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I see the definition for stability for any solution operator $E$ is that $$||Eu^{n}||^2=||u^{n+1}||^2\leq C||u^{n}||^2$$ for some constant $C$ and some pde $u_t=Lu$. However, I can show that $$||u^{n+1}||^2\leq C_1||u^{n}||^2+||f^n||^2$$ where $f$ is the part on RHS of the equation $u_t=Lu+f$. So, it might be a silly question but is that stable? It doesn't fit the definition of stability because of this additional term $||f^n||$. Thanks

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Actually the stability of the differential scheme for the nonhomogeneous PDE is derived from the stability for the homogeneous thanks to Duhamel's Principle.

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