1. **Problem Statement:** We want to solve the heat conduction equation $$4 \frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t}$$ for $$0 < x < \frac{\pi}{2}$$ and $$t > 0$$ with boundary conditions $$u_x(0,t) = 0$$ and $$u_x\left(\frac{\pi}{2},t\right) = 0$$ and initial condition $$u(x,0) = \begin{cases} 1 & 0 < x < \frac{\pi}{4} \\ 0 & \frac{\pi}{4} < x < \frac{\pi}{2} \end{cases}$$. 2. **Method: Separation of Variables** Assume a solution of the form $$u(x,t) = X(x)T(t)$$. 3. **Substitute into PDE:** $$4 X''(x) T(t) = X(x) T'(t)$$ Divide both sides by $$X(x)T(t)$$: $$4 \frac{X''(x)}{X(x)} = \frac{T'(t)}{T(t)} = -\lambda$$ where $$-\lambda$$ is a separation constant. 4. **Solve spatial ODE:** $$X'' + \frac{\lambda}{4} X = 0$$ with Neumann boundary conditions: $$X'(0) = 0, \quad X'\left(\frac{\pi}{2}\right) = 0$$. 5. **Eigenvalues and eigenfunctions:** General solution: $$X(x) = A \cos\left(\frac{\sqrt{\lambda}}{2} x\right) + B \sin\left(\frac{\sqrt{\lambda}}{2} x\right)$$ Apply $$X'(0) = 0$$: $$X'(x) = -A \frac{\sqrt{\lambda}}{2} \sin\left(\frac{\sqrt{\lambda}}{2} x\right) + B \frac{\sqrt{\lambda}}{2} \cos\left(\frac{\sqrt{\lambda}}{2} x\right)$$ At $$x=0$$: $$X'(0) = B \frac{\sqrt{\lambda}}{2} = 0 \implies B=0$$ Apply $$X'\left(\frac{\pi}{2}\right) = 0$$: $$X'\left(\frac{\pi}{2}\right) = -A \frac{\sqrt{\lambda}}{2} \sin\left(\frac{\sqrt{\lambda}}{2} \cdot \frac{\pi}{2}\right) = 0$$ For nontrivial $$A \neq 0$$: $$\sin\left(\frac{\sqrt{\lambda} \pi}{4}\right) = 0 \implies \frac{\sqrt{\lambda} \pi}{4} = n \pi, n=0,1,2,\ldots$$ So: $$\sqrt{\lambda} = 4n \implies \lambda = 16 n^2$$ Eigenfunctions: $$X_n(x) = \cos(2 n x)$$ 6. **Solve temporal ODE:** $$T'(t) + \lambda T(t) = 0 \implies T_n(t) = C_n e^{-\lambda t} = C_n e^{-16 n^2 t}$$ 7. **General solution:** $$u(x,t) = \sum_{n=0}^\infty C_n e^{-16 n^2 t} \cos(2 n x)$$ 8. **Apply initial condition:** $$u(x,0) = \sum_{n=0}^\infty C_n \cos(2 n x) = \begin{cases} 1 & 0 < x < \frac{\pi}{4} \\ 0 & \frac{\pi}{4} < x < \frac{\pi}{2} \end{cases}$$ 9. **Find coefficients $$C_n$$ using Fourier cosine series:** $$C_n = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} u(x,0) \cos(2 n x) dx$$ Since $$u(x,0) = 1$$ for $$0 < x < \frac{\pi}{4}$$ and 0 otherwise: $$C_n = \frac{2}{\pi} \int_0^{\frac{\pi}{4}} \cos(2 n x) dx = \frac{2}{\pi} \left[ \frac{\sin(2 n x)}{2 n} \right]_0^{\frac{\pi}{4}} = \frac{2}{\pi} \cdot \frac{\sin\left(\frac{n \pi}{2}\right)}{2 n} = \frac{1}{\pi n} \sin\left(\frac{n \pi}{2}\right)$$ For $$n=0$$: $$C_0 = \frac{2}{\pi} \int_0^{\frac{\pi}{4}} 1 dx = \frac{2}{\pi} \cdot \frac{\pi}{4} = \frac{1}{2}$$ 10. **Final solution:** $$u(x,t) = \frac{1}{2} + \sum_{n=1}^\infty \frac{1}{\pi n} \sin\left(\frac{n \pi}{2}\right) e^{-16 n^2 t} \cos(2 n x)$$ This series represents the temperature distribution over time and space satisfying the given PDE, boundary, and initial conditions.