(i) Consider the equation

$$\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}=\frac{\partial^{2} u}{\partial x^{2}}+f(t, x)$$

and, using the change of variables $(t, x) \mapsto(s, y)=(t, x-t)$, show that it can be transformed into an equation of the form

$$\frac{\partial U}{\partial s}=\frac{\partial^{2} U}{\partial y^{2}}+F(s, y)$$

where $U(s, y)=u(s, y+s)$ and you should determine $F(s, y)$.

(ii) Let $H(y)$ be the Heaviside function. Find the general continuously differentiable solution of the equation

$$w^{\prime \prime}(y)+H(y)=0$$

(iii) Using (i) and (ii), find a continuously differentiable solution of

$$\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}=\frac{\partial^{2} u}{\partial x^{2}}+H(x-t)$$

such that $u(t, x) \rightarrow 0$ as $x \rightarrow-\infty$ and $u(t, x) \rightarrow-\infty$ as $x \rightarrow+\infty$