Abstract
Abstract In this paper we investigate an initial-boundary value problem for the Burgers equation on the positive quarter-plane; v t + v v x - v x x = 0 , x > 0 , t > 0 , v ( x , 0 ) = u + , x > 0 , v ( 0 , t ) = u b , t > 0 , $\matrix{ {{v_t} + v{v_x} - {v_{xx}} = 0,\,\,\,x > 0,\,\,\,t > 0,} \cr {v\left( {x,0} \right) = {u_ + },\,\,\,x > 0,} \cr {v\left( {0,t} \right) = {u_b},\,\,t > 0,} \cr }$ 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.
Highlights
An initial and boundary value problem for Burgers’ equation on the positive quarter-plane will be examined
The distinct equation to be discussed is given by vt + vvx − vxx = 0, x > 0, t > 0, (1)
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,*
Summary
An initial and boundary value problem for Burgers’ equation on the positive quarter-plane will be examined . 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 y = O(1) (∈ (−u+, ∞)) and HRR(y) stays indeterminate but matching with region III* in section 2 requires. As t → ∞ with y = O(1) (∈ (0, ∞)) and HRR(y) is indeterminate but matching with region III* in section 2 requires that. LL(y) as t → ∞ with y = O(1)(∈ (0, ub)) and LL(y) is not determined, but matching region III* in section 2 requires π as y → u−b. In this subcase the asymptotic schema of the solution of QP given in regions IV*, A* and V* follows on setting. The asymptotic schema of solution to QP is complete as t → ∞
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