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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.

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804 views

Infinite solution PDE

I want to show that the following PDE of the function $u(x,y)$ has infinitely many solutions: $$ \left\{ \begin{array}{c} u_x+u_y=2xu \\ u(x,x)=e^{x^2} \\ \end{array} \right. $$ By using the ...
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1answer
131 views

Equation with non constant coefficients with the method of characteristics?

$$\dfrac{\partial u}{\partial t} +c\dfrac{\partial u}{\partial x} = 0$$ where $c$ is a constant, subject to $u(x,0) =\cosh(x)$ With the normal method I wrote down ($s$ is the parameter I'm using): $...
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2answers
506 views

How can I solve $u_{xt} + uu_{xx} + \frac{1}{2}u_x^2 = 0$ with the method of characteristics.

I am trying to solve the following PDE: $u_{xt} + uu_{xx} = -\frac{1}{2}u_x^2$, with initial condition: $u(x,0) = u_0(x) \in C^{\infty}$ using the method of characteristics. I am a beginner with the ...
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1answer
103 views

Understanding how to find the General Solution of a PDE using the method of characteristics

I've been trying and failing to understand how to find the "general solution" for PDEs, as in an answer with an arbitrary function $F(x,y,u)$, for a PDE with no boundary conditions given. I understand ...
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1answer
547 views

Method of Characteristics with 3 Partials

Use the method of characteristics to solve: $$yu_x-xu_y+u_z=1, \ u(x,y,0) = x+y.$$ I feel comfortable solving the using the method of characteristics in two dimensions, but am having trouble ...
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1answer
29 views

Defining $\{f \in F[X] \mid f'=0\}$ depending on characteristics

I want to show the following theorem about characteristics: Let $F$ be a field. Then $$\{f \in F[X] \mid f'=0\} = \Biggl\{\begin{array}{l} F, \qquad \qquad \qquad \qquad char(F)=0 \\ \{g(X^p) \...
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1answer
630 views

traffic flow problem

I am struggling with the following question: Consider the conservation law $$u_t + f(u)_x = 0, \: \text{where} \: f(u) = u(1 − u). \:\:\:\:\quad (1)$$ This conservation law describes a model of ...
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1answer
33 views

PDE Describing The Growth of Bone Matrix

$v$ - volume, $x$ - position, $t$ - time. Bone Growth is proportional to surface area of growth sites --> $ dv/dt$ ~$\pi$ $r^2$~$(r^3)$^(2/3)~$v$^(2/3). I have the equation: $v_x + v_t $= kv ^2/...
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2answers
566 views

Shock-wave solution for PDE $u_t+(u-1)u_x=2$

I want to solve the following PDE initial value problem $u_t+(u-1)u_x=2$ and $u (x,0)=\begin{cases} 1 & \text{for } x <0,\\ 1-x & \text{for } 0<x <1\\ 0 & \text{for } 1 <x \...
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1answer
142 views

Solving a discrete equation

Consider the following equation: $$-\epsilon u''(x) + \beta u'(x) = 1, \;\; x \in (0, 1)$$ $$u(0) = 0, \; u(1) = 1.$$ where $\beta > 0, \; 0 < \epsilon << \beta.$ I constructed a Finite ...
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56 views

Method of characteristics - Component transport

Problem: Assume that a Component mass Conservation Equation is given by $\phi \frac{\partial c}{\partial t} + u \frac{\partial c}{\partial x} = -rc$ , where $r=constant$. Solve the Equation using ...
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0answers
30 views

Solving the G - equation with method of characteristics

Trying to solve $$u_t = |\nabla u| \quad u(0,x) = e^{-|x|^2/2}$$ using method of characteristics. I can solve it without the absolute value, but not sure how to handle them since these solutions ...
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1answer
440 views

Method of characteristics - geometrical interpretation

I am currently studying the method of characteristics. I feel like missing a fundamental part, which I am not understanding. Consider for instance the Burgers equation, $u_t + uu_x = 0$ in $\mathbb{...
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54 views

Does this transport PDE have an analytical solution?

I am trying to solve the following PDE. I suspect it is a transport equation but I am not sure. $$ \partial_t u(x,t) + \partial_x\left[f(x,t)u(x,t)\right] = -\left(\dfrac{\phi_1}{f(x,t)} + \dfrac{\...
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1answer
348 views

Find the first shock time of Burgers' Equation

Find the maximal $t_0>0$ such that the Cauchy problem: $$uu_x+u_t=0 \qquad{} u(x,0)=e^{-x^2/2}$$ exists in $\Bbb{R}\times[0,t)$; i.e. find the first time $t=t_0$ when the shock develops. This ...
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1answer
90 views

Method of Characteristics and General Solutions

Find the general solution $u(x,y,z)$ of: $u_{x}+u_{y}+u_{z}=0$ Using the method of charactertistics. I tried to set up the characteristic equations: $x'(s)=1, y'(s)=1,z'(s)=1,w'(s)=0$ Where $w(s)...
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42 views

How can I prove that this matrix is diagonalizable?

How can I prove that for all $a \in \mathbb{R}$ the following matrix is diagonalizable? I computed the characteristic polynomial, but I couldn't decompose it into linear factors. This is the ...
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2answers
83 views

Confusion in college book on the introduction of PDE (method of characteristics)

In my course book, the method of characteristics for solving 1st order PDE of the form $P \frac{\partial u}{\partial x} + Q\frac{\partial u}{\partial y} = R$ where $u$ is a function of $x$ and $y$ is ...
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343 views

Cauchy problem for nonlinear first order hyperbolic PDE with source via method of characteristics. (Work and characteristic plots included)

Working on the following problem. $y u_x -u u_y = x \\ u(x,x)=-2x$ I've went after this with the method of characteristics. I'm using $(t,s$) as my parametrization variables. In parameterize space I ...
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49 views

Solving 1st order PDE (Finding where solution is invalid)

I'm struggling with a PDE homework question (see below) Solve the equation: $au_x + bu_y + u = 0$ With Cauchy data $u(s) = e^{−s}$ along the straight lines $(x(s), y(s)) = (cs, ds)$. The solution ...
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2answers
358 views

Shock formation in traffic flow equation

Consider the PDE initial value problem $$\frac{\partial u}{\partial t}+(1-2u)\frac{\partial u}{\partial x}=0,$$ $$u(x,0)=f(x),$$ with the initial conditions for traffic congestion: $$f(x)=\begin{...
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1answer
226 views

What is the relationship between method of characteristics and separation of variables?

The method of characteristics for a partial differential equation such as $$u_t + cu_x =0$$ Yields a solution $u(x,t) = f(x-ct)$. On the other hand, separation of variables $u(x,t) = X(x)T(t)$ ...
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4answers
941 views

What is the geometric interpretation of the solution to PDE $xu_x+yu_y=0$

I have the following PDE $$xu_x+yu_y=0$$ for which I get the characteristic function $$y=cx$$ along which the u(x,y) is constant. The general solution is $$u(x,y)=f(\frac{y}{x})$$. I understand ...
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1answer
52 views

Solving quasilinear PDE - 1D, time-dependant, convection

I have a task to solve the following quasilinear PDE (find $c(x,t)$): $$ c_x v + c_t = - v_x c $$ $c \in (0,20) , t \in (0, \infty)$ where I know function $v(x)$ to be $v(x) = \frac{3}{40}(1+\cos(\...
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1answer
208 views

Classification of solution of non-linear first order partial differential equation

A solution of the PDE $xu_x+yu_y+(u_x)^2+(u_y)^2-u=0$ represents an ellipse in the $x$-$y$ plane. an ellipsoid in the $xyu$ plane. a parabola in the $u$-$x$ plane. a hyperbola in the $u$-$y$ plane. ...
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1answer
98 views

Method of Characteristics for $(1+x^2)u_x+2u_y=y\cdot(1+u^2)$

Suppose I had a problem like $$(1+x^2)u_x+2u_y=y\cdot(1+u^2),\;\; u(1,y)=1$$ This is of the form $$a(x,y)u_x+b(x,y)u_y+c(x,y,u)$$ So I would use the Method of Characteristics: $$\frac{dx}{dt}=(1+...
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1answer
52 views

Dealing with the logarithm of absolute value when solving $u_t+xu_x=1$ by method of characteristics

I was trying to come up with a solution to the PDE IVP problem $u_t + x u_x =1$, $u(x,0) = f(x)$. Here's how I went about it: We can the following relations by applying the Method of Characteristics: ...
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1answer
54 views

Nonlinear first order pde with both IC and BC

I am trying to solve the first order problem, namely: $$ \begin{cases} u_{t}+uu_{x} = 0, \hspace{0.5cm} x>0, t>0 \\ u(0,t)=t, \hspace{0.5cm} t>0 \\ u(x,0)=x^2, \hspace{0.5cm} x>0 \end{...
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1answer
194 views

Cauchy Problem for inviscid Burgers' equation

Consider the Cauchy Problem of finding $u(x,t)$ such that $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0,x\in\mathbb{R},t>0$$ $$u(x,0) = u_0(x), x\in\mathbb{R}$$ Which choices ...
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1answer
64 views

Method of characteristics - heat convection

I have following PDE: \begin{align} v \cdot \frac{\partial T}{\partial x} + \frac{\partial T}{\partial t} &= k \cdot (T-T_0)\\ T(x,0) &= T_0\\ T(0,t) &= T_1 \end{align} It's first order ...
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1answer
66 views

Methods of Characteristics

I have a problem solving the ODE associated with the question, any help will be greatly appreciated. Use method of characteristics to solve the problem $(x-y)\dfrac{\partial u}{\partial x}+(x+y)\...
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1answer
114 views

General Methods to Solve First-Order PDE

Question is as simple as: What are the different methods for solving a first-order PDE? I'm aware of nearly all forms of Method of Characteristics - Lagrange Method, Charpit's Method. I'm ...
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74 views

A nice question related to method of characteristics

Let $ \alpha$ be real number and $h=h(x)$ be a continuous function in $\mathbb{R}$.Consider following initial value problem: $$yu_x + xu_y=\alpha u, u(x, 0) =h(x) $$ Then a) Find all points on ${(y=...
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1answer
2k views

Complete integral solution of first order PDE

Find a complete integral of $4uu_{x}-u_{y}^{3}=0$, and then that solution which satisfies $u=4at$ on $x=0$, $y=t$ Solution: The PDE $f\equiv 4up - q^{3}=0$ Thus the last pair of terms integrate to ...
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1answer
100 views

Solve: $xu_x + yu_y+ \frac{1}{2}(u^2_x+u^2_y) =u, u(x, 0) =\frac{1}{2}(1−x^2)$

In the plane find two solutions of the initial-value problem $xu_x + yu_y+ \frac{1}{2}(u^2_x+u^2_y) =u, u(x, 0) =\frac{1}{2}(1−x^2)$. I think we get to use the method of characteristics But I am not ...
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1answer
178 views

Solve the PDE using characteristics method

Consider the PDE $$3 \frac{\partial{u}}{\partial{t}} + t^2\frac{\partial{u}}{\partial{x}} = 0: t \lt 0 $$ with the initial condition $$u(x,0) = f(x): 0 \lt x \lt 1$$ Determine the characteristics ...
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1answer
46 views

Function of first integrals pde

Show that if $F$ and $G$ are first integrals of the characteristic system of $u_{t} + c(x,t,u)u_{x} = g(x,t,u)$ then $\Psi(F(x,t,u),G(x,t,u))=0$ defines the solution $u=u(x,t)$ of this equation. It ...
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1answer
242 views

Complete integral of pde without independent variables

Show that the complete integral of pde $F(u,p,q)=0$ ($p=u_{x}$ and $q=u_{y}$) is $$ f(x,y,u,a,b) = x + ay + b - \int\frac{du}{g(u,a)}, $$ where the function $p=g(u,a)$ is computed from the ...
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3answers
9k views

Show that the characteristics equation for a $2\times 2$ matrix is $\lambda^2 - \mathrm{tr}(A)\lambda^2 + \det(A)=0$

Can someone help me show that the characteristics equation for a $2\times 2$ matrix, say $A$, can be written as: $$ \lambda^2 - \mathrm{tr}(A)\lambda + \det(A) = 0,$$ where $\mathrm{tr}(A)$ is the ...
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1answer
60 views

Solving an Inhomogeneous $1$st Order PDE using Method of Characteristics

I wanna solve the equation $$u_x+u_y+u=\exp(x+2y), \quad u(x,0) = 0$$ I have just learned method of characteristics. But I don't know how to deal with $u$ term and inhomogeneous term simultaneously. ...
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1answer
153 views

Extension of the method of characteristics

Solve the pde (advection equation): $$ u_t + c u_x = 0, \hspace{0.5cm} t>0, \hspace{0.5cm} x\in \mathbb{R} $$ with the condition given on an arbitrary curve $t=\tau (x)$, that is: $$ u(x,\tau (x))= ...
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1answer
253 views

Method of characteristic for PDE $xu_x + yu_y = u$

I'm studying method of characteristic. This is a homework. I'm very new to this topic so hope anyone can give a direction. Given a PDE : $xu_x + yu_y = u$. a) Solve it, where the initial curve ...
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1answer
61 views

How to show that characteristic curves for a PDE fill the plane?

In a question I found my characteristics curves for my PDE to be $y=-\cos(x)+C$. When I sketched some of the curves I could see that I would fill the $x,y$. But the question asked explain why your ...
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2answers
143 views

Method of characteristics - finding the particular solution using initial conditions

I am trying to use the method from my previous question to solve this PDE: $$ 3u_x + 2u_t = \cos x $$ with initial condition $u(x,0) = x^2$. So I need to solve these: \begin{align} \frac{dx}{ds} &...
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2answers
194 views

Method of characteristics - eliminating variables

I am trying to follow a guide for the method of characteristics; quoting the first example: We use the method of characteristics to solve the problem $ 2u_x - u_y = 0, \;\; u(x, 0) = f(x) $ ...
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1answer
133 views

Analyze : $u_t-u^2u_x +cu =0, u(x,0)=g(x)$

Analyze : $u_t-u^2u_x +cu =0 $, $ u(x,0)=g(x)$. From This we have following $$\begin{align} \frac{dt}{ds} &=1 \\ \frac{du}{ds} &=c \\ \frac{dx}{ds} &=-u^2 \end{align}$$ then how to ...
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1answer
142 views

The Burgers equation $u_y + u u_x = 1$ with $u=0$ on the parabola $y^2=2x$

For the PDE $u_y + u u_x = 1$, sketch a plot of $\Gamma$ and a few representative curves, including the envelope curve. Conditions: $u=0$ on the curve $y^2=2x$, and $y,x>0$. Express $u$ as a ...
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0answers
25 views

integral of solution between two characteristic curves

Suppose we are given a pde: $\frac{\partial u}{\partial t} +\frac{\partial {(b(x)u)}}{\partial x}=0$. Let $u(t,x)\in C^1(\mathbb{R}^2)$ be a solution and $x=X_1(t)$ and $x=X_2(t)$ be two ...
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1answer
143 views

Is there a solution to this unidirectional wave equation, with initial value $v=f(x)$ and $x=t^2$

unidirectional wace equation: $$\frac{du}{dt}+c\frac{du}{dx}=0$$ The initial value $u=f(x)$ is given on the parabola $x=t^2$. Is there a solution to this problem, discuss why the solution is unique ...
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1answer
73 views

Show that the PDE has vertical lines as a family of characteristic curves for $y=0$.

Show that the PDE given by $\dfrac{\partial ^2u}{\partial y^2}-y\dfrac{\partial ^2u}{\partial x^2}$ has vertical lines as a family of characteristics curves for $y=0$. Taking $S=0,R=-y,T=1$ we ...