Ordinary Differential Equations ISC-5315 { 1 Ordinary differential equation 1 References • Sachin Shanbag, lecture notes (see class website) • Wikipedia (see class website) 2 Basics Explicit differential equation (1) dy y0 = = F [t] (Quadrature) (1) dt Explicit differential equation (2) y0 = F (y; t) (2) Implicit differential equation F [y; y0; t) = 0 (We will not look these) (3) 3 Higher order ODEs can be reduced to a system of first order ODEs, for example y00(t) + y(t) = 0 (4) with initial conditions y(0) = 0; y0(0) set z = y0, then we can express as a system of two ODEs z0(t) + y(t) = 0 (5) y0(t) − z(t) = 0 with initial conditions z(0) = 0; y(0). In general an n-order ODE can be reduced to a system of n 1-order ODEs. 4 Systems of equations 0 0 1 0 1 0 1 y1 a11 a12 ::: a1m y1 0 By C Ba21 a12 ::: a1mC By2C B 2C = B C B C (6) @:::A @ ::: ::: ::: ::: A @:::A 0 yn an1 an2 ::: anm yn y0 = Ay (7) y(0) = y0 (8) 1 Peter Beerli; March 31, 2011 Ordinary Differential Equations ISC-5315 { 2 5 Boundary value problems (We will not look at boundary value problems, only at initial value problems in ACS 1) Boundary value problems (BVP) y00(t) + y(t) = 0 with y(0) = 0 y(π=2) = 2 (9) Initial value problem (IVP) y00(t) + y(t) = 0 with y(0) = 0 y0(0) = 0 (10) Typically only scalar equations are used 6 Uniqueness Multiple solutions to some ODEs exist, for example y0(t) = py(t) with y(0) = 0 Solution 1: y(t) = 0 (11) Solution 2: t2 y(t) = (12) 4 7 Existence Solutions may not exist for all values of t y0(t) = [y(t)]2 with y(0) = Solution: 1 1 y(t) = exists only for t 6= (13) −1 − t 8 Lipschitz continuity In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: For every pair of points on the graph of this function, their secant line-segment's slope has absolute-value no greater than a definite real number; this constant, which is the same for all the secants, is called the function's \Lipschitz constant" (or \modulus of uniform continuity"). 2 Peter Beerli; March 31, 2011 Ordinary Differential Equations ISC-5315 { 3 A function n+1 n f(x; t): R ! R (14) is Lipschitz continuous if, for all (y1; t) and (y2; t), there exists an L, such that jf(y ; t) − f(y ; t)j 2 1 ≤ L: (15) jy2 − y1j Figure 1: For a Lips- If f(y; t) can be differentiated with respect to y, then chitz continuous function, there is a double cone @f(y; t) L = max (16) (shown in white) whose D @y vertex can be translated along the graph, so that Examples the graph always remains entirely outside the cone. f(y; t) = y2; y 2 [−3; 7] ) L = 14 (17) p f(y; t) = y2 + 5; y 2 R ) L = 1 (18) Small L implies a slowly varying function of y. 8.1 Indicator of uniqueness and existence It informs us about uniqueness and existence. A theorem without proof: If f(y; t) is Lipschitz continuous, then the initial value 0 y (t) = f(y; t) y(0) = y0 (19) has a unique differentiable solution on [a; b] for each y0, which depends continuously on y0 jy(t) − y^(t)j ≤ eLtjy(0) − y^(0)j: (20) Bounded perturbation of the ODE, such as 0 y (t) = f(t; y^ + r(t; y^)y ^(0) =y ^0 (21) with r(t; y^) ≤ kMk (22) then M jy(t) − y^(t)j ≤ eLtjy(0) − y^(0)j + (eLt − 1) (23) L Example: Recall that we had multiple solutions for the ODE y0(t) = py(t) y(0) = 0: (24) the function is discontinuous near zero @f(t; y) 1 = (25) @y 2py(t) Therefore the function is not Lipschitz continuous. 3 Peter Beerli; March 31, 2011 Ordinary Differential Equations ISC-5315 { 4 8.2 Lipschitz continuity examples 8.2.1 Continuous functions that are Lipschitz continuous: p The function f(x) = x2 +5 defined for all real numbers is Lipschitz continuous with the Lipschitz constant K = 1, because it is everywhere differentiable and the absolute value of the derivative is bounded above by 1. Likewise, the sine function is Lipschitz continuous because its derivative, the cosine function, is bounded above by 1 in absolute value. The function f(x) = jxj defined on the reals is Lipschitz continuous with the Lipschitz constant equal to 1, by the reverse triangle inequality. This is an example of a Lipschitz continuous function that is not differentiable. More generally, a norm on a vector space is Lipschitz continuous with respect to the associated metric, with the Lipschitz constant equal to 1. 8.2.2 Continuous functions that are not (globally) Lipschitz continuous: The function f(x) = x2 with domain all real numbers is not Lipschitz continuous. This function becomes arbitrarily steep as x approaches 1. It is however locally Lipschitz continuous. p The function f(x) = x defined on [0; 1] is not Lipschitz continuous. This function becomes infinitely steep as x approaches 0 since its derivative becomes infinite. 8.2.3 Differentiable functions that are not (globally) Lipschitz continuous. The function f(x) = x3=2 sin (1=x) with x 6= 0 and f(0) = 0, restricted on [0; 1], gives an example of a function that is differentiable on a compact set while not locally Lipschitz because its derivative function is not bounded. See also the first property below. 9 Numerical solutions of ODEs Consider an explicit scalar ODE 0 y (t) = f(y; t) y(0) = y0: (26) We can discretize the t domain ft0; t1; :::; tk; :::; tng. We compute approximate numerical solutions at the gridpoints tk yk = y(tk) ∼ Yk = Y (tk): (27) where Yk are the exact solutions evaluated at grid points. We explore simple methods based on Taylor series and Euler methods. 9.1 Taylor series 0 Given y (t) = f(y; t) with initial values y(t0) = y0 we can consider a Taylor series expansion of y around t0 (t − t )2 y(t) = y + (t − t )y0(t ) + 0 y00(t ) + ::: (28) 0 0 0 2! 0 4 Peter Beerli; March 31, 2011 Ordinary Differential Equations ISC-5315 { 5 recall that df @f @f @y y00(t) = = + (chain rule) (29) dt @t @y @t @f @f y00(t) = + f (30) @t @y A general expression for order-n method h2 hn y(t ) = y(t ) + hy0(t ) + y00(t ) + ::: + y(n)(t ) (31) i i−1 i−1 2! i−1 n! i−1 the error is O(hn+1) hn+1 = jY − y j = y(n+1)(ξ) withξ 2 [t ; t ] (32) G i i (n + 1)! i−1 i Example: y(t) = t − y y(0) = 1; (33) differentiate y0 repeatedly y0(t) = t − y y0(0) = −1 y00(t) = 1 − y0 y00(0) = 2 y(3)(t) = −y00 y(3)(0) = −2 y(4)(t) = −y(3) y(4)(0) = 2 4th order Taylor series solution t2 t4 y(t) = 1 − t + (2) + fract33!(−2) + (2) (34) 2! 4! t3 t4 = 1 − t + t2 − + (35) 3 12 Exact solution Y (t) = t + 2e−t − 1 (36) A comparison of the exact solution with the approximate Taylor series solution is shown in figure 2. In summary we can attest that the Taylor approximation delivers great results for small h values. The differentiation of f(t; y) can be laborious; it is problem specific and need to be repeated for dif- ferent f(t; y), can be automated. The error analysis of the Taylor approximation is straightforward and one can get an upper bound of the error. In practice only low order Taylor series formulae are used (Runge-Kutta formulae take their inspiration from higher order Taylor series). 5 Peter Beerli; March 31, 2011 Ordinary Differential Equations ISC-5315 { 6 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.0 0.2 0.4 0.6 0.8 1.0 Figure 2: The blue curve represent the exact formula (36) and the red dashed curve reflects the Taylor approximation (35). 9.2 Euler method 9.2.1 Forward Euler method 0 Given y (t) = f(y; t) with initial values y(t0) = y0 we can discretize the first derivative with time step h = tn − tn−1 (37) as y − y y0 = n n−1 = f(t ; y ) = f (38) h n−1 n−1 n−1 and then use a time marching algorithm yn = yn−1 + fn−1 (39) (40) Example: Say, we have f(t; y) = −2t3 + 12t2 − 20t + 8:5; y(0) = 1 (41) Using Mathematica we can construct the following function for the Euler method: euler[y0_,h_]:=Module[ {yy={},ii={},i=0}, AppendTo[yy,y0]; While[i<4, AppendTo[yy,yy[[-1]]+h (8.5-20 i+12 i^2-2 i^3)]; AppendTo[ii,i]; i+= h 6 Peter Beerli; March 31, 2011 Ordinary Differential Equations ISC-5315 { 7 ]; AppendTo[ii,i]; Transpose[{ii,yy}] ] and with different intervals sizes we get different accuracies. Figure 3 shows results for an h = 0:01 and h = 0:1 in comparison with the exact solution. 5 4 3 2 1 2 3 4 Figure 3: The blue curve represent the exact formula and the blue dashed curve reflects the Euler approxi- mation with h = 0:01, and green with h = 0:1 We see that the accuracy of the Euler approximation is quite good with small intervals but stinks with large intervals, looking at the algorithm, we that it is equivalent to Riemann's sum (quadra- ture).