Section 6.2 Separation of valuables and the heat conduction in a rod or Vibrating String

Objective:

1. The connection of separation of valuables and the heat conduction in a rod or spring problem

2. Determinate when a partial differential equation is separable.

The fundamental problem of heat conduction is to find $u(x,t)$ satisfying $$\begin{array}{rl}{u}_t(x,t)& =a^2{u}_{xx}(x,t),t>0\\ u(0,t)& =0,u(L,t)=0,t>0\\ u(x,0)& =f(x),0\le x\le L.\end{array}$$ With respect to the time variable $t$, this is an initial value problem; an initial condition is given and the differential equation governs what happens later. With respect to the spatial variable $x$, it is a boundary value problem; boundary conditions are imposed at each end of the bar and the differential equation describes the evolution of the temperature in the interval between them. We begin by assuming that the solution $u(x,t)$ has the form product of two functions $X(x)$ and $T(t)$, $u(x,t)=X(x)T(t)$ then we have $u(x,0)=X(x)T(0)=f(x)$. We can translate the problem into a system of differential equations $$\begin{array}{rl}X”(x)+\lambda X(x)& =0,X(0)=X(L)=0\\ T^{\prime }(t)+a^2\lambda T(t)& =0.\end{array}$$

Proof:

Another fundamental problem of heat conduction is to find $u(x,t)$ satisfying  $$\begin{array}{rl}{u}_t(x,t)& =a^2{u}_{xx}(x,t),t>0\\ u_x(0,t)& =0,u_x(L,t)=0,t>0\\ u(x,0)& =f(x),0\le x\le L.\end{array}$$ We can translate the problem into a system of differential equations $$\begin{array}{rl}X”(x)+\lambda X(x)& =0,X^{\prime }(0)=X^{\prime }(L)=0\\ T^{\prime }(t)+a^2\lambda T(t)& =0.\end{array}$$

Proof:

Our vibrating string problem is  $$\begin{array}{rl}{u}_{tt}(x,t)& =a^2{u}_{xx}(x,t),t>0\\ u(0,t)& =0,u(L,t)=0,t>0\\ u(x,0)& =f(x),u_t(x,0)=g(x),0\le x\le L\end{array}$$ We can translate the problem into a system of differential equations $$\begin{array}{rl}X”(x)+\lambda X(x)& =0,X(0)=X(L)=0\\ T”(t)+a^2\lambda T(t)& =0.\end{array}$$ Where the initial conditions of $T(t)$ will depend on $f(x)$ and $g(x)$. If $f(x)=0$ then $T(0)=0$, and if $g(x)=0$ then $T^{\prime }(0)=0$.

Proof:

It is clear the method mention above is a powerful tool to reduce multi-variables partial differential equations into a single variable differential equation. This method is called separation of valuables. 

Example 1: Use the method of separation of valuables to transform the given equations into single variable differential equations. $u_{xx}+t{u}_t=0$.

Exercise 1: Use the method of separation of valuables to transform the given equations into single variable differential equations. $x{u}_{xx}+u_t=0$.

Example 2: Use the method of separation of valuables to transform the given equations into single variable differential equations. $x{u}_{xx}+t{u}_t=0$.

Exercise 2:  Use the method of separation of valuables to transform the given equations into single variable differential equations. $t{u}_{xx}+x{u}_t=0$.

GroupWork:

Use the method of separation of valuables to transform the given equations into single variable differential equations. 

1. $u_{xx}+u_{xt}+u_t=0$.

2. $x{u}_{xx}+t{u}_{xt}+t{u}_t=0$.

3. $u_{xx}+(x+t)u_{tt}=0$.

4. $u_{xx}+u_{tt}+xu=0$.