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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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General solution of a first order PDE with zeroth order term

So I have got the following equation: $$x\frac{\partial u}{\partial x} - 2 \frac{\partial u}{\partial y} = 2u$$ I have tried to solve the following way. I was taught that LHS can be thought of as the ...
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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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PDE IVP - Characteristics, why is my method wrong?

Solve the IVP $$ \begin{cases} u_t + cu_x = 1 & c \in \mathbb{R} \\ u(x,0) = \sin x \end{cases}$$ To solve this, I have used characteristics as follows: Note that $$\frac{\...
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Solving the problem : $z z_x + z_y = 0, \quad z(x,0) = x^2$

Exercise : For the problem : $$\begin{cases} zz_x + z_y = 0 \\ z(x,0) = x^2\end{cases}$$ derive the solution : $$z(x,y) = \begin{cases} x^2, \quad y = 0\\ \frac{1+2xy - \sqrt{1+4xy}}{2y^2}, \...
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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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Solve the Cauchy problem for the linear PDE

I have the linear PDE $$yu_x - xu_y = 0 \qquad x^2 + y^2 < a^2 \\ u(0,y) = (a^2-y^2)^{\frac{1}{2}} \qquad y \in(-a,a)$$ where $a > 0$ is a constant. So what I have done is to say that $$a_1 ...
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Existence and uniqueness of PDE IVP : $z z_x + y z_y = x$, $C : x=t, y=t \; ; \; t >0$

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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I.V.P. : $ x^2 z_x + y^2 z_y = z^2 \; ; \; z=1 $ on the initial curve $C : y=2x $

Exercise : Solve the Initial Value Problem (I.V.P.) $$\begin{cases} x^2 z_x + y^2 z_y = z^2 \\[.5em] z=1 \text{ on } C =\{(x,y): y=2x \} \end{cases}$$ where $C$ is the initial curve of the ...
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Use characteristic curve method to solve the problem $u_t(x,t)+u_x(x,t)+u(x,t) = 0$

Use characteristic curve method to solve the problem: $$u_t(x,t)+u_x(x,t)+u(x,t) = 0 ~~\text{for } 0 < x < \infty, t > 0$$ $$u(x,0) = \cos(2x)$$ $$u(0,t) = 1$$ Note that if I am ...
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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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Is this nonlinear system “solvable”?

I don't really mean solvable, what I mean is: is it possible to rewrite $w$ in terms of $y$ and $z$? $$w =e^{3s}(1+r) -1$$ $$y = se^{2s}$$ $$z = 2se^{2s}(1+r) +r$$ I have fooled around with this ...
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Why Does the Characteristic Equation of a Differential Equation Depend on the Solution?

I have learnt about the characteristic equations of differential equations (D.E.) only informally and recently observed that the way they are defined seems to depend not only on the equation, but on ...
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Method of Characteristics for traffic flow equation

I've recently been studying for qualification exams for my master's program. I've run into a few problems that I'm stuck on and hope that I can get some help here. We consider the hyperbolic ...
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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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Find the domain of definition for this quasilinear equation

$$ x\frac{\partial u}{\partial x} +y\frac{\partial u}{\partial y}= 2u \qquad (1)$$ find, in explicit form, the solution of (1) which satisfies the condition: $$ u =x^3 \quad \text{on} \quad y=x+1 \...
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Unique general solution for linear PDE with method of characteristic

Consider the linear PDE $y u_x + x u_y = 0$. Applying the method of characteristics, we find that a general solution is given by $u(x,y) = f(x^2 - y^2) $ for some function $f$. My question is: can ...
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Why did I get 1/0 in the equation of characteristics?

Consider the partial differential equation $u_{xy}+u_x+u_y=3x$. Obtain the equation of its characteristics and reduce to the canonical form. Attempt The equation of its characteristics is given by $\...
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Proof methods of characteristic roots for recurrences

The textbook stated that:..." a recurrence had a unique solution once we specify the values of the first p terms, $a_0,a_1,...,a_{p-1}$... in general, a recurrence had... solutions of sequences of the ...
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Domain of definition of first order linear PDE

Consider the first order PDE $$x\frac{\partial u}{\partial y}-y\frac{\partial u}{\partial x}=u, \quad u(x,0)=f(x).$$ I've tried solving this PDE using the method of characteristics. The first time I ...
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Characteristic infinite field

Do you know how to solve these questions? $K$ is a field of characteristic $0$. Let $L$ be an extension of $K$ with $[L : K] = 2$. Prove that there exists $u \in L$ such that $L = K(u)$ and $u^2 \in ...
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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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Discontinuous solution using the method of characteristics

Consider the first order PDE given by $$\frac{\partial u}{\partial x}+2y^{\frac{1}{2}}\frac{\partial u}{\partial y}=xy,$$ with the Cauchy data for $0\leq x \leq 2$ given by $$u(x,0)=f(x)=\begin{cases}...
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Solve pde problem

For each of the following PDE. (a) solve the characteristic equation (b) define a transformation of the PDE. And obtain the transformed equation. (c) find the general solution of the transformed ...
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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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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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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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Charpit method: non-linear PDE

I have a question: $$p^{2}x+q^{2}y = z.$$ I formed the Charpit auxiliary equation as follows $$ \frac{\mathrm{d}x}{2px} = \frac{\mathrm{d}y}{2py} = \frac{\mathrm{d}z}{2(p^2x + q^2y)} = \frac{\mathrm{...
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How to take this exterior derivative of the expression $du - \sum_i p_i dx_i$?

I am reading the wikipedia page about applying the method of characteristics in the fully nonlinear case. We have the fully nonlinear equation $$ \tag{1} F(x_1, \cdots, x_n , u, p_1, \cdots, p_n) = 0,$...
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Solving PDE using Lagrange method of characteristics

The problem in this question is found in section 1.3.2 "Determining the parametric structure of models" by D.J. Cole et al. I have a linear, 6-dimensional PDE: $$ -\frac{\partial f}{\partial \phi_3}\...
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Interpretation of graph of PDE

1.Suppose we the PDE $u_{x}(x,y)=u_{y}(x,y)$. Does this simply mean we are looking for a function whos partial w.r.t $x$ and $y$ are the same att every point and then we have some additional boundary ...
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Solution of Burgers' equation

Can you help me to solve this problem and explain the method used? $u\equiv u(x,t)$ with $x\in\mathbb{R}$ $$ u_t+\left( \frac{1}{2}u^2\right)_x=0 $$ with initial data $$u_0(x)=u(x,0)= \begin{cases} ...
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The meaning of non-characteristic boundary data for a PDE

Consider the PDE and initial condition \begin{align} u_x + u^2 u_y &= 1 \\ u(x,0) &= 1, \end{align} where $u = u(x,y)$ and $(x,y) \in \mathbb{R}^2$. I've solved this with the method of ...
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Domain solutions on partial differential equations

$$(1+x^2)\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=0$$ integrating to give: $\arctan x=\ln y+A$ I managed to find the general solution as $u(x,y)=F(A)=F(\ln y -\arctan x)$ I ...
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$\left\{\begin{array}{lll} f_{t}+xf_{y}=0\\ f|_{t=0}=f_{0}(x,y) \end{array}\right.$

Given $f\in C^{1}({\bf R}^{3})$ , then find the solution of the following pde : $$\left\{\begin{array}{lll} f_{t}+xf_{y}=0\\ f|_{t=0}=f_{0}(x,y) \end{array}\right.$$ I tried to use the ...
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Characteristic curve Partial Differential Equations

I have the partial differentiation question $$du/dx -3xdu/dy = 0$$ and initial conditions $$u(x,0)=cos2x$$ I have $$A=y+3x$$ and $$u(x,y)=B$$ from integration. So I said $$u(x,y)=F(y+3x)$$ thus $$u(...
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Method of characteristics - solution doesn't seem consistent with original PDE

As an example, a problem from the book Numerical solutions of Partial Differential Equations: Finite difference methods by G.D. Smith. The PDE: $$ \frac{\partial{U}}{\partial{x}} + \frac{x}{\sqrt{U}}\...
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Another attempt at solving a PDE with the method of characteristics

I want to use the method of characteristics to obtain the solution to this PDE, $$\frac{\partial F}{\partial t}=\left(z-t\right)\left(\beta z-\gamma\right)\frac{\partial F}{\partial z}$$ which I've ...
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Finding shock curves, Burgers equation

I'm working on this problem: $u_{t}+uu_{x}=0\\ u(0,x)=-x \mathbb{1}_{[a,b]}$. For the case when $a<b=0$ I want to find a shock curve starting from point $(t,x)=(1,0)$. On the field without ...
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Transport equation $u_t+c(x)u_x=0$

Let c $\in$ $C^1$$\mathbb(R)$. Consider the transport equation. $u_t+c(x)u_x=0$. Prove that if $x=f(x)$ is a characteristic curve, then so are all horizontally translated curves $ x=f(t+a)$ for any $...
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Entropy solution - Burgers' equation

Why doesn't the following problem have a solution for $t\ge1$? $u_{t}+uu_{x}=0\\ u(0,x)=-x$. The characteristics don't intersect and they cover the whole space above t=1.
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Entropy solution of the Burgers' equation 2

I'm trying to construct a solution to the following problem: $u_{t}+uu_{x}=0\\ u(0,x)=-x \mathbb{1}_{[a,b]}$. For the case when $0<a<b$ I try to find a shock curve starting from point $(t,x)=(...
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solution for a pde on a curve.

Let assume that $ u_t+uu_x=0 $ has $C^1$ solutions in two domains which is disjointed by the curve $x=\phi(t)$. Also assume that $u$ is continuous but $u_x$ has jump discontinuty on the curve. Show ...
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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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Solving a PDE with Method of Characteristics.

I am currently trying to solve a PDE using Method of Characteristics, but am having a bit of trouble. The PDE is given by: $$xu_x+yu_y+zu_z=1$$ With Initial Conditions: $$u=0 \text{ on } x+y+z=1$$ Now ...
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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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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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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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1answer
304 views

Method of Characteristics(Advection equation with initial and boundary condition)

Solve, using the Method of Characteristics, the equation $\frac{\partial \rho}{\partial t} + \frac{\partial \rho}{\partial x}=-\mu\rho$ for $x,t>0$ with the conditions $\rho(x,0)=f(x)$ and $\rho(0,...
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Using method of characteristics to find general solution of PDE $x^3 u_x + y u_x = 4 + 2x^2u$

Use the method of characteristics to find the general solution $u(x; y)$ of the partial differential equation $$ x^3 \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial x} = 4 + 2 x^2 u $$
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Solving the partial differential equation using the method of characteristic to find the general solution

The question is solve the following partial differential equation using the method of characteristic $\frac{\partial{u}}{\partial{x}}-\frac{\partial{u}}{\partial{t}}=2t$, with the condition $u=u(x,t)...