Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
19 views

Characteristic Vector Question

I have been asked the following questions on a tutorial worksheet and am not sure how to answer. "There is a natural relationship between sets and bit strings which is called the characteristic ...
1
vote
1answer
33 views

Shock of Bruger equation $u_t+uu_x=0$ at $t=0$

Consider the Burgers equation $u_t+uu_x=0$ with the initial condition $$u_0(x) = \begin{cases} u_l,x\leq0\\ u_r,x>0 \end{cases}$$ My attempt to solve this: Using the method of characteristics we ...
0
votes
0answers
12 views

Show from the principle of conservation of mass, that if the shock moves at a constant speed, then $Q(T) − Q(0) = sT(u_l − u_r ).$

Consider the Burgers equation $u_t+uu_x=0$ with the initial condition $$u_0(x) = \begin{cases} u_l,x\leq0\\ u_r,x>0 \end{cases}$$ Consider the region in the $xt$−plane given by $[−1, X] × [0, T]$ ...
-1
votes
0answers
17 views

How is solve If A and B are similar n a matrices, then show that A and B have the same characteristic equation and therefore have the same eigenvalues

If A and B are similar n a matrices, then show that A and B have the same characteristic equation and therefore have the same eigenvalues.
2
votes
1answer
38 views

Method of characteristics non-linear PDE

Consider the following initial-value problem: $$xu_x-uu_t=t$$ $$u(1,t)=t$$ I've come to the follow characteristic equations: $$\frac{\mathrm{d}x}{\mathrm{d}\tau }=x,\,\,\,\frac{\mathrm{d}t}{\...
1
vote
0answers
39 views

Solving definite integral in two variables.

Solving a PDE with the following boundary problem with arbitrary constant $b$: $$u(0,t)=F(t)=b\int_0^\infty u(a,t)\mathrm{d}a$$ Hint given in the question is as follows: Split this integral in two ...
0
votes
1answer
11 views

PDE characteristics clarification of paper

I am curious how the following circled part is found. I have been trying to figure it out for the past half hour or so and I can't find the computation that gives that number $x(t)=\sqrt{2t}$. It ...
1
vote
2answers
45 views

Solve PDE using method of characteristics with non-local boundary conditions.

Given the population model by the following linear first order PDE in $u(a,t)$ with constants $b$ and $\mu$ : $$u_a + u_t = -\mu t u\,\,\,\,\,a,t>0$$ $$u(a,0)=u_0(a)\,\,\,a≥0$$ $$u(0,t)=F(t)=b\...
0
votes
0answers
21 views

Proving that a ring with some properties is commutative

A is a ring with the next properties: a) the order of $1$ is p (prime) in the group $(A,+)$ b) there exists $B \subset A$ with $p$ elements such that : for all $x,y \in A$, exists $b \in B$ which ...
1
vote
1answer
15 views

$k\in \mathbb Z,\ (k,p)=1$ the element $k\cdot1_A$ is invertible

Let $A$ be a ring. The order of $1_A$ in (A,+) is p (prime). For $k\in \mathbb Z,\ (k,p)=1$ the element $k\cdot1_A$ is invertible. I tried to prove this. $(k,p)=1 \to \exists m,n\in \mathbb Z\ s.t \...
1
vote
0answers
28 views

PDE-find general solution and solve initial value problem [closed]

I stuck with method of characteristics here, how to find the general solution and IVP? ${u_{xx} + 4u_{xy}+3u_{yy}}={0}, -\infty <x<+\infty, t >0 $ $ u(x,0)=0, -\infty <x<+\infty, $ ...
2
votes
3answers
40 views

Method of Characteristics for $u_t + uu_x = -2u$

Consider the following quasi-linear PDE : $u_t + uu_x = -2u$, with the boundary condition $u(0,t) = e^{-t}$. Show, using the method of characteristics, that the solution to this boundary value ...
0
votes
1answer
39 views

Method of Characteristics - Second order derivatives - Need help finding

given that $\zeta _{(x,y)}= \ln y+\frac{1}{x}$ and $\phi_{(x,y)}= 4\ln y-\frac{1}{x}$ , I've been asked to find $\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial^2 u}{\...
1
vote
0answers
21 views

Method of Characteristics - Can we have multiple answers

Determine the characteristics of the equation $z=p^2-q^2$, and find the integral surface which passes through the parabola $4z+x^2=0$, $y=0$. I solved the question and my answer was: The ...
2
votes
0answers
65 views

Solve wave equation with non-constant wave speed using method of characterstics?

I am trying to get a better understanding of wave pulses in a domain with a non-constant wave speed. I am trying to solve either one of the two equations: $$\frac{\partial^2u}{\partial t^2}-c(x)^2\...
2
votes
2answers
74 views

Solving an inhomogeneous Burgers' equation with the method of characteristics

I am trying to solve the PDE $$u_t+5uu_x=u,$$ subject the boundary condition $u(0,t)=e^{14t}.$ I first start by defining the set of characteristic equations, $$\frac{dt}{1}=\frac{dx}{5u}=\frac{du}{...
0
votes
2answers
40 views

Using the characteristics to get the canonical form of a pde

I've been asked to consider this parabolic equation. $ 3\frac{∂^2u}{∂x^2} + 6\frac{∂^2u}{∂x∂y} +3\frac{∂^2u}{∂y^2} - \frac{∂u}{∂x} - 4\frac{∂u}{∂y} + u = 0$ I calculated the characteristic ...
1
vote
1answer
33 views

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 ...
1
vote
2answers
72 views

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 ...
2
votes
2answers
50 views

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 $$u_t + u_x = 0,\; u(x,t) = x \;\text{ on }\; x^2+t^2=1.$$ ...
1
vote
1answer
33 views

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 ...
0
votes
0answers
32 views

First integrals and Lagrange’s method for PDE

If we have a linear first order partial differential equation one way to solve the equation is to think that if $U$ is the solution of the equation and we define a vector filed using the coefficients ...
1
vote
1answer
50 views

Burgers' Equation with Initial and Boundary Conditions

Consider a first-order PDE: $$u_t + (1 + 2u)u_x = 0$$ valid on $$0 \leq x \leq \infty$$ $$0 \leq t \leq \infty$$ with Initial condition: $$u(x, 0) = 0$$ and boundary condition: $$u(0,t) = \begin{...
0
votes
1answer
28 views

Find general solution to PDE using characteristic equation

Just want a check to a question I've attempted. I have to find a general solution to this pde using the characteristic equation: $ \frac{∂u}{∂x} - 4\frac{∂u}{∂y} - 3u = 0$ So I set $a=1, b=-4$ and $...
0
votes
0answers
31 views

Solve PDE with characteristic curves

I'm new to these sorts of questions and the wording of what it wants, the question is: Consider the first-order PDE: $\frac{∂u}{∂x} + 2 \frac{∂u}{∂y} = 0$ Find the characteristic curves in the form $...
4
votes
4answers
76 views

One-way wave equation IBVP

Plese help me to find the solution of te following equation. For values of $x$ in the interval $[-2,3]$ and $t>0$ we consider the one way wave equation $$u_t+u_x=0$$ with initial data \begin{...
0
votes
1answer
37 views

Characteristic curves of a pde

I am new to these sorts of questions, and the method of characteristics. I've been asked to consider the equation: $ \frac{∂u}{∂x} + xy^{3}\frac{∂u}{∂y} = 0$ I need to find the characteristic curves ...
2
votes
1answer
55 views

Solving nonlinear 1D advection pde with MoC

I would like to solve the 1D nonlinear advection equation with the method of characteristics. Here is my notation: \begin{equation} \begin{cases} \rho_t + (1+\rho)\rho_x = 0\\ \rho = \rho(x,t); \quad ...
0
votes
0answers
14 views

Uniqueness of solutions to PDEs via method of characteristics

I'm revising the method of characteristics for my upcoming exam on PDE's and I am a bit confused by an apparent ambiguity that I am always able to find two different forms of solutions to first order ...
1
vote
3answers
93 views

How to solve this partial differential equation $\frac{\partial p(k,t)}{\partial t}+k\frac{\partial p(k,t)}{\partial k}+k^2p(k,t)=0$

How to solve this partial differential equation $$\frac{\partial p(k,t)}{\partial t}+k\frac{\partial p(k,t)}{\partial k}+k^2p(k,t)=0$$ I'm a beginner to PDE, I think I need to construct the ...
2
votes
1answer
22 views

Question about the differentiability of solution on base characteristics curve.

Let $u(x, t)$ be a function that satisfies the PDE: $u_t+uu_x = 1, x \in \mathbb{R}, t > 0$, and the initial condition $u\big(\frac{t^2}{4}, t\big) = \frac{t}{2}$. Then show that the IVP has ...
0
votes
1answer
41 views

Uniqueness: linear first order pde with constant coefficients

Let us say I find the characteristic lines of some easy PDE $a U_x + b U_y = 0$ to be $bx-ay=c$, where $b, a, c$ are constants. Now, we say the solution must be constant along those lines, so it HAS ...
-2
votes
2answers
56 views

Partial Differential Equations Question: Find an explicit expression for the solution of the IVP [closed]

Find an explicit expression for the solution of the IVP $$ \begin{cases} u_{t}(x,t)+u_{x}(x,t)+u(x,t)=e^{t+2x}\\ \\ u(0,x)=0, \end{cases} $$ by using the method of characteristics
1
vote
0answers
32 views

Characteristics for nonhomogeneous wave equation $y_{tt}=y_{xx} + f$

Consider the initial- and boundary-value problem $$\eqalign{ & {y_{tt}} = {y_{xx}} + f(t,x){\text{ }}{\text{, (t}}{\text{,x)}} \in {\text{(0}}{\text{,}}\infty {\text{)}} \times {\text{(0}}{\text{...
1
vote
1answer
34 views

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 ...
2
votes
1answer
66 views

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 ...
0
votes
1answer
27 views

Solve $u_x + 4xu_y = 1 + u^2$ for $u(0,y)=y$

I got a weird result so I'm not sure I did this right Let the initial condition be $u(0, y_0) = y_0 $ for some $y_0$ By the method of characteristics let $$\frac{dx}{ds} = 1 \to x = s + A$$ $$x(s=0)...
0
votes
2answers
55 views

Are both answers for $xu_x + yu_y = 0$ valid?

Solving this problem by the method of characteristic curves we have to solve the ODE $$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{y}{x}$$ which gives us $$C = \ln(y/x)$$ where $C$ is constant. ...
0
votes
1answer
36 views

Characteristic coordinates $ξ(x, y)$ and $η(x, y)$ for $xu_{xx} + u_{yy} = 0$ when $x<0$

How would I determine the characteristic coordinates for $xu_{xx} + u_{yy} = 0$? This PDE reads $au_{xx} + 2b u_{xy} + cu_{yy} = 0$ with $a=x, b=0, c=1$. The polynomial equation $a\lambda^2 -2b\...
0
votes
2answers
49 views

Quasilinear PDE using method of characteristics

The equation is: $yu_x+uu_y=-xy$ with initial conditions $u=y$ on $x=0$ I first find that $\frac{dx}{y}=\frac{dy}{u}=-\frac{du}{xy}$ Solving $\frac{dx}{y}=\frac{dy}{u}$ we get, $ux=\frac{1}{2}y^2+...
2
votes
0answers
41 views

Exact solution for two coupled non-homogeneous transport equations

I want to solve the following system $$\eqalign{ & {y_t} = -{y_x} + z{\text{ in (0}}{\text{,T)}} \times {\text{(0}}{\text{,1)}} \cr & {z_t} = {z_x} + y{\text{ in (0}}{\text{,T)}} \times {...
0
votes
1answer
49 views

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 ...
1
vote
1answer
97 views

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 ...
1
vote
1answer
81 views

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 $t > 0$ ...
2
votes
2answers
81 views

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 ...
1
vote
2answers
73 views

Method of characteristics for $f f_x + f_y = 1$. Where is the solution valid?

Suppose we have a PDE that can be solved with the method of characteristics \begin{align} F(\nabla u, u , x) = 0 \text{ in $U$}\\ u|_\Gamma = g \text{ on $\Gamma$ } \end{align} Where $\Gamma \...
0
votes
2answers
50 views

Characteristic Curve of a PDE

Question from a Exam: Consider the pde $xu_{xx}+2x^2u_{xy}=u_x-1$. Find the characteristic curves of the above. Can someone please tell me how are these types of problems handled? I dont want ...
1
vote
0answers
35 views

existence of solutions for Cauchy problems

Consider the equation $$(1-\cos x)u_{tt} - u_{tx} - u_{xx} = 0$$ with Cauchy data $$u(x,0) = f(x), u_t(x,0) = g(x),\text{ for } f,g\in\mathcal{C}^2$$ What compatibility condition do $f$ and $g$ have ...
1
vote
1answer
51 views

charpits method to solve $u_x^2 + yu_y = u$

$u_x^2 + yu_y = u$ subject to $u(x,1) = 1 +x^2 /4$ for $-\infty < x < \infty $ Setting $p = du/dx$ and $q = du/dy$, I get $p^2 +yq = u$ and so I am able to write the diffeq as $F(p,q,y,u) = p^2 ...
2
votes
2answers
71 views

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 ...