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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Method of Characteristics for the Equation $\;\left(x+\alpha y\right)u_{xx} + u_{yy} = 0\:$

Below is Exercise $3.6$ (p.$91$) of "Applied Partial Differential Equations" by Ockendon et al., $2^\mathrm{nd}$ ed.: Show that, if $$\big(x+\alpha y\big)\dfrac{\partial^2 u}{\partial x^2} + \...
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85 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\...
4
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1answer
91 views

Solve transport equation $\frac{\partial\phi}{\partial t}+\phi\frac{\partial\phi}{\partial x}=0$ using method of characteristics

I'm trying to solve the following transport equation $$\frac{\partial\phi}{\partial t}+\phi\frac{\partial\phi}{\partial x}=0$$ subject to the initial condition $$\phi(x,0)=f(x)=\left\{ \...
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0answers
40 views

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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130 views

Why is the domain of dependence for a system of hyperbolic PDE’s an interval on the $x$ -axis?

I’m looking at the hyperbolic system ${{\mathbf{u}}_t} + {\mathbf{A}}({\mathbf{u}},x,t){{\mathbf{u}}_x} = {\mathbf{h}}({\mathbf{u}},x,t)$ $\quad$ (1) where ${\mathbf{u}}(x,t) \in {\mathbb{R}^n},\;\;...
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34 views

Existence and uniqueness of PDE IVP : $z z_x + y z_y = x$, $C : x=t, y=t \; ; \; t >0$ and $z=t$ over $C$.

Exercise : Consider the equation $$z z_x + y z_y = x$$ and the initial curve $$C : x=t, y=t \; ; \; t >0$$ Decide whether there is a unique solution, no solution or infinitely many ...
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53 views

Show that no new maxima or minima can develop in smooth solutions to the conservation equation

I would like to show that no new maxima or minima can develop in smooth solutions to $ \frac{\partial \phi(x, t)}{\partial t} + \frac{\partial }{\partial x}(f(\phi(x, t)) = 0 $ My solution: Since ...
3
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1answer
142 views

Solving a first order non linear PDE with the method of characteristics

We have to find the function $u(x,y)$ for the following system: $u_xu_y = xy$ $u(x,y) = y+1$ for $x=y$ Using the method of characterstics I get: $F(x,y,u,u_x,u_y) = u_xu_y-xy = 0$ Defining $p = ...
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72 views

Solve inhomogenous Wave Equation with Method of Characteristics

Solve, using the methods of characteristics: $$\begin{cases}u_{tt}(x,t)=u_{xx}(x,t)\\ IC: u(x,0)=u_t(x,0)=0\\ BC: u(0,t)=h(t),\ u(L,t)=0\end{cases}$$ I tried to use the substitution $u(x,t)=v(x,t)+...
3
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1answer
52 views

Describing the solution to a nonlinear PDE

Given the pde $$xu_x+yu_y+uu_z=0$$ where $$u(x,y,0)=xy$$ for $x>0$ and $y>0$, the solution, gotten from the method of characteristics is $$u(x,y,z) = xye^{\frac{-2x}{u}}$$ My question is, how ...
3
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1answer
239 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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15 views

Charpit PDE, rays parallel or perpendicular to boundary

The rays are all possible curves $(x(\tau), y(\tau))$. The derivation of the equations is clear and the condition of defining $p_0, q_0$ is $\frac{dx_0}{ds}\frac{dy_0}{d\tau} - \frac{dy_0}{ds}\frac{...
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2answers
35 views

Finding an Integral Surface

Consider finding the integral surface of $$x^2 p + xy q = xyz-2y^2$$ which passes through the line $x=y e^y$ in the $z=0$ plane. Attempt In Lagrange's subsidiary form $$\frac{dx}{x^2}=\frac{dy}{...
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51 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 {...
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2answers
73 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 ...
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63 views

Unidirectional Non-Linear Wave Motion PDE Problem: $u_t + uu_x = 0$, with $u(x, 0) = f(x)$

I'm completely stuck on the following problem: Consider an example of unidirectional non-linear wave motion: $u_t + uu_x = 0$, with $u(x, 0) = f(x)$. i) Show that the characteristic ...
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79 views

Finding the characteristic curves of a PDE

The Question: In the region $y>0$, reduce the PDE $$y \frac{\partial ^2u}{\partial x^2} = \frac{\partial ^2u}{\partial y^2}$$ to canonical form, and sketch the characteristic curves. My Attempt:...
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1k views

How to Solve this Lagrange's Linear equation and find the integral surface?

I need help in solving this problem Find the general solution of the partial differential equation $$x(z+2a)p+(xz+2yz+2ay)q=z(z+a),$$ where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\...
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76 views

Principal Minor and Non-zero Eigen values in Diagonal Matrix

While reading the definition of pseudo determinant in here, I've found the following : $$pdet(L) : = \sum_{I\in[n] , |I| = r}det(L_{I,\ I}) = \prod_{i=1}^r\lambda_i$$ where $L_{I,\ I}$ denotes the ...
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72 views

Finite Field has Finite Characteristic

I want to prove that for finite $\mathbb{F}$, there exists $n\in\mathbb{Z}^+$ such that $$ \underbrace{1+1+...+1}_{\text{n times}} = 0 $$ Proof: by the field axioms, there exists $1\in\mathbb{F}$. ...
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275 views

Solving Heat Equation using Method of Characteristics

Given the equation, $$u_{xx}=\frac{1}{\kappa}u_t$$ How would I go about solving using Method of Characteristics? I know I end up with the PDE being transformed into something along the lines of $$u_{\...
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82 views

Partial Differential Equations - Transport Equation & Characteristics

So I am doing an introductory course to PDEs and I have been introduced to the method of characteristics. Now I have sort-of learnt the method used to solve the transport equation, where the ...
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39 views

Solving a PDE with Method of Characteristics

I am currently trying to solve a PDE but am having difficulties. The PDE is: $$(y+u)u_x+yu_y=x-y, \>\>\>\>u(x,1)=1+x$$ Now I found the Characteristic Equations: $$\dot x(s) = y+z, \>\&...
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241 views

1st Order Nonlinear PDE: Understanding Envelopes and Monge Cones

I have a question about envelopes of surfaces. In a book I am reading the following: Suppose $S_a$ is a one parameter family of surfaces in $R^3$ given by $z=w(x,y;a)$ where $w$ depends smoothly on ...
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136 views

Finding the Riemann Invariants of a system of 2 PDEs

I've been asked to find the Riemann Invariants for the system: $$ \begin{pmatrix} \cos(v) & 0 \\ 0 & \cos(v) \end{pmatrix} \begin{pmatrix} u_x \\ v_y \end{pmatrix} + \begin{pmatrix} \sin(...
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142 views

Solving a system of semilinear coupled partial differential equations

I would like to know if it is possible to find the solution to the following system of partial differential equations: \begin{equation} \partial_x \mathbf{u}+B(x,y) \cdot \partial_y \mathbf{u}-\mathbf{...
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89 views

Traffic Flow problem space time diagram question

Traffic with constant density, $\rho_0$, is stopped by a red light at $x=0$. If $v(\rho)=1-\rho$, calculate what happends behind the light. Unfortunately the main focus of this example in my notes ...
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1answer
420 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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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)\...
2
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2answers
216 views

Solve the quasilinear initial value problem by the method of characteristics.

I am trying to solve $u_{t}+uu_{x}=0, \quad x\in\mathbb{R},\quad t>0$ with the initial values $u(x,0)= \begin{cases} 1-x^2, & |x| \leq 1 \\ 0, & |x| > 1 \end{cases}$ I also need ...
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1answer
43 views

Method of Characteristics for parametric problems

I am considering the equation $u_t(t,x,\eta)+V(t,x,\eta)\cdot u_x(t,x,\eta)=0$ where $V:[0,T]\times\mathbb{R}^n\times[0,1]^D\to \mathbb{R}^n$. For the parameter independent case, one would solve ...
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58 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 ...
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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 ...
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0answers
37 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{...
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36 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 ...
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27 views

Compute the Characteristics of a PDE

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 ...
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1answer
116 views

Method of characteristic: why a characteristic curve cannot intersect the initial curve in two points

Let consider a general linear first order PDE in $xy$ space, ie $$a(x,y)u_x+b(x,y)u_y=c(x,y)u+d(x,y),\qquad (x,y)\in U$$ where $U$ is an open and connected subset of $\mathbb{R}^2$ and $a,b,c,d \in C^...
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21 views

A first order hyperbolic problem using method of characteristics

Consider the initial value problem for the equation $u_t+au_x=f(t,x)$ with $u(0,x)=1$ and $f(t,x)=1$ if $x\ge 0$ and $0$ otherwise. Assume that $a$ is positive. Show that the solution is given by $u(t,...
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Method of characteristics example: using different change of variables to textbook, characteristic speed, and example clarification

The following method of characteristics example is from chapter 4 of Essential Partial Differential Equations by Griffiths, Dold, and Silvester: Example 4.2 (Half-plane problem) Solve the PDE $...
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44 views

Solve partial differential equation by Charpit's equations

Solve partial differential equation, $$F=xp^2-ypq+y^2q-y^2z=0,\;\;\;\; p = \frac{\partial z}{\partial x},\;\;\;\;q = \frac{\partial z}{\partial y}$$ My attempt: $F_x = p^2, \\F_y = -pq+2yq-2yz, \\...
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1answer
63 views

Coincidence of the Characteristic projection and the Data Curve

While studying first order Semi-linear PDE and the method of characteristics, I found this part in the note that I just cannot comprehend. From what I’ve understood, the solution surface is swept ...
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1answer
42 views

Find where solution is unique using Method of Characteristics

Find (in implicit form) the solution $u(x, y)$ of the equation: $$u_x+uu_y=x$$ when $$u(0, y) = 1 + \sin y \qquad \text{for} \qquad y > 0 $$ Show that your solution is uniquely determined in the ...
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0answers
43 views

Introduction to characteristic surfaces and bicharacteristics

I am currently studying the propagation of contact discontinuities in systems of hyperbolic PDE (multidimensional and transient). I have found that the concept of characteristics is helpful in ...
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0answers
43 views

Question about a PDE and its Characteristics

I am trying to solve a problem using Method of Characteristics but I'm having trouble solving it. The question is: Suppose $u(x,y)$ is a smooth function which is constant on any curve of the form $...
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1answer
36 views

Having trouble solving a PDE with method of characteristics

I am currently trying to solve a problem I have already solved, but am trying to solve it the way our professor solved it. The PDE is given by: $$yu_x-2xyu_y=2xu, \>\>\>\>\>\>\>...
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0answers
27 views

Why is the family of characteristics for a second order linear/semilinear parabolic PDE given by this equation?

Consider a PDE of the form $$ a(x,y) \frac{\partial^2 u}{\partial x^2} + 2b(x,y) \frac{\partial^2 u}{\partial x \partial y} + c(x,y) \frac{\partial^2 u}{\partial y} + d(x,y) \frac{\partial u}{\partial ...
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1answer
220 views

How to solve the following Cauchy problem.

Solve the following Cauchy problem for a first order PDE: $$(2x_1 + x_2)u_{x_1} + (x_2 + 1)u_{x_2} = u^2, \ \ u(x_1, 1) = x_1^2 + 1, \ \ x_1 \ge 0, x_2 \ge 1$$ and find an implicit conldition over $...
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1answer
118 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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0answers
29 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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0answers
332 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 ...