Section 6.7 The heat condition in a rod

Objective:

1. The solution set of a homogeneous boundary value problem associated to the heat conduction in a rod

We are ready to find the solution of heat conduction problems. 

Case (1): When the bar at the both ends are kept at the $0$, we need to solve the boundary value problem: $$\begin{array}{rl}{u}_t(x,t)& =a^2{u}_{xx}(x,t),t>0\\ u(0,t)& =0,u(L,t)=0,t>0\\ u(x,0)& =f(x),0\le x\le L.\end{array}$$ We assume $u(x,t)=X(x)T(t)$, then translate the problem into solving $$\begin{array}{rl}X”(x)+\lambda X(x)& =0,X(0)=X(L)=0\\ T^{\prime }(t)+a^2\lambda T(t)& =0.\end{array}$$ The only non-trivial solutions of $X^{″}(x)+\lambda X(x)=0,X(0)=X(L)=0$ are ${\lambda }_n=(\frac{n\pi }{L}{)}^2$ and $X_n=c_n\text{sin}(\frac{n\pi }{L}x)$. Hence we need to solve $T^{\prime }(t)+a^2(\frac{n\pi }{L}{)}^{2}T(t)=0.$ Using separable method, we have $T_n(t)=k_n{e}^{-(\frac{n\pi a}{L}{)}^{2}t}$ with $k_n$ constant. We let  $$u_n(x,t)=e^{-(\frac{n\pi a}{L}{)}^{2}t}\text{sin}(\frac{n\pi }{L}x)$$ and write  $$u(x,t)=\sum _{n=1}^{\mathrm{\infty }}{d}_n{u}_n(x,t)=\sum _{n=1}^{\mathrm{\infty }}{d}_n{e}^{-(\frac{n\pi a}{L}{)}^{2}t}\text{sin}(\frac{n\pi }{L}x).$$ We use the initial condition $u(x,0)=f(x)$ to solve for $d_n$. $$u(x,0)=f(x)=\sum _{n=1}^{\mathrm{\infty }}{d}_n{u}_n(x,0)=\sum _{n=1}^{\mathrm{\infty }}{d}_n\text{sin}(\frac{n\pi }{L}x).$$ We see that $f(x)$ is a sine Fourier series and hence $d_n=\frac{2}{L}{\int }_0^{L}f(x)\text{sin}(\frac{n\pi }{L}x)dx$. 

Example 1: Suppose the temperature distribution function $u(x,t)$ of a rod is given by the initial-boundary value problem $$\begin{array}{rl}{u}_t(x,t)& =25u_{xx}(x,t),t>0\\ u(0,t)& =0,u(4\pi ,t)=0,t>0\\ u(x,0)& =20\text{sin}(2x)-40\text{sin}(4x),0\le x\le 4\pi .\end{array}$$

(a) What is the physical meaning of its boundary conditions?

(b) State the general form of its solution. Then find the particular solution of the initial-boundary value problem.

(c) What is $\underset{t\to \mathrm{\infty }}{lim}u(2\pi ,t)$?

Exercise 1: Suppose the temperature distribution function $u(x,t)$ of a rod is given by the initial-boundary value problem $$\begin{array}{rl}{u}_t(x,t)& =16u_{xx}(x,t),t>0\\ u(0,t)& =0,u(5,t)=0,t>0\\ u(x,0)& =10\text{sin}(2\pi x)-30\text{sin}(5\pi x),0\le x\le 5.\end{array}$$

(a) What is the physical meaning of its boundary conditions?

(b) State the general form of its solution. Then find the particular solution of the initial-boundary value problem.

(c) What is $\underset{t\to \mathrm{\infty }}{lim}u(3,t)$?

Case (2): When the bar at the both ends are kept at the $T_1$ and $T_2$, we need to solve the boundary value problem: $$\begin{array}{rl}{u}_t(x,t)& =a^2{u}_{xx}(x,t),t>0\\ u(0,t)& =T_1,u(L,t)=T_2,t>0\\ u(x,0)& =f(x),0\le x\le L.\end{array}$$ We assume $u(x,t)=v(x)+w(x,t)$, where $v(x)=Ax+B,$ and $v(0)=T_1$, $v(L)=T_2$, and $w(x,t)$ satisfies $$\begin{array}{rl}{w}_t(x,t)& =a^2{w}_{xx}(x,t),t>0\\ w(0,t)& =0,w(L,t)=0,t>0\\ w(x,0)& =f(x)-v(x),0\le x\le L.\end{array}$$ then by case (1), we have $$w(x,t)=\sum _{n=1}^{\mathrm{\infty }}{d}_n{e}^{-(\frac{n\pi a}{L}{)}^{2}t}\text{sin}(\frac{n\pi }{L}x).$$ We use the initial condition $w(x,0)=f(x)-v(x)$ to solve for $d_n$. Hence $d_n=\frac{2}{L}{\int }_0^L(f(x)-v(x))\text{sin}(\frac{n\pi }{L}x)dx$. 

Example 2: Suppose the temperature distribution function $u(x,t)$ of a rod is given by the initial-boundary value problem $$\begin{array}{rl}{u}_t(x,t)& =25u_{xx}(x,t),t>0\\ u(0,t)& =40,u(2,t)=0,t>0\\ u(x,0)& =40-20x+20\text{sin}(2\pi x)-40\text{sin}(4\pi x),0\le x\le 2.\end{array}$$

(a) What is the physical meaning of its boundary conditions?

(b) State the general form of its solution. Then find the particular solution of the initial-boundary value problem.

(c) What is $\underset{t\to \mathrm{\infty }}{lim}u(1,t)$?

Exercise 2: Suppose the temperature distribution function $u(x,t)$ of a rod is given by the initial-boundary value problem $$\begin{array}{rl}{u}_t(x,t)& =16u_{xx}(x,t),t>0\\ u(0,t)& =20,u(8,t)=60,t>0\\ u(x,0)& =20+5x+10\text{sin}(2\pi x)-30\text{sin}(5\pi x),0\le x\le 8.\end{array}$$

(a) What is the physical meaning of its boundary conditions?

(b) State the general form of its solution. Then find the particular solution of the initial-boundary value problem.

(c) What is $\underset{t\to \mathrm{\infty }}{lim}u(4,t)$?

Case (3): No exchange of heat going into bar or out of the bar, and the temperature of the bars at any position $x$ at time $0$ is given by $f(x)$. We need to solve the boundary value problem: $$\begin{array}{rl}{u}_t(x,t)& =a^2{u}_{xx}(x,t),t>0\\ u_x(0,t)& =0,u_x(L,t)=0,t>0\\ u(x,0)& =f(x),0\le x\le L.\end{array}$$ We assume $u(x,t)=X(x)T(t)$, then translate the problem into solving $$\begin{array}{rl}X”(x)+\lambda X(x)& =0,X^{\prime }(0)=X^{\prime }(L)=0\\ T^{\prime }(t)+a^2\lambda T(t)& =0.\end{array}$$ The only non-trivial solutions of $X^{″}(x)+\lambda X(x)=0,X^{\prime }(0)=X^{\prime }(L)=0$ are case (a) $\lambda =0$, and $X(x)=B\ne 0$; (b) ${\lambda }_n=(\frac{n\pi }{L}{)}^2$ and $X_n=c_n\text{cos}(\frac{n\pi }{L}x)$. Hence we need to solve

(a) $T^{\prime }(t)=0$; (b) $T^{\prime }(t)+a^2(\frac{n\pi }{L}{)}^{2}T(t)=0.$ Using separable method, we have (a) $T(t)=k$, constant; (b) $T_n(t)=k_n{e}^{-(\frac{n\pi a}{L}{)}^{2}t}$ with $k_n$ constant. We let  $$u_n(x,t)=e^{-(\frac{n\pi a}{L}{)}^{2}t}\text{cos}(\frac{n\pi }{L}x)$$ and write  $$u(x,t)=\frac{c_0}{2}+\sum _{n=1}^{\mathrm{\infty }}{d}_n{u}_n(x,t)=\frac{c_0}{2}+\sum _{n=1}^{\mathrm{\infty }}{d}_n{e}^{-(\frac{n\pi a}{L}{)}^{2}t}\text{cos}(\frac{n\pi }{L}x).$$ We use the initial condition $u(x,0)=f(x)$ to solve for $d_n$. $$u(x,0)=f(x)=\frac{c_0}{2}+\sum _{n=1}^{\mathrm{\infty }}{d}_n{u}_n(x,0)=\frac{c_0}{2}+\sum _{n=1}^{\mathrm{\infty }}{d}_n\text{cos}(\frac{n\pi }{L}x).$$ We see that $f(x)$ is a cosine Fourier series and hence $d_n=\frac{2}{L}{\int }_0^{L}f(x)\text{cos}(\frac{n\pi }{L}x)dx$, $c_0=\frac{2}{L}{\int }_0^{L}f(x)dx$.

Example 3: Suppose the temperature distribution function $u(x,t)$ of a rod is given by the initial-boundary value problem $$\begin{array}{rl}{u}_t(x,t)& =25u_{xx}(x,t),t>0\\ u_x(0,t)& =0,u_x(4,t)=0,t>0\\ u(x,0)& =40+20\text{cos}(2\pi x)-40\text{cos}(4\pi x),0\le x\le 4.\end{array}$$

(a) What is the physical meaning of its boundary conditions?

(b) State the general form of its solution. Then find the particular solution of the initial-boundary value problem.

(c) What is $\underset{t\to \mathrm{\infty }}{lim}u(2,t)$?

Exercise 3: Suppose the temperature distribution function $u(x,t)$ of a rod is given by the initial-boundary value problem $$\begin{array}{rl}{u}_t(x,t)& =16u_{xx}(x,t),t>0\\ u_x(0,t)& =0,u_x(5,t)=0,t>0\\ u(x,0)& =20+10\text{cos}(2\pi x)-30\text{cos}(5\pi x),0\le x\le 5.\end{array}$$

(a) What is the physical meaning of its boundary conditions?

(b) State the general form of its solution. Then find the particular solution of the initial-boundary value problem.

(c) What is $\underset{t\to \mathrm{\infty }}{lim}u(3,t)$?