Numerical Solution of Differential Equations I: Problem Sheet 1 1. Let the real function u(t), defined for t [0, ) satisfy the differential equation u = du = f (t, u), t > 0, dt with initial condition u(0) given. Verify that the following functions satisfy a Lipschitz condition with respect to u, uniformly in t, for 0 t , u R: a) f (t, u) = 2u(1 + t)4 ; 2 b) f (t, u) = e(1+t) tan1 u ; c) f (t, u) = 2u(1 + u2 )1 (1 + e|t| ) and for each case determine a bound for the truncation error when using Euler's method to approximate u(t) at equally spaced points tn = nt, where t > 0 and n = 0, 1, 2, . . .. In (a) you may assume that the solution u is bounded with |u| umax for some positive umax R. 2. Suppose that m is a fixed positive integer. Show that the initial value problem u = u2m/(2m+1) , u(0) = 0 , has infinitely many continuously differentiable solutions. Why does this not contradict Picard's Theorem? 3. Van der Pol's equation u (1 u2 )u + u = 0 subject to the initial conditions u(0) = a1 and u (0) = a2 , where a1 and a2 are given real numbers, and > 0 a parameter, models electrical circuits connected with electronic oscillators. Rewrite the equation as a coupled system of two first-order differential equations with appropriate initial conditions. Formulate Euler's method for this system, when = 1, a1 = 1/2 and a2 = 1/2, on the interval [0, T ], for some T > 0 using n points with uniform spacing t = 1/(n 1). Evaluate algebraically the Euler approximation to u(t) and u (t) at the point t = t. Use matlab to calculate the solution using Euler's method and graph the results for T = 20 and n = 101, 1001, 10001 for = 1, = 5. 4. Consider the initial value problem u = log log(4 + u2 ) , t [0, 1] , u(0) = 1 , and the sequence (Un )N n=0 , N 1, generated by the explicit Euler method Un+1 Un = log log(4 + Un2 ) , n = 0, . . . , N 1 , U0 = 1 , t using the time points tn = nt, n = 0, . . . , N, with spacing t = 1/N. Here log denotes the logarithm with base e. a) Let Tn denote the truncation error of Euler's method at t = tn for this initial value problem. Show that |Tn | t/(4e). b) Verify that |un+1 Un+1 | (1 + tL)|un Un | + t|Tn | , n = 0, . . . , N 1 , where L = 1/(2 log 4). c) Let en = un Un . Prove by induction that T |en | (1 + tL)n |e0 | + [(1 + tL)n 1] . L d) Find a positive integer N0 , as small as possible, such that max |un Un | 104 0nN whenever N N0 . 5. [2005 Finals] Consider the initial value problem u = f (u), u(0) = 1, where f (u) = tan1 (1 + u2 ). [You may assume that this problem has a unique solution t u(t), defined for all t R and that the functions u and u are defined and continuous for all t R.] a) [8 marks] Show that |u| 4 for all t R. Show further that the function f satisfies the following Lipschitz condition: 1 |f (u) f (v)| |u v| u, v R. 2 b) [8 marks] The implicit Euler approximation Un to un = u(tn ) on the mesh {tn : tn = nt, n = 0, 1, . . .} of uniform spacing t (0, 1], is obtained from the formula Un Un1 = f (Un ). n = 1, 2, 3, . . . , U0 = 1. t Let g(u) = u tf (u). Show that the function g is strictly monotonic increasing and limu g(u) = . By rewriting Euler's method as g(Un ) = Un1 , deduce that, given Un1 R, the Euler approximation Un is uniquely defined in R c) [9 marks] Show that the truncation error Tn of the implicit Euler method applied to the initial value problem under consideration satisfies |Tn | t, n = 1, 2, 3, . . . . 8 Show further that 2 2t |un Un | |un1 Un1 | + |Tn |, n = 1, 2, . . . , 2 t 2 t and deduce that i t \u0011n h\u0010 1+ 1 t, n = 1, 2, . . . . |un Un | 4 2 t Show that if t [25(e 1)]1 , then Un approximates un to within 102 for all t n tn [0, 1]. [You may use without proof that (1 + 2t ) etn for all t (0, 1] and all n 0.]