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.

Filter by
Sorted by
Tagged with
0
votes
1answer
45 views

A first-order linear pde

I am interested in solving the following pde $$a u v \partial_u f - (b + u v) \partial_v f = 0$$ over the reals, possibly with some restriction on the range of $a,b\in\mathbb{R}$ if necessary. I only ...
0
votes
0answers
25 views

Nonhomogeneous Burgers equation weak (or Integral) Solutions and Rankine-Hugoniot

We have the following equation the nonhomogeneous Burgers equation with initial condition \begin{align*} uu_x +u_y&= 1\\ u(x,0)&=x\quad x\in\mathbb{R} \end{align*} corresponding to the ...
1
vote
0answers
24 views

Find all characteristic curves of PDE $ u_{t}-(x^2-1)u_{xx}=0$ for $(x,t) \in \mathbb{R}^2$.

I am trying to solve the following problem from the 2021 Analysis of PDE Tripos exam: Find all the characteristic curves to the equation $$ u_{t}-(x^2-1)u_{xx}=0. $$ When $x \neq 1,-1$, the definition ...
2
votes
1answer
48 views

Solve $u_{x_1}u_{x_2} = u$ with $u(0,x_2)=x_2^2$

I'm solving the non-linear IVP $$u_{x_1}u_{x_2} = u, \qquad u(0,x_2)=x_2^2$$ The general PDE has the form $F(p,q,z)=pq-z=0$. The characteristic differential equations are \begin{align*} \frac{dx_1}{...
0
votes
0answers
50 views

Solving the Euler equation $\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+\frac{\partial p}{\partial x}=0$

In solving the one dimensional Euler equation in fluid dynamics, we have two functions. What should I do If I want to solve it without considering that one of them is constant? $$\frac{\partial u}{\...
0
votes
1answer
47 views

$u_x + u^2u_y = 1$ with $u(x,0)=1$

How can we solve $u_x + u^2u_y = 1$ with $u(x,0)=1$? I have reached a dead end. I'm using the characteristic relations $$\frac{dx}{dt}=1,\enspace \frac{dy}{dt}=u^2,\enspace \frac{du}{dt}=1$$ The first ...
1
vote
1answer
49 views

Rankine-Hugoniot for Burgers equation with Lorentzian initial data

We have the following equation with initial condition \begin{align*} u_y + uu_x &= 0\\ u(x,0)&=\frac{1}{1+x^2}\quad x\in\mathbb{R} \end{align*} let's say $g(x)=\dfrac{1}{1+x^2}$ and ...
0
votes
1answer
58 views

Solving $uu_{x_1}+u_{x_2}=u$ with boundary conditions

I'm trying to solve the problem \begin{cases} uu_{x_1}+u_{x_2}=u & (x_1,x_2)\in U\\ u(x_1,x_2)=2x_1, &(x_1,x_2)\in\Gamma \end{cases} where $U:=1<x_1<2,x_2>0$ and $\...
1
vote
1answer
45 views

How to apply method of characteristics to this PDE?

I want to solve the following PDE using the method of characteristics $$ u_{tt} - 2tu_{xt} +t^2 u_{xx} - u_x =x$$ The hint I was given is the following: Find $a$ such that $$ u_{tt} - 2tu_{xt} +t^2 u_{...
0
votes
0answers
47 views

Second-order PDE with non-constant coefficients: method of characteristics

I have formulated the following PDE during my research: $$ \rho(W - K + \delta R)\partial_W U + \mu R\, \partial_R U - \gamma = 0\ , $$ where $U = U(W,R)$, and all other symbols in the above are ...
0
votes
0answers
49 views

Burgers equation with method of characteristic [duplicate]

This question is example 1 page 139 Evans PDE book 2nd edition. Consider the initial value problem: \begin{cases} u_t+\left(\frac{u^2}{2} \right)_x =0 & \text{in }\mathbb{R}\times(0,\infty) \\ \...
0
votes
1answer
51 views

Solving PDE by separation of variables method

I'm trying to solve $$u_x - 2u_y = u$$ with initial condition $u(x,0)=6e^{-3x}$. I begin by trying to find a solution of the form $$u(x,y)=X(x)Y(y)$$ substituting this into the PDE yields the relation ...
1
vote
0answers
32 views

Reduce PDE to canonical form

Task I have the following equation: $$x^2u_{xx}-2u_{xy}+u_{yy}=0$$ which I need to reduce to its canonical form in the region where it is hyperbolic. My solution The region is the following: $x\in(-1,...
2
votes
1answer
69 views

How can I solve this 1st order non-homogeneous PDE?

I've been trying to solve the initial value problem: $$u_t+2u_x-3u_y+4u_z+u=e^{xyz}$$ $$u(x,y,z,0)=\sin(xyz)$$ I've tried to approach the problem by characteristics' method but the non-homogeneous ...
0
votes
0answers
38 views

Largest value of time for which solution of PDE can be continued to $[0, t)$

It's a promblem from Arnold ODE book. Find the largest value of $t$ for which the solution of the Cauchy problem $$ u_{t}+u u_{x}=-\sin x,\left.\quad u\right|_{t=0}=0 $$ can be continued to $[0, t)$. ...
0
votes
0answers
35 views

Inconsistent Solution Techniques for Quasilinear PDE

In this video, the final step in solving the quasilinear PDE is to write $c_1 = f(c_2)$. The example used is the Burger Equation $uu_x + u_y = 0$, and the solution is $u = f(x - uy)$. However, in ...
-1
votes
1answer
74 views

Finding characteristic curve of PDE

The characteristic curves of PDE $(2x+u)u_x + (2y+u)u_y = u$ passing through $(1,1)$ for any arbitrary initial values prescribed on a non characteristic curve are given by: $x=y$ $x^2 + y^2 = 2$ $x ...
0
votes
0answers
16 views

Solve transport equation when initial data is along $x = -t$.

Problem. Consider $c > 0$ and the PDE $u_t - cu_x =1$, $u(x,-x) = \phi(x)$. Solve, if possible, using the method of characteristics. My Attempt. I write the system of ODEs so that $\frac{dt}{d\tau} ...
1
vote
1answer
59 views

Trouble solving a Cauchy's problem, what went wrong?

I am trying to solve the following Cauchy's problem: $$u_x-3u_y=\sin x + \cos y \\ u(t,t)=p(t) $$ Following the example given here and here. I guess the system of equations associated with it is: $$\...
0
votes
0answers
26 views

In the context of solving PDEs with Method of Characteristics, what are unions of curves?

So I was trying to wrap my head around the method described by this article here https://www.iist.ac.in/sites/default/files/people/IN08026/MoC_0.pdf. Now the issue I have is what they stated, that ...
0
votes
0answers
31 views

Geometric Indicators in Solutions to Different Types of First Order PDEs

I have been graphing solutions to two input, first order PDEs as three-dimensional surfaces with characteristic curves printed upon them, in order to better understand the solution logic in this ...
2
votes
2answers
53 views

Making Sense of Method of Characteristics Solution Geometry

How can I make sense of the following surface as insight into understanding the method of characteristics? The partial differential equation this initially came from was $2xu_x + u_y = 0$, which has ...
0
votes
0answers
23 views

Why are first integrals constant along characteristic curves?

Why are first integrals constant along characteristic curves? My lecturer in a PDE course seems to be using this fact. I found a nice explanation of what a first integral is here: Why are they called &...
1
vote
1answer
30 views

Characteristics method, an implicit solution

I am solving the following PDE: $$xy(u_x-u_y)=(x-y)u$$ using the characteristic curves method. To obtain the characteristic curves I have to solve: $$\dfrac{dx}{xy}=\dfrac{dy}{-xy}=\dfrac{du}{(x-y)u}$$...
1
vote
1answer
43 views

Characteristic Equation Intgeration PDE

I've got this PDE: $$2u^3u_x+ u_y= x,\\ u(x,0) =\sqrt x$$ I've gotten the first steps out the way but I have the problem of the differential integration: $$\frac{\partial x}{\partial \tau } = 2u^3 \\ \...
0
votes
1answer
56 views

Maximum lifetime of solution to exponential Burgers equation

Consider the Burgers Equation $$\left\{\begin{array}{ccl} u_t + [e^u]_x & = & 0;\\ u(x,0) & = & u_0(x). \end{array}\right.$$ a) Study the behavior of the characteristic curves in the ...
0
votes
0answers
9 views

question for the charateristic method part of evans

In evans page 90 , the last sentence is " We will sometimes refer to $\mathbf{x}(\cdot)$ as the projected characteristic: it is the projection of the full characteristics $(\mathbf{p}(\cdot),z(\...
1
vote
0answers
13 views

characteristic system for the linear PDE when some partial derivatives does not appear

The first step to solve the linear Partial Differential Equation $$ f_1(x_1,\dots,x_n)\frac{\partial\omega(x_1,\dots,x_n)}{\partial x_1}+\cdots+f_n(x_1,\dots,x_n)\frac{\partial\omega(x_1,\dots,x_n)}{\...
2
votes
1answer
31 views

Help find my mistake in using method of characteristics to find general solution to wave-equation?

I obtained sum of unidirectional waves, and not the sum of a forward and backward wave. Where did I go wrong? $$u_{tt}=c^2u_{xx}$$ $$\iff u_{tt}-c^2u_{xx}=0$$ $$\iff (\dfrac{\partial}{\partial t} - c ...
0
votes
0answers
26 views

Numerical solution of Hyperbolic PDE using Characteristics

I am studying numerical solutions of hyperbolic pde's by the method of characteristics. Suppose I have a PDE $$u_{xx}-u_{yy}=0$$ with initial conditions $u(0,t)=u(1,0)=0$ and initial conditions as $u(...
1
vote
1answer
25 views

How to Find $u(x,y)$ for the PDE $u_x + 2u_y + (2x − y)u = 2x^2 + 3xy − 2y^2$ using Method of Characteristics and Method of Integrating factors?

$$dx=\dfrac{dy}{2}=\dfrac{du}{(-2x+y)u+2x^2+3xy-2y^2}$$ $$\dfrac{dx}{dy}=\dfrac{1}{2} \implies x=\dfrac{y}{2}+A$$ By sagemath software, $$\dfrac{du}{dy}=\dfrac{(-2x+y)u+2x^2+3xy-2y^2}{2}=1.0 \, A^{2} ...
1
vote
2answers
35 views

Why is initial curve not assumed to be coincident with characteristics

I was reading the method of characteristic for first order p.d.e's. $a\frac{\partial u}{\partial x}+b\frac{\partial u}{\partial y}=c$. Now it said to find out the solution we try to find out a ...
0
votes
0answers
28 views

Orthogonal basis in 2-dimensional vector spaces

Let $V$ be a $2$-dimensional vector space over a finite field $F_p$. Given a non-degenerated symmetric bilinear form, what may I ask to $F_p$ to guarantee that there exists an orthogonal basis of $V$? ...
1
vote
2answers
130 views

Solve $uu_x+u_y=-\frac{1}{2}u$ with characteristics

I'm stuck trying to solve $uu_x+u_y=-\frac{1}{2}u$ satisfying $u(x,2x)=x^2$, this solution also needs to be in some parametrised form with $x(r,s), y(r,s),u(r,s)$. So far I have this: $\dfrac{dx}{ds}=...
1
vote
0answers
46 views

Burgers' equation initial conditions for global existence

given IVP: $\begin{cases} u_t+uu_x=-u \\ u(x,0) = f(x) \end{cases}$ using method of characteristics ($\dot F=\frac{dF}{ds}$): $\dot T(\sigma,s) = 1 \quad T_0=0\\ \dot X(\sigma,s)=U(\sigma,s) \quad X_0=...
2
votes
2answers
36 views

$yu_x+xu_y=u$ with two conditions using method of characteristics

$\begin{cases} yu_x+xu_y=u\\u(x,0) = x^3 \\ u(0,y)=y^3\end{cases}$ $\dot X(\sigma,s)=Y \quad X(\sigma,0)=\sigma \\ \dot Y(\sigma,s)=X \quad Y(0,s)=s \\ \dot U(\sigma,s)=U \quad U(\sigma,0)=\sigma^3\...
0
votes
1answer
55 views

How to Find $u(x,y)$ for the PDE $u_x + 2u_y + (2x − y)u = 2x^2 + 3xy − 2y^2$ using Method of Characteristics?

Lagrange-Charpit equations: $$dx=\dfrac{dy}{2}=\dfrac{du}{(-2x+y)u+2x^2+3xy-2y^2}$$ $$\dfrac{dx}{dy}=\dfrac{1}{2} \implies x=\dfrac{y}{2}+A, A \in \mathbb{R}$$ By sagemath software a substitution for ...
3
votes
2answers
61 views

Solving the One Way Wave Equation with Boundary Condition and Initial Condition

I want to solve the one way $1$D wave equation with the following IC and BC: $$ u_t+au_x=0; \quad 0\leq x\leq1, \quad t\geq0 $$ $$ u(x,0)=u_0(x) \quad\quad u(0,t)=g(t) $$ Previously, with a question ...
5
votes
1answer
85 views

PDE: Solving using the Method of characteristics

I am trying to solve this PDE using Method of characteristics: $$(u+e^x)u_x+(u+e^y)u_y=u^2-e^{x+y}$$ I don't know how the next equation is called in English, but it is used to solve the PDE: $$\frac{...
-1
votes
1answer
90 views

Show that equation $u_x + u_t =0$ has no solution [duplicate]

I need to prove that the Partial Differential Equation $u_x + u_t = 0 $ where $u(x,t) = x$ on $x^2 + t^2 = 1$ has no solution. I solved this equation with method of characteristics and got one value ...
-1
votes
2answers
37 views

can there exists subring of nonzero characteristic if ring has zero characteristic?

Question Let $R$ be a ring such that $\operatorname{char}(R)$ is $0$ and $S$ is subring of $R$ then $\operatorname{char}(S)$ is $0$. My attempt: If $\operatorname{char}(R)=0$ then $\nexists n\in\...
2
votes
1answer
75 views

Field must be perfect or characteristic p

Problem statement: Given field $F$, if for any field extension $M/F$, $[M:F]$ is divisible by a fixed prime $p$, show that $F$ is either perfect or have characteristic $p$. Previously in this ...
2
votes
3answers
144 views

Linear first order PDE $u_x+u_t=u$ with the method of characteristics

I miss something so that I can understand how my teacher finds the solution. I will write the exercise as it is written first. $$u_x+u_t=u, x\in \mathbb{R}, t>0$$ $$ u(x,0)=\cos x$$ Then I have: $$\...
1
vote
1answer
53 views

Are Riemann invariants & eigenspaces uniquely defined?

I am trying to understand some derivations surrounding the Riemann Invariants for the system: $$ \begin{pmatrix} u_t \\ \eta_t \end{pmatrix} + \begin{pmatrix} u & 1 \\ \eta & u \end{...
0
votes
0answers
44 views

How do the characteristics of a transport equation determine where we need to impose boundary conditions?

Consider the one-dimensional linear transport equation $$u_t+u_x=0\;\;\;\text{in }(0,\infty)\times\Omega;\tag1$$ $$u(0,\;\cdot\;)=u_0\;\;\;\text{in }\Omega.\tag2$$ If $\Omega=\mathbb R$, these ...
3
votes
1answer
84 views

Correctly solving this PDE using the method of characteristics

I am trying to solve $$ PDE: - \frac{\partial f}{\partial t} + x \frac{\partial f}{\partial x} = x \\ IC : f(t_{max},x) = h(x) $$ using the method of characteristics. I attempted to follow the usual ...
0
votes
0answers
49 views

First Order Equations and characteristics

I am trying to solve $u_t +x^2 u_x = 0\hspace{0.5 cm} x>0, t\in \mathbb{R}$ $u(0,t)=g(t)\hspace{0.5 cm} t\in \mathbb{R}$ and by the method of the characteristics I have found that it has as a ...
2
votes
1answer
58 views

Solving a first order PDE

Consider the PDE $$u_t + uu_x +\alpha u = 0; \quad u(x,o)=u_0(x)$$ Whose characteristics are: $$\begin{cases}dx/dt = u; \quad x(0) = \xi \\ du/dt = -\alpha u; \quad u(0)=u_0(\xi)\end{cases}$$ I can ...
3
votes
2answers
75 views

Finding a general solution of pde

$(x^2-y^2-u^2)\cdot u_x+(2xy)\cdot u_y=2xu$ how can I solve this partial dif. equation. I try to use Langarange methods This is my solution; $$\frac {dx}{x^2-y^2-u^2}=\frac {dy}{2xy}=\frac {du}{2xu}= ...
0
votes
1answer
33 views

Solve the equation $xu_x+tu_y=-tu$ with $u(x,0)=x$

Using the characteristic method to solve $$xu_x+uu_y=-tu, \quad\infty <x<\infty\;\; t>1\\ u(x,o)=x$$ here $\frac{dx}{x}=\frac{dt}{t}=-\frac{u}{t}$ Consider $\frac{dt}{t}=-\frac{u}{t}\implies ...

1
2 3 4 5
12