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

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,$\matrix{ {{v_t} + v{v_x} - {v_{xx}} = 0,\,\,\,x > 0,\,\,\,t > 0,} \cr {v\left( {x,0} \right) = {u_ + },\,\,\,x > 0,} \cr {v\left( {0,t} \right) = {u_b},\,\,t > 0,} \cr }$ 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