Problem 2 (50 points). Partial differential equation: parabolic PDF. In a fluidized bed, by injecting tracer particles at the center of the fluidized bed (Figure 1.) and monitoring the dispersion of the tracer, the radial dispersion coefficient of solids can be determined. The governing partial differential equation of the dispersion model for the particle tracer in cylindrical coordinates given by: 12 =D at ror R at t = 0, Initial codition : 0 Sr SR1, C = 100% Boundary conditions: t> 0 r = 0, = 0 ar t> 0 r = R2, 0 or Figure 1. A cylindrical bed column. where c is the concentration of the particle tracer and ris the radial position. R1 = a is the initial radius of the particle tracer covered, and R2 = 100 mm is the radius of the bed column. Dis the solid dispersion coefficient, 2 x10 m/s. Q1) Drive the finite difference equations using the central finite difference with the first-order approximation: use index i for temporal variablet and j for spatial variable r