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- Homework Statement
- please see below

- Relevant Equations
- please see below

Sorry the problem is a bit long to read. thank you to anyone who comments.

We consider the initial value problem for the Burger's equation with viscosity given by

$$\begin{cases} \partial_t u-\partial^2_xu+u\partial_xu=0 & \text{in}\quad (1,T)\times R\\\quad \quad \quad \quad \quad u(0,\cdot)=g & \text{on}\quad \{t=0\}\times R \end{cases}$$

(1) We know that ##u(0,x)=g## so the substitution of the Burger equation gives $$u(0,x)=\boxed{g=\frac{-2}{\phi(0,x)}\frac{\partial}{\partial x}\phi(0,x)}$$ which is a separable ODE for ##\psi:=\phi(0,x)##. \\\\To solve the ODE, we re-write the equation

$$\Rightarrow \frac{\partial}{\partial x}\Big[ln(\phi(0,x))\Big]=-\frac{g}{2}$$

and integrate

$$\Rightarrow \int\frac{\partial}{\partial x}\Big[ln(\phi(0,x))\Big]dx=-\int \frac{g}{2} dx$$

$$\Rightarrow ln(\phi(0,x))=-\int \frac{g}{2} dx$$

then exponentiate

$$\Rightarrow \phi(0,x)=\boxed{exp(-\int\frac{g}{2}dx)}$$

We consider the initial value problem for the Burger's equation with viscosity given by

$$\begin{cases} \partial_t u-\partial^2_xu+u\partial_xu=0 & \text{in}\quad (1,T)\times R\\\quad \quad \quad \quad \quad u(0,\cdot)=g & \text{on}\quad \{t=0\}\times R \end{cases}$$

(1) We know that ##u(0,x)=g## so the substitution of the Burger equation gives $$u(0,x)=\boxed{g=\frac{-2}{\phi(0,x)}\frac{\partial}{\partial x}\phi(0,x)}$$ which is a separable ODE for ##\psi:=\phi(0,x)##. \\\\To solve the ODE, we re-write the equation

$$\Rightarrow \frac{\partial}{\partial x}\Big[ln(\phi(0,x))\Big]=-\frac{g}{2}$$

and integrate

$$\Rightarrow \int\frac{\partial}{\partial x}\Big[ln(\phi(0,x))\Big]dx=-\int \frac{g}{2} dx$$

$$\Rightarrow ln(\phi(0,x))=-\int \frac{g}{2} dx$$

then exponentiate

$$\Rightarrow \phi(0,x)=\boxed{exp(-\int\frac{g}{2}dx)}$$