# Questions tagged [linear-pde]

This tag is for questions relating to linear partial differential equations, in which the dependent variable (and its derivatives) appear in terms with degree at most one

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### General solution of PDEs using Green's function

I have a question regarding the form of the general solution to a PDE in terms of its Green's function. For example, consider the heat equation: \frac{\partial u}{\partial t}-\Delta u=...
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### On the existence of global classical non-zero solutions of a linear elliptic equation

Does the equation $$-\Delta u +u=0$$ have any non-zero classical, i.e., $C^2$, solutions on $\mathbb{R}^d$? How about if $\mathbb{R}^d$ is replaced with half-space? How about solutions of polynomial ...
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### Solution of Poisson's equation with a linear term

Consider the partial differential equation $$\Delta u = au - b,~~~~~ x \in \Omega \subset \mathbb{R}^{n},~a,b>0$$ $$\frac{\partial u}{\partial n} = 0~~ \text{on}~ \partial \Omega.$$ Using some ...
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### Condition for existence of solutions of a system of first order PDE

Consider (locally) the following system of linear first order PDEs: $$\forall\,1\leq j<k\leq n,\quad \frac{\partial F_k}{\partial x_j}-\frac{\partial F_j}{\partial x_k} = \nu_{jk}(x),$$ for the ...
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### Is there a name for the PDE resembles a "reversed" heat/diffusion equation $\frac{\partial^{2} u}{\partial t^{2}} = \frac{\partial u}{\partial x}$?

Is there a name for the second-order linear Partial Differential Equation of the form $\frac{\partial^{2} u}{\partial t^{2}} = \frac{\partial u}{\partial x}$ which resembles the heat/diffusion ...
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