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Questions tagged [cauchy-problem]

Use this tag for questions about partial differential equations that satisfy certain conditions given on a hypersurface in the domain.

2
votes
1answer
171 views

Deducing the existence of a PDE by constructing it inductively via its Taylor series expansion

Consider the differential equation $$u_{t t} = c^2 u_{x x} - k u, \qquad k, u > 0$$ together with the Cauchy data $$u(x, 0) = e^x, u_t(x, 0) = 0$$ I now want to find a solution $u: \mathbb R^...
3
votes
1answer
426 views

Find the solution of the Cauchy problem $u_t-uu_x=0$.

For the Cauchy problem $u_t-uu_x=0, ~x\in \mathbb{R}, t>0$ with $u(x,0)=x,~x\in \mathbb{R} $, which of the following statements is true? the solution $u$ exists for all $t>0$. the ...
2
votes
2answers
157 views

Explanation to PDE $u_{tt}-c^2u_{xx}=0$ solution

Exercise: Solve the Cauchy problem $u_{tt}-c^2u_{xx}=0$ with conditions $u(x,0)=g(x)$ and $u_t(x,0)=h(x)$, where $g(x)=0,\ h(x)=\begin{cases} 0,\ x<0 \\ 1,\ x\ge0 \end{cases}$. Please ignore this ...