please send complete handwritten solution for Q4 Q5 Q6 Separation of variablesThe lateral radiation of heat from a bar of length L insulted at both ends can be modelled withthe linear partial differential equation (PDE)Ju² u= D.- au for 0 0,(1)Ət მე2with the conditionsux(0, t) = ux (L, t) = 0 and u(x,0) = f(x),where L, a, DER are positive constants.1. Seek a separation solution of the form u(x, t) = X(x)T(t) to showT' a X"+ = = k,(2)DT D Xwhere k denotes the separation constant.2. Use equation (2) to derive two ordinary differential equations (ODEs), one in space x andone in time t.3. Determine the boundary conditions for the ODE that depends on x.4. Find the non-trivial solutions of X(x) = Xn(x) and corresponding values of the separationconstant k = kn, for n = 0, 1, 2, 3....Hint: you may need to consider each of the three cases k=-p², k = 0, k = p² (p = 0) tofind all the non-trivial solutions5. Find the solutions T(t) = Tn(t), for n = 0, 1, 2, 3 ....6. Use the superposition principle for linear PDEs to write down the solution for u that satisfiesthe boundary conditions ur (0, t) = u(L, t) = 0.7. Determine expressions for the unknown coefficients in your solution for u that satisfies theinitial condition u(x,0) = f(x).8. Find the solution when f(x) = 1 + cos πx, for values of L = 1, a = 1 and D = 1.9. In a single figure, plot the solution found in Question 8 at times t = 0, 0.25, 0.5, 2. Brieflydescribe the behaviour as t increases. What is the constant value temperature of theenvironment that surrounds the bar?

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The lateral radiation of heat from a bar of length L insulted at both ends can be modelled with the linear partial differential equation (PDE) du 8²u - au for 0 0, (1) Ət əx² with the conditions ux (0, t): = ux (L, t) = 0 and u(x,0) = f(x), where L, a, DER are positive constants. 1. Seek a separation solution of the form u(x, t) = X(x)T(t) to show T' Ο X" + = = k₁ DT D X (2) where k denotes the separation constant. 2. Use equation (2) to derive two ordinary differential equations (ODES), one in space x and one in time t. 3. Determine the boundary conditions for the ODE that depends on x. 4. Find the non-trivial solutions of X(x) = Xn(x) and corresponding values of the separation constant k = kn, for n = 0, 1, 2, 3.... = -p², k = 0, k = p² (p = 0) to Hint: you may need to consider each of the three cases k=-1 find all the non-trivial solutions 5. Find the solutions T(t) = Tn(t), for n = 0, 1, 2, 3 .... 6. Use the superposition principle for linear PDEs to write down the solution for u that satisfies the boundary conditions uz (0, t) = ux (L, t) = 0. = D