Consider backward differentiation formula (BDF). Prove that the k-step BDF method is stable if and only if k≤6 holds.

1. Consider backward differentiation formula (BDF).(a) Prove that the k-step BDF method is stable if and only if k≤6 holds.(b) Prove that the k-step BDF method is A-stable if k= 1,2.(c) Draw the regions of absolute stability of the k-step BDF with k= 1,2,3,4,5,6, respectively.2. Try to construct 3-stage Runge Kutta method (A, b, c) of Guasstype.(a) Discuss the existence and uniqueness of the solution pro-duced by the method when it is applied toy′(t) =f(y(t)),where f satisfies the Lipschitz condition.(b) Verify that the method is of order 6.(c) Prove that the method is symplectic.3. Consider the initial value problem{dy(t)dt=Ay(t) +φ(t), t≥0,y(0) = (0,0,···,0,0)T∈Rm−1,(0.1)where A=m2−2 11−2 1………1−2 11−2∈R(m−1)×(m−1),φ(t) =m2(1,0,···,0,−1)T∈Rm−1.(a) Find the all eigenvalues ofA.(b) Givenm= 10ßsolve (0.1) by using Euler method with step-sizeh= 0.0045,0.01, respectively. Observe the numerica lresults att= 1 and present your comments.(c) Given m= 10ßsolve (0.1) by using implicit midpoint method with step size h= 0.0045,0.01, respectively. Compare with1

the numerical results of case (b), find out their differencesand present your explanation.(d) Givenh= 0.001ßsolve (0.1) withm= 10,100 by usingEuler method, respectively. Observe the numerical resultsatt= 1 and present your comments.4. Consider the initial value problemdudt=u(v−2) :=a(u, v),dvdt=v(1−u) :=b(u, v),u(0) =u0, v(0) =v0,(0.2)wheret >0. Let stepsizeh= 0.1. Solve (0.2) by using thefollowing three methods:(a) Explicit Euler method with (u0, v0) = (1,1)∂(b) Implicit Euler method with (u0, v0) = (4,4)∂(c)un+1=un+ha(un, vn+1),vn+1=vn+hb(un, vn+1),(u0, v0) = (5,2).(0.3)Draw figures of the numerical results with respect to the abovethree methods in phase space, respectively, wheret∈[0,10000].Observe the figures and explain the phenomena observ