Section 6.1 Introduction to Two-Point Boundary Value Problems

Objective:

1. Definition of a Two-Point Boundary Value Problem

2. Applications for multi-valuables differential equations

In this section, we give an introduction on Two-Point Boundary Value Problems and the applications that we are interested in to find the solutions. Recall that a second order differential equation with initial conditions is this form: [latex]y''(x)+p(x)y'(x)+q(x)y(x)=r(x)[/latex] and [latex]y(a)=b_{0}[/latex], [latex]y'(a)=b_{1}[/latex]. At here, the initial values are give at [latex]x=a[/latex]. In a multi-valuables and partial differential equation, the initial condition could be given at two different points. A differential equation is a two-point boundary problem if the initial conditions are given at two different points. For example, [latex]y''(x)+p(x)y'(x)+q(x)y(x)=r(x)[/latex] and [latex]y(a_{0})=b_{0}[/latex], [latex]y'(a_{1})=b_{1}[/latex] where [latex]a_{0}\neq a_{1}[/latex].  We begin the study of partial differential equations with the problem of heat flow in a uniform bar of length [latex]L[/latex] , situated on the [latex]x[/latex] axis with one end at the origin and the other at [latex]x=L[/latex]. We assume that the bar is perfectly insulated except possibly at its endpoints, and that the temperature is constant on each cross section and therefore depends only on [latex]x[/latex] and [latex]t[/latex]. We also assume that the thermal properties of the bar are independent of [latex]x[/latex] and [latex]t[/latex]. In this case, it can be shown that the temperature [latex]u=u(x,t)[/latex] at time [latex]t[/latex] at a point [latex]x[/latex] units from the origin satisfies the partial differential equation \[ u_{t}(x,t)=a^{2}u_{xx} \] where [latex]0 \lt x \lt L[/latex], [latex]t \gt 0[/latex], and [latex]a[/latex] is a positive constant determined by the thermal properties. This is the heat equation. To determine [latex]u(x,t)[/latex], we must specify the temperature at every point in the bar when [latex]t=0[/latex], say [latex]u(x,0)=f(x),0\leq x\leq L.[/latex] We call this the initial condition. We must also specify boundary conditions that [latex]u(x,t)[/latex] must satisfy at the ends of the bar for all [latex]t \gt 0[/latex].  We’ll call this problem an initial-boundary value problem. 

Case (1): We begin with the boundary conditions [latex]u(0,t)=u(L,t)=0[/latex], and write the initial-boundary value problem as \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*}

In physics, this type of question means that the bar at the both ends are kept at the [latex]0[/latex], and the temperature of the bars at any position [latex]x[/latex] at time [latex]0[/latex] is given by [latex]f(x)[/latex]. According the physics, one can expect the solution of this problem will give [latex]\lim_{t\rightarrow\infty}u(x,t)=0[/latex]. We will learn how to solve this later. 

Case (2): Now the boundary conditions [latex]u(0,t)=T_{1},u(L,t)=T_{2}[/latex], and write the initial-boundary value problem as \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*}

In physics, this type of question means that the bar at the end of [latex]x=0[/latex] the temperature is kept as [latex]T_{1}[/latex], and at the end of [latex]x=L[/latex], the temperature is kept as [latex]T_{2}[/latex]. The temperature of the bars at any position [latex]x[/latex] at time [latex]0[/latex] is given by [latex]f(x)[/latex]. According the physics, one can expect the solution of this problem will give [latex]\lim_{t\rightarrow\infty}u(x,t)=v(x)[/latex], the temperature is only depending on the position of the bar and is independent of the time. We will learn how to solve this using the first case by assuming the solution is [latex]u(x,t)=v(x)+w(x,t)[/latex] where [latex]w(x,t)[/latex] satisfy the case (1) conditions. [latex]v(x)[/latex] is called the steady state solution. 

Case (3): If both ends of bar are insulated so that no heat can pass through them, then the boundary conditions are [latex]u_{x}(0,t)=0,u_{x}(L,t)=0,t \gt 0[/latex] and write the initial-boundary value problem as \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*} In physics, this type of question means that there is 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]. According the physics, one can expect the solution of this problem will give [latex]\lim_{t\rightarrow\infty}u(x,t)=C[/latex] , a constant. We will learn how to solve this later. 

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) & =40,u(4,t)=0,t \gt 0\\ u(x,0) & =40-10x+20\text{sin}(2x)-40\text{sin}(4x),0\leq x\leq4. \end{align*}

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

(b) Find its steady state solution.

(c) What is [latex]\lim_{t\rightarrow\infty}u(2,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) & =20,u(5,t)=60,t \gt 0\\ u(x,0) & =20+8x+10\text{sin}(2.5x)-30\text{sin}(5x),0\leq x\leq5. \end{align*}

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

(b) Find its steady state solution.

(c) What is [latex]\lim_{t\rightarrow\infty}u(3,t)[/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_{x}(0,t) & =0,u_{x}(4,t)=0,t \gt 0\\ u(x,0) & =40+20\text{cos}(2x)-40\text{cos}(4x),0\leq x\leq4. \end{align*}

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

(b) Find its steady state solution.

(c) What is [latex]\lim_{t\rightarrow\infty}u(2,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_{x}(0,t) & =0,u_{x}(5,t)=0,t \gt 0\\ u(x,0) & =20+10\text{cos}(2.5x)-30\text{cos}(5x),0\leq x\leq5. \end{align*}

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

(b) Find its steady state solution.

(c) What is [latex]\lim_{t\rightarrow\infty}u(3,t)[/latex]?

We consider initial-boundary value problems of Vibrating String: Suppose that an elastic string of length [latex]L[/latex] is tightly stretched between two supports at the same horizontal level. Let the [latex]x[/latex]-axis be chosen to lie along the axis of the string, and let [latex]x=0[/latex] and [latex]x=L[/latex] denote the ends of the string. Suppose that the string is set in motion so that it vibrates in a vertical plane, and let [latex]u(x,t)[/latex] denote the vertical displacement experienced by the string at the point [latex]x[/latex] at time [latex]t[/latex]. Under these assumptions, the string vibration is governed by the one-dimensional wave equation, and has the form \begin{align*} u_{tt}(x,t) & =a^{2}u_{xx}(x,t),t \gt 0,0\leq x\leq L \end{align*}  where [latex]a[/latex] is the velocity of propagation of waves. We assume that the ends of the string remain fixed, and hence [latex]u(0,t)=0,u(L,t)=0,t\geq0[/latex].

Since the wave equation is of second order with respect to [latex]t[/latex], it is plausible to prescribe two initial conditions, the initial position of the string, and its initial velocity: [latex]u(x,0)=f(x)[/latex], [latex]u_{t}(x,0)=g(x)[/latex] where [latex]f(x)[/latex] and [latex]g(x)[/latex] are given functions. In order for these four conditions to be consistent, we require [latex]f(0)=f(L)=0,g(0)=g(L)=0[/latex]. The wave equation problem is \begin{align*} u_{tt}(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),u_{t}(x,0)=g(x),0\leq x\leq L \end{align*}  where [latex]f(x)[/latex] is the initial position function and [latex]g(x)[/latex] is the initial velocity. We will learn how to solve this type of problems in the later section. 

Example 3: Suppose the displacement [latex]u(x,t)[/latex] of a piece of flexible string is given by the initial-boundary value problem \begin{align*} u_{tt}(x,t) & =25u_{xx}(x,t),t \gt 0\\ u(0,t) & =0,u(\pi,t)=0,t \gt 0\\ u(x,0) & =20\text{cos}(2x),u_{t}(x,0)=20-x^{2},0\leq x\leq\pi. \end{align*}

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

(b) What are the initial displacement and initial velocity of the string at the mid- point, [latex]x=\frac{\pi}{2}[/latex]. 

Exercise 3: Suppose the displacement [latex]u(x,t)[/latex] of a piece of flexible string is given by the initial-boundary value problem \begin{align*} u_{tt}(x,t) & =16u_{xx}(x,t),t \gt 0\\ u(0,t) & =0,u(6,t)=0,t \gt 0\\ u(x,0) & =20+10\text{cos}(\pi x)-30\text{cos}(2\pi x),u_{t}(x,0)=10+2x,0\leq x\leq6. \end{align*}

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

(b) What are the initial displacement and initial velocity of the string at the mid- point, [latex]x=3[/latex]. 

GroupWork:

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) & =5u_{xx}(x,t),t \gt 0\\ u_{x}(0,t) & =0,u_{x}(\pi,t)=0,t \gt 0\\ u(x,0) & =9+10\text{cos}(2x)-40\text{cos}(5x),0\leq x\leq\pi. \end{align*}

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

(b) Find its steady state solution.

(c) What is [latex]\lim_{t\rightarrow\infty}u(\frac{\pi}{2},t)[/latex]?

(d) Suppose the initial condition is, instead, [latex]u(x,0)=8+36\text{cos}(6x)+49\text{cos}(7x)[/latex].

Will the limit [latex]\lim_{t\rightarrow\infty}u(\frac{\pi}{2},t)[/latex] be higher than, lower than, or equal to the temperature you found in part (c)?

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) & =10u_{xx}(x,t),t \gt 0\\ u(0,t) & =40,u(10,t)=80,t \gt 0\\ u(x,0) & =40+4x+10\text{sin}(2.5x)-30\text{sin}(5x),0\leq x\leq10. \end{align*}

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

(b) Find its steady state solution.

(c) What is [latex]\lim_{t\rightarrow\infty}u(5,t)[/latex]?

(d) Suppose the initial condition was changed to [latex]u(x,0)=80+10x+20\text{sin}(\pi x)+60\text{sin}(3\pi x)[/latex].

What is [latex]\lim_{t\rightarrow\infty}u(5,t)[/latex] in this case?

3. Consider the following two heat conduction initial-boundary value problems

(A) \begin{align*} u_{t}(x,t) & =8u_{xx}(x,t),t \gt 0\\ u(0,t) & =60,u(5,t)=20,t \gt 0\\ u(x,0) & =30,0\leq x\leq5 \end{align*}

(B)  \begin{align*} u_{t}(x,t) & =8u_{xx}(x,t),t \gt 0\\ u_{x}(0,t) & =0,u_{x}(5,t)=0,t \gt 0\\ u(x,0) & =30,0\leq x\leq5. \end{align*}

(a) Which problem (A or B) models the temperature distribution of a rod with both ends kept at certain constant temperatures?

(b) What is the physical meaning of the boundary conditions of the other heat conduction problem?

(c) Suppose [latex]u_{1}(x,t)[/latex] is the solution of problem A and [latex]u_{2}(x,t)[/latex] is the solution of problem B. Compare the temperatures at the middle of the rod (that is, at [latex]x=2.5[/latex]) as [latex]t\rightarrow\infty[/latex].\\

(i) [latex]\lim_{t\rightarrow\infty}u_{1}(2.5,t) \gt \lim_{t\rightarrow\infty}u_{2}(2.5,t)[/latex]

(ii) [latex]\lim_{t\rightarrow\infty}u_{1}(2.5,t) \lt \lim_{t\rightarrow\infty}u_{2}(2.5,t)[/latex]

(iii) [latex]\lim_{t\rightarrow\infty}u_{1}(2.5,t)=\lim_{t\rightarrow\infty}u_{2}(2.5,t)[/latex]

(iv) There is not enough available information to compare them.

4. Suppose the displacement [latex]u(x,t)[/latex] of a piece of flexible string is given by the initial- boundary value problem

\begin{align*} u_{tt}(x,t) & =25u_{xx}(x,t),t \gt 0\\ u(0,t) & =0,u(1,t)=0,t \gt 0\\ u(x,0) & =x\text{cos}(\pi x),u_{t}(x,0)=0,0\leq x\leq1. \end{align*}

(a) Give a physical interpretation (of functions in the problem): (                ) is the displacement and (                 ) is the velocity of the flexible string at point [latex]x[/latex] at time [latex]t[/latex].

(b) Circle the correct answer regarding the meaning of the boundary conditions. 

(1) They guarantee the displacement is [latex]0[/latex] at the endpoints of the string. 

(2) They keep the motion of the string free and undamped.

(c) Use the boundary conditions to determine the steady-state displacement [latex]v(x)[/latex] of this string.

(d) How will you modify the initial-boundary value problem stated above (by making one change to the problem) so that its solution is equal to the steady-state displacement for all [latex]t \gt 0[/latex]? 

(i) Add a damping term into the equation. 

(ii) Set [latex]u(x,0)=0.[/latex] 

(iii) Set [latex]u_{t}(x,0)=v'(x)[/latex].

(iv) Any of the above changes would do.