# 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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### Meaning of the method of characteristics at crossing points of characteristic lines

Given a system of linear PDE's of first order such as $w^1_t + cw^1_x = 0,w^2_t - cw^2_x =0,$ one usually uses the method of characteristics to find the solution $(w^1(x,t),w^2(x,t))$ on each point of ...
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### Help in solving PDE with charasteristic equation problem

It's my first time on this sort of problem, so I am having some trouble solving this PDE: $\frac{\partial u}{\partial t}+e^{-x}\frac{\partial u}{\partial x}+\frac1y\frac{\partial u}{\partial y}=0$ and ...
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### How to find the characteristic form of coupled PDEs?

I'm reading a textbook named linear and nonlinear waves. In chap.14, a method was used to solve the coupled equations which can reduce PDEs to ODEs. But I don't know how to find the needed ...
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### Semilinear equation with characteristics

I have to solve the following equation by the method of characteristics $$\frac{\partial T}{\partial t}+v\frac{\partial T}{\partial z}=\frac{2 U}{RC_F}(T_M-T),$$ where $v,C_F,U,R,T_M$ are known ...
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### First-order inhomogeneous PDE in general form

I am solving the initial value problem: $$u_t +u_x = -\sigma(x)u+m(x), \quad u(x,0)=\phi(x), \quad u(0,t)=\gamma(t)$$ I need to get solution like this: \begin{equation*} u(x,t)=\phi(x-t) e^{-\int_{x-t}...
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### Find the solution for given PDE along unit circle

The problem assigned to me is: For the equation $$u_x^2 + u_y^2 = u^2$$ Find the solution when $u|_\Gamma$ = 1, where $\Gamma$ is unit circle at origin. I was taught about method of characteristics ...
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### Why is this the method of characteristics?

The following question refers to ref. 1, the equation are numbered alike with a slightly different notation. The author claims to solve a renormalization group (RG) equation - so the context is ...
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### General solution to PDE $xu_x + yu_y = 2xy$, why using characterics in non-parametric form I get an incorrect general solution?

I am trying to solve the PDE \begin{align} xu_x + yu_y = 2xy \end{align} using the method of characteristics. So the characteristic equations are \begin{align} \frac{dx}{x} = \frac{dy}{y} = \frac{du}{...
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### How to solve this first order PDE using method of characteristics, cannot find any resources anywhere online. [closed]

The PDE in question is $u_{x}u_{y}=1, u(x,0)=\sqrt{x}$. The process for solving a first order PDE using method of characteristics when the terms are summed is relatively well-established, but I cannot ...
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### Method of characteristic curves

I'm trying to understand the method of characteristic curves, in that purpose I have these two exercises, but i think somwhere i get totaly wrong with this I need to allocate areas of a constant type ...
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### Solving for traffic flow density as a function of time and determining when it's no longer valid

So I'm working on the following problem. When given an initial density of: \rho(x,0)= \begin{cases} \frac{\rho_{max}}{4}, & x<0\\ \rho_{max}, & 0 < x ...
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### Proof of [Lee, Theorem 22.35] The Cauchy Problem for a Hamilton-Jacobi Equation

Relevant Background (for completeness, can be skipped) The statement of Theorem 22.35 in Lee's Introduction to Smooth Manifolds is Suppose $M$ is a smooth manifold, $W \subseteq T^*M$ is an open ...
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### What is the intuition behind the method of characteristics for a second order PDE?

I understand the idea behind the method of characteristics as applied to first-order PDEs: watching how $u(x,y)$ changes along special curves $(x(s),y(s))$ simplifies the problem to a set of coupled ...
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### Characteristic function on I is uncountable

I studied characteristic function in real analysis in Richard R goldberg book for methods of real analysis and there is an exercise problem asking to prive that characteristic function on I is ...