1. **State the problem:** We want to solve the heat equation $$\frac{\partial u}{\partial t} = \alpha \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right)$$ with $$\alpha = 1$$ on the domain $$0 < x < 1, 0 < y < 1$$. Initial condition (IC): $$u(x,y,0) = 0$$. Boundary conditions (BCs): - $$u(0,y,t) = 100$$ - $$u(1,y,t) = 100$$ - $$u(x,0,t) = 0$$ - $$u(x,1,t) = 0$$ 2. **Method:** We use separation of variables and steady-state plus transient solution decomposition. 3. **Steady-state solution $$u_s(x,y)$$:** Solve Laplace's equation $$\frac{\partial^2 u_s}{\partial x^2} + \frac{\partial^2 u_s}{\partial y^2} = 0$$ with the same BCs. 4. **Steady-state BCs:** $$u_s(0,y) = 100, \quad u_s(1,y) = 100, \quad u_s(x,0) = 0, \quad u_s(x,1) = 0$$. 5. **Rewrite steady-state solution as:** $$u_s(x,y) = 100 + v(x,y)$$ where $$v$$ satisfies homogeneous BCs: $$v(0,y) = 0, \quad v(1,y) = 0, \quad v(x,0) = -100, \quad v(x,1) = -100$$. 6. **Solve for $$v(x,y)$$ using separation of variables:** Assume $$v(x,y) = X(x)Y(y)$$. 7. **Eigenvalue problem for $$X(x)$$:** $$X'' + \lambda^2 X = 0, \quad X(0) = 0, \quad X(1) = 0$$ Eigenvalues: $$\lambda_n = n\pi, n=1,2,3,...$$ Eigenfunctions: $$X_n(x) = \sin(n\pi x)$$ 8. **Equation for $$Y(y)$$:** $$Y'' - \lambda_n^2 Y = 0$$ General solution: $$Y_n(y) = A_n \sinh(n\pi y) + B_n \cosh(n\pi y)$$ 9. **Apply BCs for $$v$$ at $$y=0$$ and $$y=1$$:** $$v(x,0) = \sum_{n=1}^\infty C_n X_n(x) Y_n(0) = -100$$ $$v(x,1) = \sum_{n=1}^\infty C_n X_n(x) Y_n(1) = -100$$ Since $$Y_n(0) = B_n$$ and $$Y_n(1) = A_n \sinh(n\pi) + B_n \cosh(n\pi)$$, and the BCs are constants, we use Fourier sine series expansion of constant function $$-100$$ on $$x \in (0,1)$$: $$-100 = \sum_{n=1}^\infty b_n \sin(n\pi x)$$ where $$b_n = 2 \int_0^1 (-100) \sin(n\pi x) dx = -\frac{400}{n\pi} \text{ for odd } n, 0 \text{ for even } n$$ 10. **Construct $$v(x,y)$$:** $$v(x,y) = \sum_{n=1,3,5,...} \frac{-400}{n\pi} \frac{\sinh(n\pi y)}{\sinh(n\pi)} \sin(n\pi x)$$ 11. **Final steady-state solution:** $$u_s(x,y) = 100 + v(x,y) = 100 - \sum_{n=1,3,5,...} \frac{400}{n\pi} \frac{\sinh(n\pi y)}{\sinh(n\pi)} \sin(n\pi x)$$ 12. **Transient solution $$u_t(x,y,t)$$:** Solve heat equation with zero BCs and initial condition $$u_t(x,y,0) = -u_s(x,y)$$. 13. **Transient solution form:** $$u_t(x,y,t) = \sum_{m=1}^\infty \sum_{n=1}^\infty D_{mn} e^{-\pi^2 (m^2 + n^2) t} \sin(m\pi x) \sin(n\pi y)$$ 14. **Initial condition for transient:** $$u_t(x,y,0) = -u_s(x,y)$$ 15. **Full solution:** $$u(x,y,t) = u_s(x,y) + u_t(x,y,t)$$ This satisfies the PDE, IC, and BCs. **Summary:** The problem is solved by decomposing into steady-state and transient parts, using Fourier sine series and eigenfunction expansions.