The temperature of an exotic liquid in a pipe of length L can be described by the 1D non-dimensional heat equation 8T Ot 8²T a(T) + Q(x, t) 822 with time t, temperature T, temperature dependent thermal diffusivity a(T), and a source term describing additional heating/cooling along the pipe, Q(x, t). On the right end of the pipe, the boundary is adiabatic, 8T On the left end, the fluid has a temperature 8x (x = L, t) = 0. = 2 T(x = 0, t) = 3-sin At t=0 the temperature of the fluid is T(x, t = 0) = 2.5. The thermal diffusivity of the fluid can be modeled by a third-degree polynomial a(T) = ao + a₁T+ a₂T² + a3T³. - 2T (1) (5) For a grid of M + 1 equally spaced points along the pipe and Ar the spacing between points, do the following tasks. Tasks: 6Ar f(x-2) 6f(ri-1) + 3ƒ(x₁) + 2ƒ(x₁+1) 6Ar a) write the equation for the temperature at the last point TM+1 as a function of the neighboring points using a third order accurate finite difference approximation to Eq. (2). Potential third-order finite difference formulas for different stencils using the notation of Table 8.1 are: -2f(x-3) +9f(x₁_2) 18f(r₁_1) | 11ƒ(r.)) +0(Az³¹) +0(Ar³) 2f(x-1)-3f(r₂) | 6ƒ(F;+1) — ƒ(X₁12) +0(Ar) 6AT (2) 11ƒ(x;) + 18ƒ(x4+1) — 9ƒ(*₁+2)+2ƒ(*++3) +.0(Az³) 6A.F (3) (4) (6) (8) (9) b) write the equation for the temperature at the first point 7 using Eq. (3); c) write the equation for the right-hand-side f" of Eq. (1) using Eq. (5) at all non-boundary mesh points à using second-order central finite differences, leaving Q(r, t) as Q(x₁, t") or Q". Use only the above variable names and not any specific parameter values given in later problems.