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 [latex]0[/latex], we need to solve the boundary value problem: \begin{align*} u_{t}(x,t) & =a^{2}u_{xx}(x,t),t \gt 0\\ u(0,t) & =0,u(L,t)=0,t \gt 0\\ u(x,0) & =f(x),0\leq x\leq L. \end{align*} We assume [latex]u(x,t)=X(x)T(t)[/latex], then translate the problem into solving \begin{align*} X”(x)+\lambda X(x) & =0,X(0)=X(L)=0\\ T'(t)+a^{2}\lambda T(t) & =0. \end{align*} The only non-trivial solutions of [latex]X''(x)+\lambda X(x)=0,X(0)=X(L)=0[/latex] are [latex]\lambda_{n}=(\frac{n\pi}{L})^{2}[/latex] and [latex]X_{n}=c_{n}\text{sin}(\frac{n\pi}{L}x)[/latex]. Hence we need to solve [latex]T'(t)+a^{2}(\frac{n\pi}{L})^{2}T(t)=0.[/latex] Using separable method, we have [latex]T_{n}(t)=k_{n}e^{-(\frac{n\pi a}{L})^{2}t}[/latex] with [latex]k_{n}[/latex] constant. We let \[ u_{n}(x,t)=e^{-(\frac{n\pi a}{L})^{2}t}\text{sin}(\frac{n\pi}{L}x) \] and write \[ u(x,t)=\sum_{n=1}^{\infty}d_{n}u_{n}(x,t)=\sum_{n=1}^{\infty}d_{n}e^{-(\frac{n\pi a}{L})^{2}t}\text{sin}(\frac{n\pi}{L}x).\] We use the initial condition [latex]u(x,0)=f(x)[/latex] to solve for [latex]d_{n}[/latex]. \[ u(x,0)=f(x)=\sum_{n=1}^{\infty}d_{n}u_{n}(x,0)=\sum_{n=1}^{\infty}d_{n}\text{sin}(\frac{n\pi}{L}x). \] We see that [latex]f(x)[/latex] is a sine Fourier series and hence [latex]d_{n}=\frac{2}{L}\int_{0}^{L}f(x)\text{sin}(\frac{n\pi}{L}x)dx[/latex].
Example 1: Suppose the temperature distribution function [latex]u(x,t)[/latex] of a rod is given by the initial-boundary value problem \begin{align*} u_{t}(x,t) & =25u_{xx}(x,t),t \gt 0\\ u(0,t) & =0,u(4\pi,t)=0,t \gt 0\\ u(x,0) & =20\text{sin}(2x)-40\text{sin}(4x),0\leq x\leq4\pi. \end{align*}
(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 [latex]\lim_{t\rightarrow\infty}u(2\pi,t)[/latex]?
Exercise 1: Suppose the temperature distribution function [latex]u(x,t)[/latex] of a rod is given by the initial-boundary value problem \begin{align*} u_{t}(x,t) & =16u_{xx}(x,t),t \gt 0\\ u(0,t) & =0,u(5,t)=0,t \gt 0\\ u(x,0) & =10\text{sin}(2\pi x)-30\text{sin}(5\pi x),0\leq x\leq5. \end{align*}
(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 [latex]\lim_{t\rightarrow\infty}u(3,t)[/latex]?
Case (2): When the bar at the both ends are kept at the [latex]T_{1}[/latex] and [latex]T_{2}[/latex], we need to solve the boundary value problem: \begin{align*} u_{t}(x,t) & =a^{2}u_{xx}(x,t),t \gt 0\\ u(0,t) & =T_{1},u(L,t)=T_{2},t \gt 0\\ u(x,0) & =f(x),0\leq x\leq L. \end{align*} We assume [latex]u(x,t)=v(x)+w(x,t)[/latex], where [latex]v(x)=Ax+B,[/latex] and [latex]v(0)=T_{1}[/latex], [latex]v(L)=T_{2}[/latex], and [latex]w(x,t)[/latex] satisfies \begin{align*} w_{t}(x,t) & =a^{2}w_{xx}(x,t),t \gt 0\\ w(0,t) & =0,w(L,t)=0,t \gt 0\\ w(x,0) & =f(x)-v(x),0\leq x\leq L. \end{align*} then by case (1), we have \[ w(x,t)=\sum_{n=1}^{\infty}d_{n}e^{-(\frac{n\pi a}{L})^{2}t}\text{sin}(\frac{n\pi}{L}x).\] We use the initial condition [latex]w(x,0)=f(x)-v(x)[/latex] to solve for [latex]d_{n}[/latex]. Hence [latex]d_{n}=\frac{2}{L}\int_{0}^{L}(f(x)-v(x))\text{sin}(\frac{n\pi}{L}x)dx[/latex].
Example 2: Suppose the temperature distribution function [latex]u(x,t)[/latex] of a rod is given by the initial-boundary value problem \begin{align*} u_{t}(x,t) & =25u_{xx}(x,t),t \gt 0\\ u(0,t) & =40,u(2,t)=0,t \gt 0\\ u(x,0) & =40-20x+20\text{sin}(2\pi x)-40\text{sin}(4\pi x),0\leq x\leq2. \end{align*}
(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 [latex]\lim_{t\rightarrow\infty}u(1,t)[/latex]?
Exercise 2: Suppose the temperature distribution function [latex]u(x,t)[/latex] of a rod is given by the initial-boundary value problem \begin{align*} u_{t}(x,t) & =16u_{xx}(x,t),t \gt 0\\ u(0,t) & =20,u(8,t)=60,t \gt 0\\ u(x,0) & =20+5x+10\text{sin}(2\pi x)-30\text{sin}(5\pi x),0\leq x\leq8. \end{align*}
(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 [latex]\lim_{t\rightarrow\infty}u(4,t)[/latex]?
Case (3): No exchange of heat going into bar or out of the bar, and the temperature of the bars at any position [latex]x[/latex] at time [latex]0[/latex] is given by [latex]f(x)[/latex]. We need to solve the boundary value problem: \begin{align*} u_{t}(x,t) & =a^{2}u_{xx}(x,t),t \gt 0\\ u_{x}(0,t) & =0,u_{x}(L,t)=0,t \gt 0\\ u(x,0) & =f(x),0\leq x\leq L. \end{align*} We assume [latex]u(x,t)=X(x)T(t)[/latex], then translate the problem into solving \begin{align*} X”(x)+\lambda X(x) & =0,X'(0)=X'(L)=0\\ T'(t)+a^{2}\lambda T(t) & =0. \end{align*} The only non-trivial solutions of [latex]X''(x)+\lambda X(x)=0,X'(0)=X'(L)=0[/latex] are case (a) [latex]\lambda=0[/latex], and [latex]X(x)=B\neq0[/latex]; (b) [latex]\lambda_{n}=(\frac{n\pi}{L})^{2}[/latex] and [latex]X_{n}=c_{n}\text{cos}(\frac{n\pi}{L}x)[/latex]. Hence we need to solve
(a) [latex]T'(t)=0[/latex]; (b) [latex]T'(t)+a^{2}(\frac{n\pi}{L})^{2}T(t)=0.[/latex] Using separable method, we have (a) [latex]T(t)=k[/latex], constant; (b) [latex]T_{n}(t)=k_{n}e^{-(\frac{n\pi a}{L})^{2}t}[/latex] with [latex]k_{n}[/latex] constant. We let \[ u_{n}(x,t)=e^{-(\frac{n\pi a}{L})^{2}t}\text{cos}(\frac{n\pi}{L}x) \] and write \[u(x,t)=\frac{c_{0}}{2}+\sum_{n=1}^{\infty}d_{n}u_{n}(x,t)=\frac{c_{0}}{2}+\sum_{n=1}^{\infty}d_{n}e^{-(\frac{n\pi a}{L})^{2}t}\text{cos}(\frac{n\pi}{L}x).\] We use the initial condition [latex]u(x,0)=f(x)[/latex] to solve for [latex]d_{n}[/latex]. \[u(x,0)=f(x)=\frac{c_{0}}{2}+\sum_{n=1}^{\infty}d_{n}u_{n}(x,0)=\frac{c_{0}}{2}+\sum_{n=1}^{\infty}d_{n}\text{cos}(\frac{n\pi}{L}x).\] We see that [latex]f(x)[/latex] is a cosine Fourier series and hence [latex]d_{n}=\frac{2}{L}\int_{0}^{L}f(x)\text{cos}(\frac{n\pi}{L}x)dx[/latex], [latex]c_{0}=\frac{2}{L}\int_{0}^{L}f(x)dx[/latex].
Example 3: Suppose the temperature distribution function [latex]u(x,t)[/latex] of a rod is given by the initial-boundary value problem \begin{align*} u_{t}(x,t) & =25u_{xx}(x,t),t \gt 0\\ u_{x}(0,t) & =0,u_{x}(4,t)=0,t \gt 0\\ u(x,0) & =40+20\text{cos}(2\pi x)-40\text{cos}(4\pi x),0\leq x\leq4. \end{align*}
(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 [latex]\lim_{t\rightarrow\infty}u(2,t)[/latex]?
Exercise 3: Suppose the temperature distribution function [latex]u(x,t)[/latex] of a rod is given by the initial-boundary value problem \begin{align*} u_{t}(x,t) & =16u_{xx}(x,t),t \gt 0\\ u_{x}(0,t) & =0,u_{x}(5,t)=0,t \gt 0\\ u(x,0) & =20+10\text{cos}(2\pi x)-30\text{cos}(5\pi x),0\leq x\leq5. \end{align*}
(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 [latex]\lim_{t\rightarrow\infty}u(3,t)[/latex]?