# Initial-boundary value problem for PDE

I need a little help with solving IBVP for hyperbolic and parabolic equations like these: $$hyperbolic: \left\{\begin{matrix} \frac{\partial^2u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}+1\\ u(0,t) =u(1, t) =0\\ u(x,0) = 0\\ u_t(x,0)=x \end{matrix}\right. \\parabolic: \left\{\begin{matrix} 3\frac{\partial u}{\partial t}=4\frac{\partial^2 u}{\partial x^2}\\ u(0,t) =u(5, t) =0\\ u(x,0) = x \end{matrix}\right.$$ Don't even know where to start.

I'm not actually asking for full solution, but would appreciate it if you could give me direction. Links to tutorials and examples are also highly welcomed.

Update:

Second equation looks like 1 dimensional heat conduction equation: $u_t = c^2u_{xx}$ with $c=\frac{2\sqrt{3}}{3}$.

Ok, for the second one method of separation of variables could be applied.

We assume that $u$ can be written as a product of single variable functions of each independent variable, $u(x, t) = X (x)T (t)$. Substituting this guess into the heat equation, we find that $XT′ =c^2X′′T$. Dividing both sides by $c^2$ and $u = XT$, we then get $\frac{1}{c^2}\frac{T'}{T} = \frac{X''}{X} = \lambda$. This leads to two equations: $$T′ =c^2\lambda T\\ X'' = \lambda X$$ giving $$T(t) = Ae^{c^2λt}\\ X(x) = c_1e^{\sqrt{\lambda}x} + c_2e^{\sqrt{-\lambda}x}$$ The aim is to force our product solutions to satisfy both the boundary conditions and initial conditions.

Is it possible to use this method for the first equation?

3
2022-07-25 20:45:31
Source Share
Yes, the very same method jobs. You first require to find the eigenfunctions of the equivalent uniform trouble (allow me represent them as $e_i(x)$, these are $X(x)$ in your remedy, there is a countable variety of them), after that you represent your non - uniform term (in your instance $f(x,t)=1$) as a collection making use of the located eigenfunctions $f(x,t)=\sum f_i(t)e_i(x)$, and afterwards seek the remedy in the kind $u(x,t)=\sum T_i(t)e_i(x)$ (simply connect this expression in your formula). For the unidentified features $T_i(t)$ you'll get differential formulas which think about $f_i(t)$.