I am struggling to solve the heat equation with below boundary condition. $$\frac{\partial T}{\partial t} \ = \ \frac{\partial^2 T}{\partial x^2} \\ I.C. \ : \ T(x,0)=100 \\ B.C. \ : \ T(0,t)=20, \ \frac{\partial T(1,t)}{\partial x}=0$$ Generally when the boundary condition contains nonzero temperature or time-dependent condition, we represent $T(x,t)$ as follows. $$T(x,t) \ = \ v(x,t)+g_1(t)+\frac{x}{L}(g_2(t)-g_1(t))$$ where the boundary condition is $T(0,t)=g_1(t)$ and $T(L,t)=g_2(t)$.

But in this case one end is insulated, and this method fails. I searched for solving this kind of boundary condition, but I couldn't find any clues.

Any advice?



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