On the evolution of solutions of Burgers equation on the positive quarter-plane

Abstract: In this paper we investigate an initial-boundary value problem for the Burgers equation on the positive quarter-plane; vt + vvx − vxx = 0, x > 0, t > 0, v(x, 0) = u+, x > 0, v(0, t) = ub , t > 0, where x and t represent distance and time, respectively, and u+ is an initial condition, ub is a boundary condition which are constants (u+ ≠ ub). Analytic solution of above problem is solved depending on parameters (u+ and ub) then compared with numerical solutions to show there is a good agreement with each solutions.


Introduction
An initial and boundary value problem for Burgers' equation on the positive quarter-plane will be examined in this section. The distinct equation to be discussed is given by v t + vvx − vxx = 0, x > 0, t > 0, (1) v(x, 0) = u+, x > 0, (2) v(0, t) = u b , t > 0, and u+ is an initial condition and u b is a boundary condition (u b ≠ u+) which are both integers. Later on, the initial-boundary value problem (1)-(3) is labeled as QP, the stationary solution as SS, expansive wave as EW and wave speed as ws. The method in [7][8][9] is used to set the schema of the solution to QP. The problem parameters u+ and u b play a vital role for the large t-solution of QP. In particular, the attractor of the solution to QP is given by: (i) TW with positive ws while −u b < u+ < u b with u b > 0. (ii) SS when (−u+ > u b > u+ with u+ < 0) or when 0 u+ > u b . (iii) The schema containing a conjunction of a EW and SS while u b < 0 and u+ > 0. (iv) An EW when u+ > u b 0.
The methodology in [7] (see [6] as well) has been applied to complete the large t asymptotic schema of QP. We start with the case (i) where −u b < u+ < u b with u b > 0.

Asymptotic schema while −u b < u + < u b , u b > 0
We start with the first which we denote as I*, in this region as t → 0 with = O(1) (0 < ∞). In the region II*, . The solution of QP as t → 0 is solved asymptotically and now completed with equations (4) and (5) bringing an irreversible resemblance to the QP as t → 0. The appearance of equation (5) of region II* for x ≫ 1 as t → 0 requires a new region that is labeled as region III*, region III* v(x, t) = u+ + exp  (6)) v(y, t) = u+ + e −NN(y,t) as t → ∞, NN(y, t) = n * 0 (y)t + n * 1 (y) ln t + n * 2 (y) where y = O(1)(> 0) as t → ∞ and n * 0 (y) > 0, n * 1 (y) and n * 2 (y) are functions to be identified. On substituting (7) and (8) into equation (1) and after some computations a one-parameter family of linear solutions has occurred, n * 0 (y) = A + [y − (A + + u+)], y > 0 (9) for any A + ∈ R, together with the associated envelope solution Combining of (9) and (10) which stay continuous and differentiable also provides an envelope touching solutions. The solution (9)-(10) is given either by the envelope solution or by the family of envelope touching solutions 2 4 , y u+ + 2A + , for each A + > 0. Each case will have to be tackled in turn one at a time.
First the case −u b < u+ < u b with u b > 0 and later on we must think the subcases such as 0 < u+ < u b , u+ = 0 and −u b < u+ < 0 distinctly.
(ii) u+ = 0. In this case equation (13) is reversible as y → 0 + and to continue the large t-asymptotic schema of QP we refer to a new region as region A2,* as t → ∞ with x = O(1) ( 0). On substituting (14) into (1) the leading order problem is found to be (b) If n * 0 (y) is taken as in the appearance (12) then in region IV(a)* we obtain v(y, t) = u+ + exp The function H LL (y) is still indeterminate but by comparing it to region TR* as y → u − b we find There are three different cases to consider.
and I refer to a final region B*.
3 Asymptotic schema when −u + > u b > u + , u + < 0 The asymptotic schema of QP as t → 0 and as x → ∞ (t = O(1)(> 0)) in this section is followed directly like in section 2 and is not repeated here. In region IV(a)*  (1) )︂ e −u 2 as t → ∞ with x 0. The schema in this case is now completed.

Asymptotic schema while u b < u + < 0, u b < 0
The large t-asymptotic schema in this case follows closely that summarized in Section 3. In region IV(a)*  (1) )︂ e −u 2

Asymptotic schema when u b < 0, u + = 0
The asymptotic solution of QP as t → ∞ differs from that seen in Sections 3 and 4. In region IV* v(y, t) = − exp }︂ as t → ∞ with y = O(1) (∈ (0, ∞)) and H RR (y) is not determined but matching with region III* in section 2 requires √ π (> 0) in this case. As y → 0 this equation is reversible and we refer to a new region as region V*. Therefore, in region V*

∞)) and H RR (y) is indeterminate but matching with region III* in section 2 requires that
√ π (> 0) in this case. We need to consider the subcases: (i) u+ > u b > 0, and (ii) u b = 0, separately. We start with the first subcase as t → ∞ with y = O(1) (∈ (u b , u+)). We note that this equation is reversible as y → u + b and we must refer to another one that is called region B*, located at y = u b . In region B we have as t → ∞ with η * = O(1). As η * → −∞ from the localised region B* into region VI*, v(y, t) = u b + exp 0, u b )) and LL(y) is not determined, but matching region III* in section 2 requires We conclude that the equation must be reversible as y → 0 + . We introduce a final asymptotic region, in region C*. v( (ii) u b = 0. In this subcase the asymptotic schema of the solution of QP given in regions IV*, A* and V* follows on setting A = 2u+ √ π and u b = 0 that given above after some calculation region C* has (1). Now the asymptotic schema of solution to QP is complete as t → ∞.

Asymptotic schema while u b < 0 u + > 0
The difference from Section 6 is that region B* has,

Numerical solutions
Here we use the numerical method summarized in [10] to solve QP. We consider numerical simulation of QP for each case in turn: It appears as though the numerical solution overlaps quickly to the awaited TW where we assume the overlap to be exponential in t as t → ∞.    (ii) −u+ > u b > u+ with u+ < 0.

Conclusion
The entire asymptotic schema of QP have been obtained as t → ∞ over whole parameter measures. The type of large t-attractor that occurs as t → ∞ is controlled by u+ and u b . In the second Section a TW evolved as t → ∞ in the QP. The grade of overlap of the QP onto the TW is exponential in t, being of O(t −3/2 e − A 2 4 t ) as t → ∞. In the third section it is important to note that the grade of overlap of the QP is exponential in t as t → ∞. In the fourth section, as in third section, the grade of overlap of the QP is exponential in t as t → ∞ has been obtained. In the section follwing section 4 and throughout the regions an irreversible approximation was obtained. In section 6 the solution displayed the appearance of an EW for x 0 and where u+ > u b 0. In Section 7, as in Section 5, the same irreversible approximation was found throughout the regions. In Section 8, numerical solutions of QP are represented that affirm and support the asymptotic analysis presented in the sections mentioned above. In all case the numerical simulations are in good agreement with the theory as t → ∞. In conclusion, Equation (1)-(3) arises in mathematical chemistry, biology, physics and play a vital role in related areas.