for the system (1) with the initial data can be written as { (𝜌𝑙 (𝑥), 𝑣𝑙 (𝑥)), for 𝑥 < 0, 𝑈 (𝑥, 0) = (20) (𝜌𝑟 (𝑥), 𝑣𝑟 (𝑥)), for 𝑥 > 0,

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where 𝜌𝑟 (𝑥), 𝑣𝑟 (𝑥), 𝜌𝑙 (𝑥) and 𝑣𝑙 (𝑥) are arbitrary functions. We set 𝜌𝐿 = lim− 𝜌𝑙 (𝑥), 𝑥⟶0

𝑣𝐿 = lim− 𝑣𝑙 (𝑥),

𝜌𝑅 = lim+ 𝜌𝑟 (𝑥), 𝑥⟶0

𝑣𝑅 = lim+ 𝑣𝑟 (𝑥), 𝑥⟶0

𝑥⟶0

(21)

where 𝜌𝐿 ≠ 𝜌𝑅 and 𝑣𝐿 ≠ 𝑣𝑅 . In order to solve the GRP (20), we consider the solution (20) and substitute this into the constraint (18) which gives the following constraint equation for the initial conditions 𝑑𝜌0 (𝑧) 𝑑𝑣0 (𝑧) + 𝜆0 𝜌−1 = −𝛼𝜌0 (𝑧). 0 𝑑𝑧 𝑑𝑧

RK Chaturvedi et al.: Preprint submitted to Elsevier

(22) Page 5 of 8

Journal Pre-proof According to method of differential constraints, the initial data (20) satisfies the constraint equation (22). Then on integrating (22) along with the initial data (20) yields the following implicit forms 𝑥 ⎧𝑣 (𝑥) = 𝑣 (𝑥) − 𝜆 (ln 𝜌 (𝑥) − ln 𝜌 ) − 𝛼 𝜌 (𝜉)𝑑𝜉, 𝐿 0 𝑙 𝐿 ⎪ 𝑙 ∫0 𝑙 𝑥 ⎨ ( ) ⎪𝑣𝑟 (𝑥) = 𝑣𝑅 (𝑥) − 𝜆0 ln 𝜌𝑟 (𝑥) − ln 𝜌𝑅 − 𝛼 𝜌 (𝜉)𝑑𝜉, ⎩ ∫0 𝑟

(23)

re-

along with ) 𝜆 ( 𝑧 = 𝑥 + 𝛼𝜌0 1 − 𝑒𝛼𝑡 . 𝑙 Case II- For 𝑥 > 0, we get { 𝜌(𝑧, 𝑡) = 𝜌𝑟 (𝑧)𝑒−𝛼𝑡 , 𝑣(𝑧, 𝑡) = 𝑣𝑟 (𝑧)𝑒−𝛼𝑡 − 𝜆0 (1 − 𝑒−𝛼𝑡 ),

pro

of

Now we characterize the solution (19) to solve the governing system (1) along with (20). Case I- For 𝑥 < 0, we get { 𝜌(𝑧, 𝑡) = 𝜌𝑙 (𝑧)𝑒−𝛼𝑡 , 𝑣(𝑧, 𝑡) = 𝑣𝑙 (𝑧)𝑒−𝛼𝑡 − 𝜆0 (1 − 𝑒−𝛼𝑡 ),

(24)

(25)

𝑥𝑙 (𝑡) =

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along with ) 𝜆 ( 𝑧 = 𝑥 + 𝛼𝜌0 1 − 𝑒𝛼𝑡 . 𝑟 Since 𝑣𝑙 (𝑥) and 𝑣𝑟 (𝑥) are given by (23) and we obtain that 𝑧 = 𝑥 at 𝑡 = 0. Therefore the solution (24) exists for 𝑧 < 0, i.e. 𝑥 < 𝑥𝑙 (𝑡), where ) 𝜆0 ( 𝛼𝑡 𝑒 −1 . 𝛼𝜌𝑙

(26)

) 𝜆0 ( 𝛼𝑡 𝑒 −1 . 𝛼𝜌𝑟

(27)

Similarly, the solution (25) exists for 𝑥 > 𝑥𝑟 (𝑡), where 𝑥𝑟 (𝑡) =

In order to connect smoothly the left state (24) to the right state (25), we integrate the differential constraints (17) subject to the following initial conditions

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𝑥(0) = 0, 𝜌 = 𝑆(𝑎), 𝑣 = 𝑉 (𝑎), ∀ 𝑎 ∈ [0, 1], 𝑆(0) = 𝜌𝐿 , 𝑆(1) = 𝜌𝑅 , 𝑉 (0) = 𝑣𝐿 , 𝑉 (1) = 𝑣𝑅 .

(28)

Then the solutions are

⎧ −𝛼𝑡 ⎪𝜌(𝑥, 𝑡) = 𝑆(𝑎)𝑒 , ( ) ⎪𝑣(𝑥, 𝑡) = 𝑉 (𝑎)𝑒−𝛼𝑡 − 𝜆0 1 − 𝑒−𝛼𝑡 , ⎨ ⎪𝑥 = − 𝜆0 (𝑒𝛼𝑡 − 1) . ⎪ 𝛼𝑆(𝑎) ⎩ RK Chaturvedi et al.: Preprint submitted to Elsevier

(29)

Page 6 of 8

Journal Pre-proof On substitution of (29) in (18), yields (30)

𝑉 ′ = 𝜆0 𝑆 −1 𝑆 ′ . On integrating (30) along the conditions (28), we get ( ) 𝑉 (𝑎) = 𝑣𝐿 + 𝜆0 ln 𝑆(𝑎) − ln 𝜌𝐿 ,

(31)

of

along with 𝑣𝐿 − 𝜆0 ln 𝜌𝐿 = 𝑣𝑅 − 𝜆0 ln 𝜌𝑅 .

(32)

re-

pro

Now, in order to confirm the existence of the solution (29), we need 𝑑𝜆∕𝑑𝑎 > 0 which implies that 𝜌𝐿 > 𝜌𝑅 if 𝜆0 = 1 and 𝜌𝐿 < 𝜌𝑅 if 𝜆0 = −1. Therefore the central state (29) exists in the region 𝑥𝑙 (𝑡) ≤ 𝑥(𝑡) ≤ 𝑥𝑟 and it connects the left state (26) with right state (27) provided that the condition (32) is satisfied. On eliminating 𝑆(𝑎) from (29)1 and (29)3 and substituting (31) in (29)2 yields the required result { ) 𝜆 ( 𝜌(𝑥, 𝑡) = 𝛼𝑥0 𝑒−𝛼𝑡 − 1 , )) ( ( (33) 𝑣(𝑥, 𝑡) = 𝑣𝐿 + 𝜆0 ln 𝑆(𝑎) − ln 𝜌𝐿 𝑒−𝛼𝑡 − 𝛼𝑥𝑆(𝑎)𝑒−𝛼𝑡 ,

5. Conclusion

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) 𝜆 ( where 𝑆(𝑎) = 𝛼𝑥0 1 − 𝑒𝛼𝑡 . Equation (33) is required solution of GRP for the governing system (1). In the present work, we consider the hyperbolic system of first order PDEs describing 𝑝−system with linear damping and use differential constraint method to characterize the solution of this model. Within the framework of this method, we have obtained the consistency conditions for the governing model. Compatibility of the solution of the constraint equation and governing system is also discussed. The solution of the governing model is obtained in section 3 which is characterized for an arbitrary function as initial data to determine the exact solution for generalized Riemann problem. The solution obtained for the GRP consists of two states; left state and right state which are connected smoothly by the solution. This solution describes the rarefaction wave of the homogeneous case. 6. Acknowledgments

References

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The authors are grateful to the referee for the valuable suggestions and comments to improve the quality of the manuscript. [1] M. Mei, Best asymptotic profile for hyperbolic p-system with damping, SIAM Journal on Mathematical Analysis 42 (1) (2010) 1–23. [2] K. Nishihara, T. Yang, Boundary effect on asymptotic behaviour of solutions to the p-system with linear damping, journal of differential equations 156 (2) (1999) 439–458. [3] T. Luo, R. Natalini, T. Yang, Global bv solutions to a p-system with relaxation, Journal of Differential Equations 162 (1) (2000) 174–198. [4] T. Yang, C. Zhu, Existence and non-existence of global smooth solutions for p-system with relaxation, Journal of Differential Equations 161 (2) (2000) 321–336.

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Journal Pre-proof Highlights A generalized Riemann problem for 2x2 hyperbolic system with damping is analyzed.

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Differential constraint method is used to obtain the exact solution of hyperbolic p-system.

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A generalized Riemann data is taken as discontinuous initial condition.

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Exact solution of generalized Riemann problem is obtained.

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Behavior of solution profiles of generalized Riemann problem is discussed.

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