Can someone tell me what type of equation is this and how would one go about solving it. \begin{align} &u_{xx}-u_{tt}=u+\sin \left(\frac{\pi x}{4}\right),\quad x\in (0,2),\, t>0, \\ &u(0,t)=2,\quad u(2,t)=t-2,\quad t>0, \\ &u(x,0)=2,\quad u_{t}(x,0)=x^{3},\quad x\in (0,2). \end{align} I know a method of fourier separation of variables and of expanding $u_{xx}-u_{tt}=f(x,t)$ the right hand side into a sine series. One thing I believe would be useful is to shift the conditions to $u(0,t)=0~and~u(2,t)=0$ , I dont know how to proceed further.
Anyone has any clue?
Thank you so much!
Let $u(x,t)=v(x,t)+f(x)+tg(x)$; then $$ v_{xx}+f''(x)+tg''(x)-v_{tt}=v+f(x)+tg(x)+\sin\left(\frac{\pi x}{4}\right). \tag{1} $$ Now choose $f$ and $g$ such that $$ f''(x)=f(x)+\sin\left(\frac{\pi x}{4}\right),\qquad f(0)=2, \qquad f(2)=-2, \tag{2} $$ and $$ g''(x)=g(x), \qquad g(0)=0, \qquad g(2)=1. \tag{3} $$ This reduces the original PDE to $$ v_{xx}-v_{tt}=v, \tag{4} $$ $$ v(0,t)=0=v(2,t), \tag{5} $$ $$ v(x,0)=2-f(x), \tag{6} $$ $$ v_t(x,0)=x^3-g(x). \tag{7} $$ In Physics, Eq. $(4)$ is known as Klein-Gordon equation. You can use the method of separation of variables to solve it.