How to Solve a Partial Differential Equation with a Laplacian Operator?

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To solve the equation Δf = cos(κ·r) with the Laplacian operator, one can utilize the properties of the Laplacian in Cartesian coordinates. The radius vector r is defined as r = xex + yey + zez, where ex, ey, and ez are unit vectors in the x, y, and z directions, respectively. A particular solution can be derived, which helps in formulating the general solution. The discussion emphasizes the importance of understanding the Laplacian's behavior in different coordinate systems. Overall, the approach involves leveraging known solutions and mathematical techniques to address the equation effectively.
Petar Mali
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Can you give me idea how to solve equation?

\Delta f=cos(\vec{k}\cdot\vec{r})

where \Delta is Laplacian and \vec{k}=\vec{const}
 
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What is r ?
 
Radius vector!

\vec{r}=x\vec{e}_x+y\vec{e}_y+z\vec{e}_z
 
An obvious particular solution allows to express the general solution (attachment)
 

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