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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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 ...
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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 ...
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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 ...
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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}{...
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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 ...
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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) \\ \...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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 ...