Answer Happy

The following four non-linear equations, i=1 i=2 i=3 i=4 FI(T) = [ko + (T₁+ To) ]-To- [2 ko + (To+ 2-T₁+T2) ] T₁ + [ko + (T₁+T2)]-T2 2 12(1) TD := [ ko + 2 (T₂+ T₁) ] T₁ - [2 ko + 2 (T₁ + 27T2+T3) ]-T2 + [ko + 2 (12+T3)]-T3 13CTD= [ko + £4(T) = [ ko + hat form a set of non-linear simultaneous equations of the form, (7)=0, result from the numerical solution by the finite difference method of the heat conduction problem in a bar (see below) whose thermal conductivity varies linearly with temperature, that is k(T)=k, +aT To (T3 +12)] 12-[2 ko + (T2+2-T3 + Ta) ]-T3 + [ko + (T3 + Ta)]-Ta (T3 + 2-T₁ + TL)]-Ta₁ + [ko + 2 (T4 + TL)]-TL (T4+T3) ] T3 -[2-ko + i=0 i=1 i=2 i=3 i=4 i=5 TL X=L X=0 Ax A. Solve the non-linear problem for temperatures, T,, using the Newton-Raphson method. The Jacobian matrix must be evaluated at each iteration step of the Newton- Raphson algorithm, and the resulting linear system for the update vector may be solved by
using the intrinsic Octave linear solver (such as the linsolve routine). You are to evaluate the elements of the Jacobian by using a first order forward finite differencing approximation with a step size of Ax = 105 to evaluate the partial derivatives. Take the following input parameter values for the problem: ko 10.25 a 0.25 To = 450 TL-25 Finally set up the N=4 non-linear equations, and use the following initial guess to solve the non-linear problem: NN (400 300 200 100 Set the convergence factor 105 for both the iterative norm and the residual norm. Set the maximum number of iterations to K=100, and report: 1. The initial elements of the Jacobian (report 1 decimal places and chop). 2. The temperatures, T,,i=1,2...4 (report 1 decimal places and chop). 3. The initial residual norm (report 1 decimal places and chop). B. Using the same values for the linear coefficient k but setting the non-linear coefficients a=0 to zero, solve the linear problem and report: 4. The temperatures, T,,i=1,2...4 (report 1 decimal places and chop).