# Questions tagged [characteristics]

The method of characteristics is a way of solving certain partial differential equations by reducing them to ordinary differential equations. It is most often used for 1st order equations. Use with the (pde) tag.

331 questions
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### Why does the cardinality of the vector space over a finite field of characteristic $p$ have to be a power of $p$?

In a lecture note that I have, it is written that if $F$ is a field of $q$ elements of characteristic $p$, then $q = p^m$ for some $m>0$. To show this, observe that $F$ is a vector space ...
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### Smooth solutions of $u_t - x u u_x = 0$ deduced from characteristics

Consider the equation $u_t - x u u_x = 0$. with cauchy data $u(x,0) = x$. Solving this equation I see the characteristics are given by $x= r e^{-rt}$ for some $r$ and the solution is defined ...
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### Solution for $u_t+u_x=0$ using characteristics

P. Dravek and G. Holubova, Elements of Partial Differential Equations, Section 3.4 Exercise 22: Show that the initial value problem $$u_t + u_x = 0,\; u(x,t) = x \;\text{ on }\; x^2+t^2=1.$$ ...
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### Solving $u_t + cu_x = k$ by method of characteristics

Given the 1st order linear PDE $$u_t + cu_x = k$$ with initial condition $u(x,0)=\mathrm{cosh}2x$, I am required to find a solution using the method of characteristics. Characteristic equations are ...
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### Basic question about a first-order linear equation

I am just learning PDE. My lecture notes say the following: Consider the IVP $$\begin{cases} u_t + a u_x = 0 \\ u(x,0) = \phi(x) \end{cases}$$ where $a \in \mathbb{R}$. Our goal is to reduce this ...
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### Finding the time when the speed of discontinuity becomes time-dependent in traffic flow

I am trying to use the following conservation law: $$u_t+f(u)_x=0 \ \ \ \ \text{where} \ \ \ f(u)=u(1-u).$$ IC: $u(x,0)=\frac{1}{4}$ for BC: $u(0,t)=1$ for $t>0$. I found the solution ...
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### Behavior of the solution of the eikonal equation

Consider the nonlinear ﬁrst-order initial-value problem: $$(u_t )^2 + (u_x )^2 = 1$$ with initial condition $u(x, 0) = {−\sqrt{1+x^2}}$. Find its solution for all $t>0$ using the method of ...
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### Characteristic curves for second-order Tricomi equation

Consider the Tricomi equation $$yu_{xx} + u_{yy} = 0$$ Find ordinary differential equations describing the real characteristic curves and solve these ODEs to obtain equations for the ...
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### Solve IVP of $(u_t )^2 + (u_x )^2 − u^2 = 0$ using method of characteristics

Consider the nonlinear ﬁrst-order initial-value problem: $$(u_t )^2 + (u_x )^2 − u^2 = 0$$ with initial condition $u(x, 0) = Ae^{−\sqrt{1+x^2}}$. (a) Find its solution for all $t > 0$ ...
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### Shock formation condition in IVP of $u_t + uu_x + \alpha u = 0$

Consider $u_t + uu_x + \alpha u = 0$ for $t > 0$, all $x$ where $\alpha > 0$ is a constant. Find the characteristic equations for the equation with initial data $u(x, 0) = f(x)$ given. Show ...
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### Domain of definition for $u_x + uu_y = 1$

How do i find the domain of definition for $u_x + uu_y = 1$ with $u = x/2$ on $y=x$ , $0 \leq x \leq 1$ I parametrise by letting $y=s$ , $x=s$ , $u=s/2$ , $0 \leq s \leq 1$ at $t=0$ The ...
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### Nonhomogeneous Semi-Linear PDE with the Characteristic Method

I need to solve this semi-linear PDE: $u_x - 3u_y = \sin (y) + \cos (x)$ The initial condition provided is: $u (t,t)= t^2$ I need to use the Characteristic Method. I learned the method from this ...
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### Uniqueness of solution based on characteristic curves

I have a pde $$\begin{cases} u_t − xu_x = 2u & x\in\mathbb{R}, t>0\\ u(x, 0) = \frac{1}{1+x^2} \end{cases}$$ I've solved it using method of characteristics ($u=\frac{1}{1+x^2e^{2t}}e^{2t})$ ...
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### Solve the pde $u_xu_y=u$

I want to solve the following pde $$\left\{\begin{array}{cc} u_xu_y=u \mbox{ on \Omega:=\{(x,y)|x>0\}} \\ u(0,y)=y^2 \end{array}\right.$$ I supposed that $u$ was a polynomial of two variables ...
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### Question about characteristics and classification of second-order PDEs

I am currently reading through the book 'Computational Techniques for Fluid Dynamics', by C.A.J. Fletcher. Chapter 2 discusses classification of PDEs by finding the number and nature of their ...
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### Solve this Semi-Linear PDE (Partial Differential Equation) with the Characteristic Method

I need to solve this linear PDE: $3u_x - 4u_y = y^2$ The initial condition provided is: $u (0,y)= sin(y)$ I need to use the Characteristic Method. I learned the method from this video. I have ...
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### Inviscid Burgers equation

Burgers Equation Consider the initial value problem for Burger's equation \begin{align}\begin{cases} u_{t} + u u_{x} = 0 \\ u(x,0) = \phi(x) \end{cases} \end{align} \tag{1} our ...
Let $\displaystyle u_x^2+u_y^2=n_0^2$ be given, with the initial condition that $u(x,2x)=1$ and $n_0\in\mathbb{R}$ I want to find a solution using the methods of characteristics. I computed the ...