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The boundary condition is boundedness at infinity.
We decompose the ___domain '''R'''<sup>2</sup> into two overlapping subdomains H<sub>1</sub> = (<nowiki>[− ∞,1]</nowiki> × '''R''' and H<sub>2</sub> = <nowiki>[0,+ ∞]</nowiki>) × '''R'''. In each subdomain, we will be solving a BVP of the form:
:''u''<sup>( ''j'' )</sup><sub>''xx''</sub> + ''u''<sup>( ''j'' )</sup><sub>''yy''</sub> = ''f'' in H<sub>''j''</sub>
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