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Chapter 5: Problem 3

Consider the Cauchy problem $$ \begin{array}{c} u_{t}+u u_{x}=0, \\ u(x, 0)=\left\\{\begin{array}{cc} 1 & x \leq 0, \\ \cos ^{2} x & 0 \leq x \leq \pi / 2, \\ 0 & x \geq \pi / 2 . \end{array}\right. \end{array} $$ (a) Determine the characteristics and show that they have an envelope of two branches. (b) Plot the picture of chracteristics and their envelope with Mathematica. (c) Find a weak solution.

Short Answer

Expert verified
Characteristics: \( \frac{dx}{dt} = u \). Weak solution involves handling shocks at boundaries.

Step by step solution

01

Introduction to the Cauchy Problem

The given PDE is a Cauchy problem for a first-order non-linear equation: \( u_t + u u_x = 0 \). This is a hyperbolic PDE and can be treated using the method of characteristics.
02

Find Characteristics

For a PDE of the form \( u_t + a(u) u_x = 0 \), the characteristics are given by solving \( \frac{dx}{dt} = u \). For our problem, \( u_t + u u_x = 0 \) implies \( a(u) = u \). So, the characteristics satisfy \( \frac{dx}{dt} = u \). We also have \( \frac{du}{dt} = 0 \), which implies that \( u \) is constant along the characteristic curves.
03

Solve Initial Curve

We use the initial condition \( u(x, 0) = \{ 1, \cos^2(x), 0 \} \) for different intervals. On \( x \leq 0 \), the characteristic is \( u = 1 \), so the lines are \( x - t = \text{const} \). On \( 0 \leq x \leq \pi/2 \), \( u(x, 0) = \cos^2(x) \). Therefore, characteristics in this region are described by \( x - \cos^2(x)t = \text{const} \). For \( x \geq \pi/2 \), we have \( u = 0 \), so the lines are vertical, \( x = \text{const} \).
04

Identify the Envelope

The envelope of characteristics results from intersecting the various branches. The line \( x - t = 0 \) from the first region and \( x - \cos^2(x)t = 0 \) from the second region can be considered. The intersection of these provides the envelope. It forms when two characteristics cross and can be solved if required to find explicit points of crossings.
05

Plot the Characteristics (with Mathematica)

Using Mathematica, we plot the characteristics lines for the relationships derived:- For \( x \leq 0 \), draw lines of the form \( x - t = \text{const} \).- For \( 0 \leq x \leq \pi/2 \), lines are \( x - \cos^2(x)t = \text{const} \).- For \( x \geq \pi/2 \), lines of \( x = \text{const} \).These plots illustrate the envelope at the interface where characteristics intersect.
06

Find a Weak Solution

A weak solution accommodates the discontinuities and can be expressed in terms of shock waves, where \( u(x, t) \) is allowed to have jumps possibly determined via Rankine-Hugoniot conditions. In this problem, designate where \( u \) jumps at \( x = 0 \) and \( x = \pi/2 \). Compute speeds of potential shocks based on constant characteristics across each region.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Characteristics Method
The characteristics method is a powerful tool for solving partial differential equations, particularly first-order nonlinear equations. This method involves transforming a PDE into a family of ordinary differential equations (ODEs) which are easier to solve. For the equation \( u_t + u u_x = 0 \), we're dealing with a hyperbolic PDE, where the method of characteristics allows us to track how information propagates along curves in the \( (x,t) \)-plane.
  • Characteristics curves: These are curves along which the PDE becomes an ODE. For our problem, they are determined by the equation \( \frac{dx}{dt} = u \), indicating that \( x \) changes over time at a rate given by \( u \).
  • Constant Solution: Along these characteristic curves, \( u \) remains constant, i.e., \( \frac{du}{dt} = 0 \).
  • Initial Conditions: These are vital for determining the specific form of the characteristics and ensuring compatibility with the problem's initial setup.
Understanding these aspects allows us to trace how the solution evolves from the initial state, leading to further applications such as shock formation or wave propagation.
Hyperbolic Partial Differential Equations
Hyperbolic partial differential equations are a class of PDEs suited to describing wave propagation and convection processes, often arising in physics and engineering problems. They are characterized by real eigenvalues, leading to wave-like solutions.
  • Wave Nature: Solutions to hyperbolic PDEs, like our \( u_t + u u_x = 0 \), tend to propagate along characteristic lines, making them ideal for modeling phenomena like compressible fluid flow.
  • Cauchy Problem: This is a common setup involving hyperbolic PDEs, particularly relevant when initial conditions are known and solutions are desired for subsequent time evolution.
  • Stability and Speed: Their hyperbolic nature requires careful consideration of stability and propagation speed, crucial for scenarios like predicting how disturbances move through a medium.
By recognizing the hyperbolic nature of these equations, students can better appreciate their applicability to real-world dynamic systems and processes.
Weak Solution
A weak solution to a differential equation is a generalized kind of solution that allows for discontinuities, making it especially useful for problems involving shocks or other abrupt changes.
  • Discontinuities Management: In our Cauchy problem, the weak solution handles the potential jumps in \( u \) at points like \( x = 0 \) or where characteristics intersect, suggesting a change in solution behavior.
  • Rankine-Hugoniot Conditions: These are utilized to determine how the discontinuities or shocks propagate based on conservation laws, defining the shock speeds in terms of characteristic information.
  • Interpretation: Even if \( u \) doesn't have a classical solution form, the weak solution ensures that the PDE is satisfied in integral or averaged sense, accommodating more complex behaviors.
Implementing weak solutions is key to managing cases where straightforward solutions fail due to the inherent complexities of the problem. This is particularly vital in fields like fluid dynamics and signal processing.
Initial Conditions
Initial conditions specify the state of the system at \( t = 0 \) and guide the evolution of the solution over time. They are crucial for a well-posed Cauchy problem, where they dictate how the characteristics are arranged originally.
  • Determining Characteristics: The initial value \( u(x, 0) \) guides the slope of the characteristic lines. For instance, constant initial values yield straight-line characteristics, while variable ones can lead to more complex configurations.
  • Region-Specific Behavior: Since the initial conditions in our problem vary by region, they result in differing characteristic structures, creating a need to consider each subregion鈥檚 dynamics individually.
  • Consistency: The provided initial conditions must align properly with the PDE to ensure a consistent and meaningful solution across all \( x \) values.
Appropriately handling initial conditions is crucial for the accurate prediction and depiction of the solution's progression, especially in environments subject to changing dynamics across different regions.

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Most popular questions from this chapter

(a) Consider the Cauchy problem for the advection-diffusion equation $$ \begin{aligned} u_{t}+a u_{x} &=\varepsilon u_{x x}, \quad x \in \mathbf{R}, t>0, \\ u(x, 0) &=u_{0}(x), \quad x \in \mathbf{R}, \end{aligned} $$ with \(u_{0}(x) \in L^{1}(\mathbf{R})\). Making a change of variables \(v(x, t)=u(x+a t, t)\) it reduces to Cauchy problem for the diffusion equation Shock Waves and Conservation Laws $$ \begin{aligned} v_{t} &=\varepsilon v_{x x}, & x \in \mathbf{R}, t>0, \\ v(x, 0) &=u_{0}(x), & x \in \mathbf{R} \end{aligned} $$ (b) Show that the solution \(u_{\varepsilon}(x, t) \in C^{\infty}(\mathbf{R} \times(0, \infty))\) of \((5.27)\) is $$ u_{\varepsilon}(x, t)=\frac{1}{2 \sqrt{\varepsilon \pi t}} \int_{-\infty}^{\infty} u_{0}(\xi) e^{-\frac{1}{4 \varepsilon t}(x-a t-\xi)^{2}} d \xi $$ and \(u_{\varepsilon}(x, t) \sim u_{0}(x-a t)\) as \(\varepsilon \rightarrow 0\).

Show that the equation \(u_{t}+u u_{x}=\varepsilon u_{x x}\) has a traveling wave solution of the form \(u_{\varepsilon}(x, t)=w(x-a t)\), where \(w\) satisfies the equation $$ \varepsilon w^{\prime}(y)+a w(y)=\frac{1}{2} w^{2}(y)+C . $$ Verify that the function $$ w(y)=a-\sqrt{a^{2}-2 C} \tanh \frac{1}{2 \varepsilon} \sqrt{a^{2}-2 C} y, \quad C \leq \frac{a^{2}}{2}, $$ satisfies the equation (5.32). Determine the behavior of this solution as \(\varepsilon \rightarrow 0\).

Consider the problem $$ \begin{aligned} u_{t}+u u_{x}+a u &=0_{1}, \\ u(x, 0) &=u_{0}(x) . \end{aligned} $$ Show that the characteristics of the problem are $$ x=x_{0}+\frac{1-e^{-a t}}{a} u_{0}\left(x_{0}\right) . $$ Discuss the question of breaking of solutions.

(a) Find the characteristics and the solution of the problem $$ \begin{array}{c} u_{t}+u u_{x}=0, \\ u(x, 0)=\left\\{\begin{array}{lc} \alpha a+\beta & x \leq a \\ \alpha x+\beta & a \leq x \leq b, \\ \alpha b+\beta & x \geq b, \end{array}\right. \end{array} $$ (b) Show that: if \(\alpha \geq 0\) the solution is differentiable for \(t \geq 0\) if \(\alpha<0\) the solution is continuous for \(0 \leq t<-\frac{1}{\alpha}\).

(a) Let \(u(x, t)=|x|, \quad B=\left\\{(x, t): x^{2}+t^{2}<1\right\\}\). Verify that \(u\) has weak derivatives $$ \frac{\partial u}{\partial x}=\operatorname{sgn} x=\left\\{\begin{array}{cl} 1 & \text { if } x>0 \\ -1 & \text { if } x<0 \end{array}, \quad \frac{\partial u}{\partial t}=0\right. $$ in \(B\). (b) Let \(u(x, t)=\operatorname{sgn} x, \quad B=\left\\{(x, t): x^{2}+t^{2}<1\right\\} .\) Show that \(u\) has not a weak derivative \(\frac{\partial u}{\partial x}\) in \(B\).
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