Question: Consider the heat conduction problem in a thin, uniform rod of length L. The temperature distribution u(x, t) in the rod is governed by the one-dimensional heat equation: Ju(x,t) Ət a Pu(x,t) მე-2 where: u(x, t) represents the temperature at position at and time t, • is the thermal diffusivity constant, . x = [0, L) and t≥ 0. Part A: Solving the Heat Equation with Boundary Conditions 1. Solve the heat equation using separation of variables for the following boundary and initial . conditions: Boundary conditions: (0,t) = 0 and u(L,t) = 0 (Dirichlet boundary conditions), Initial condition: u(x, 0) = f(x), where f(x) is a given temperature distribution along the rod at t=0. 2. Find the general solution for u(x, t) in terms of the Fourier series expansion of f(x), and provide the explicit form of the temperature distribution as a sum of the eigenfunctions. 3. Analyze the long-term behavior of the temperature u(x, t) as t→ ∞. Under what conditions on f(x) will the rod eventually reach a uniform temperature? Part B: Non-Homogeneous Heat Equation with External Heating Now, suppose the rod is subjected to a non-uniform internal heat source along its length. The heat equation becomes: Ju(x,t) Ət Ju(x,t) მე2 +q(x), where q() is a continuous function representing the heat generated per unit length at position 2. 1. Solve the non-homogeneous heat equation for the same boundary conditions as in Part A and with q(x) being a constant heat source, i.e., q(x) = q. 2. Extend your solution to the case where q(2) is a sinusoidal heat source, i.e., q(x) = go sin (). 3. Discuss the physical interpretation of the solution and how the internal heating affects the temperature distribution along the rod over time. Part C: Wave Equation for a Vibrating String Consider now a vibrating string of length L, where the displacement y(x, t) of the string satisfies the one-dimensional wave equation: where: Fy(x,t) Ət² y(x,t) მე-2 . y(x, t) is the vertical displacement of the string at position and time t • c is the wave speed, • x = [0, L] and t≥ 0. 1. Solve the wave equation using the method of separation of variables, with the following boundary and initial conditions: • Boundary conditions: y(0,t) = 0 and y(L, t) - 0 (the string is fixed at both ends), • Initial conditions: y(x, 0) = g(x) and 1 Əy(x,t) Ət - h(x), where g(x) is the initial It-0 displacement of the string and h(x) is the initial velocity distribution. 2. Provide the general solution for y(x, t) in terms of a Fourier series, and show how the initial conditions determine the coefficients in the series expansion. 3. Investigate the effect of the initial velocity h(x) on the overall motion of the string. Under what conditions will the string exhibit standing waves? Part D: System of Coupled Differential Equations Consider a system of two coupled first-order linear differential equations describing the population dynamics of two species interacting in a predator-prey relationship: dx(t) dt - = x(t)(a - by(t)), dy(t) - -y(t)(c - dx(t)), dt where: • x(t) is the prey population at time t, . y(t) is the predator population at time t, • a, b, c, and d are positive constants representing the interaction rates between the species. 1. Analyze the equilibrium points of the system by setting 40 and 40, and find the population sizes at equilibrium. 2. Use the linearization method to analyze the stability of the equilibrium points by approximating the system near the equilibrium. 3. Discuss the nature of the solutions for different values of a, b, c, and d, and describe possible behaviors such as oscillations, extinction, or coexistence of the species.