Chapter 8.4, Problem 1E

Show that for any constant c, the function $u(x,t)=c$ is an equilibrium solution of Burgers’ equation $u_t+u{u}_x=D{u}_{xx}$.

To determine

To show: for any constant c,the function $u\left(x,t\right)=c$ is an equilibrium $u_t+u{u}_x=D{u}_{xx}$ solution of the burger’s equation .

Explanation of Solution

Concept used:

A solution to the linear wave equation can be obtained as a special case of the non-linear wave equation (1). When $c\left(u\right)=constant$ , the characteristic curves $x=ct+\xi$ sets the solution u with $u\left(x,t\right)=F\left(\xi \right)=F\left(x-ct\right).$

for any constant c,the function $u\left(x,t\right)=c$ is an equilibrium solution of the burger’s equation $u(x,t)=cand{u}_t+u{u}_x=D{u}_{xx}$

Calculation:

As it’s known from the theorem that A solution to the linear wave equation can be obtained as a special case of the non-linear wave equation (1). When $c\left(u\right)=constant$ , the characteristic curves $x=ct+\xi$ sets the solution u with $u\left(x,t\right)=F\left(\xi \right)=F\left(x-ct\right).$

As in the problem mention for any constant c, the function $u\left(x,t\right)=c$ is an equilibrium solution of the burger’s equation $u(x,t)=cand{u}_t+u{u}_x=D{u}_{xx}$

Since all partial derivatives are zero, Burgers' equation  $u_t+u{u}_x-D{u}_{xx}=0$ is automatically satisfied.