# Draw figures of the numerical results with respect to the above three methods in phase space, respectively, where t∈[0,10000]. Observe the figures and explain the phenomena observed.

Numerical analysis of dynamic system(Master’s degree 2020)(All calculation results or figures obtained by computer must beattached with the program codes)(2 pages in total. Test paper and answer paper shall be submitted together ú)1. Consider backward differentiation formula (BDF).(a) Prove that thek-step BDF method is stable if and only ifk≤6 holds.(b) Prove that the k-step BDF method is A-stable ifk= 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 Guass type.(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 of A.(b) Given m= 10ßsolve (0.1) by using Euler method with step-size h= 0.0045,0.01, respectively. Observe the numerical results at t= 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 differences and present your explanation.(d) Given h= 0.001ßsolve (0.1) with m= 10,100 by using Euler 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)where t >0. Let step size h= 0.1. Solve (0.2) by using the following 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 above three methods in phase space, respectively, where t∈[0,10000]. Observe the figures and explain the phenomena observed.