(6.1) -The Linear Shooting Method

Transcription

1 (6.1) -The Linear Shooting Method Consider the boundary value problems (BVPs) for the second order differential equation of the form (*) y fx,y,y, a x b, ya and yb. Under what conditions does a boundary value problem have a solution or has a unique solution? 1. Existence and Uniqueness of Solutions of BVPs: Theorem Suppose that f is continuous on the set D x,y,y ; a x b, y, y and the partial derivatives f y and f y are continuous on D. If f y x,y,y, for all x,y, y in D, and there exists a constant M such that f y x,y,y M for all x,y,y in D, then the boundary value problem (*) has a unique solution. Example Consider the following boundary value problem: y e xy siny, 1 x 2, y1 y2 Determine if the boundary value problem has a unique solution. Rewrite y e xy siny so fx,y,y e xy siny Check conditions: fx,y,y e xy siny, f y x,y,y xe xy,andf y x,y,y cosy are continuous on D x,y,y ; 1 x 2, y, y. (i) f y x,y,y xe xy ond. (ii) f y t,y,y cosy 1 M. So, the boundary value problem has a unique solution in D. Example Consider the linear boundary value problem of the form: y pxy qxy rx, a x b, ya, yb Under what condition(s) does a linear BVP have a unique solution? For the linear boundary value problem, fx,y,y pxy qxy rx. Observe that f y x,y,y qx, f y x,y,y px. f y x,y,y and f y x,y,y are continuous on D if and only if px, qx and rx are continuous for a x b. Now we check conditions in (i) and (ii). (i) f y x,y,y qx fora x b. (ii) Since f y is continuous on a, b, f y is bounded. So, if px, qx and rx are continuous for a x b, andqx fora x b, then this linear boundary value problem has a unique solution. 1

2 2. The Linear Shooting Method: Consider the linear boundary value problems of the form: y pxy qxy rx, a x b, ya, yb where px, qx and rx are continuous and qx fora x b. Consider the solutions of the following two initial-value problems: (i) y pxy qxy rx, a x b, ya, y a (**) (ii) y pxy qxy, a x b, ya, y a 1 say, y 1 x and y 2 x. Letyx be the following linear combination of y 1 x and y 2 x : yx y 1 x y 1b y 2 x y 2 b Then yx is the solution of the boundary value problem. Check: y x y 1 x y 1b y y 2 b 2 x pxy 1 x qxy 1 x rx y 1b pxy y 2 b 2 x qxy 2 x px y 1 x y 1b y y 2 b 2 x qx y 1 x y 1b y 2 x rx y 2 b So, yx is a solution of y pxy qxy rx. Check the boundary conditions: ya y 1 a y 1b y 2 a y 1b y 2 b y 2 b yb y 1 b y 1b y 2 b. y 2 b This suggests that a boundary value problem can be solved by solving two (independent) initial-value problems in (**). Review: Solve a second-order initial-value problem: y fx,y,y, ya, y a 1. Let y 1,and y 1. Then above second-order differential equation for y becomes the following system of two first-order differential equation in and : f 1 x,,, a x b, a, a 1. Example Set Rewrite the differential equation y 2y 4y te 2t as a system of 2 1st-order differential equations. y y y y 2y 4y te 2t 4 2 te 2t The system is: u te 2t te 2t 2

3 Example Rewrite the initial-value problem for the system of 2 second-order differential equations y 1 y 1 2y 2 2y 1 y 2 t y 2 3y 1 y 1 4y 2 2sint, y 1 1, y 2 1, y 1 2, y 2 3 Set as an initial-value problem for a system of 4 first-order differential equations. y 1 y 1 y 1 2y 1 y 2 y 1 2y 2 t 2 2 t y 2 y 2 y 2 y 1 4y 2 3y 1 2sint 4 3 2sint The system of 4 1st-order linear differential equations is: 2 2 t or in matrix-vector notation: 4 3 2sint , 1, 2, 1, 3; 1 t 2, 1 2sint 3 Example Solve the boundary value problem: y 4 x y 2 x y 2 2 x lnx, 2 1 x 2, y1 1, y2 ln2. 2 Exact solution is: yx 1 x 2 2 4x 3 2 x2 x 2 lnx. Note that it is a linear boundary value problem where px 4 x, qx 2 x, rx 2 2 x lnx 2 continuous on 1,2. Since qx, we cannot say if this boundary value problem has a unique solution. Now we solve the following two initial-value problems: 3

4 y(x i )-y i (i) y 1 4 x y 1 2 x 2 y 1 2 x 2 lnx, 1 x 2, y , y 1 1 (ii) y 2 4 x y 2 2 x 2 y 2, 1 x 2, y 2 1, y Set up the initial-value problem for a system of 4 1st-order differential equations: 4 x 2 x 2 2 x 2 lnx, 1 1 2, 1, 1, x 2 x 2.8 Shooting Method: y"=-4/x*y-2/x 2 *y+2*ln(x)/x 2, y(1)=1/2,y(2)=ln2 1-5 Shooting Method: y"=-4/x*y-2/x 2 *y+2*ln(x)/x 2, y(1)=1/2,y(2)=ln y=y 1 +(beta-y 1 (2))/y 2 (2)*y IVP: y 1 "=-4/x*y 1-2/x 2 *y 1 +2*ln(x)/x 2, y 1 (1)=1/2,y 1 (1)= IVP: y 2 "=-4/x*y 2-2/x 2 *y 2, y 2 (1)=, y 2 (1)= x help ode45 ODE45 Solve non-stiff differential equations, medium order method. [TOUT,YOUT] ODE45(ODEFUN,TSPAN,Y) with TSPAN [T TFINAL] integrates MatLab program for this example lect6_1_ex1.m: clf alpha1/2; betalog(2); a1; b2; [xv,yv]ode45( funsysa,[a b],[alpha;;;1]); plot(xv,yv(:,1), r-.,xv,yv(:,3), m ) hold nlength(yv(:,1)); y1nyv(n,1); y2nyv(n,3); yvsolyv(:,1)(beta-y1n)/y2n*yv(:,3); 4

5 truesol-1./(xv.^2).*(2-4*xv3/2*xv.^2-xv.^2.*log(xv)); plot(xv,yvsol, b-,xv,truesol, k- ) title( Shooting Method: y"-4/x*y-2/x^2*y2*ln(x)/x^2, y(1)1/2,y(2)ln2 ) text(1.2,.4, IVP: y_1"-4/x*y_1-2/x^2*y_12*ln(x)/x^2, y_1(1)1/2,y_1(1) ) text(1.3,.1, IVP: y_2"-4/x*y_2-2/x^2*y_2, y_2(1), y_2(1)1 ) text(1.2,.6, - yy_1(beta-y_1(2))/y_2(2)*y_2 ) axis([1 2.8]) hold off figure(2) semilogy(xv,abs(yvsol-truesol), b-. ) title( Shooting Method: y"-4/x*y-2/x^2*y2*ln(x)/x^2, y(1)1/2,y(2)ln2 ) ylabel( y(x_i)-y_i ) xlabel( x ) MatLab function: funsysa.m function yvfunsysa(t,y); yv(1,1)y(2,1); yv(2,1)-4/t*y(2,1)-2/(t^2)*y(1,1)2*log(t)/(t^2); yv(3,1)y(4,1); yv(4,1)-4/t*y(4,1)-2/(t^2)*y(3,1); Notes: ode45.m is an MatLab building-in function which solves initial-value problems for systems of n first-order differential equations: u n f 1 x,,...,u n f n x,,...,u n, a x b, a 1,...,u n a n. It is called by: [xv,uv]ode45( funsysa,[a b],[ 1,..., n ]); where funsysa.m is a user-provided Matlab program that x a evaluates functions of the system at x. The outputs are the vector xv x 1 x N 1 x N b and u n uv 1 u n1 N u nn. That is, for a x b, y 1 x, 1,..., N,..., y n x u n, u n1,..., u nn. 5

6 Exercises: 1. For each of the following boundary value problems, determine (1) if it is linear; and (2) if it has a unique solution. (1) y 1 x y 3 x y lnx 2 x 1, 1 x 2, y1 y2 (2) y y 3 yy,1 x 2, y1 1, y (3) y y 2y lnx 3 1 x,2 x 3, y2 1 2 ln2, f3 1 3 ln3 2. Consider the boundary value problem: y y, x b, y, yb B. Find choices for b and B so that this boundary value problem has (1) no solution; (2) exactly one solution; (3) infinitely many solutions. 3. Rewrite the initial-value problem for the system of 2 second-order differential equations y 1 y 1 y 2 2y 1 y 2 e 2t y 2 3y 1 y 1 3y 2 2cost, y 1 1, y 2 3, y 1 1, y 2 3 as an initial-value problem for a system of 4 first-order differential equations. 4. Consider the boundary value problem: y y 2y cosx, x 2, y.3, y 2.1. a. determine if it is linear; b. determine if it has a unique solution; c. approximate the solution by the Shooting Method for linear boundary value problem with h 8 ; and d. compare the approximation with the true solution yx 1 1 sinx 3cosx by plotting the absolute values of the differences. 5. For each of the following boundary value problems, a. determine if it is linear; b. determine if it has a unique solution; c. approximate the solution by the Shooting Method for linear boundary value problem with given h; and d. plot the graph of approximation of the solution yx. (1) y 4 x y 2 x y 2 2 x lnx, 1 x 2, y1 1, y2 ln2, h (2) y 2y y xe x x, x 2, y, y2 4, h.2 6. Let p represent the electrostatic potential in r between two concentric metal spheres of radii R 1 and R 2 where R 1 R 2. The potential of the inner sphere is kept constant at V 1 volts, and the potential of the 6

7 outer sphere is volts. The potential in the region between the two spheres is governed by Laplace equation, which, in this particular application, reduces to d 2 p dr 2 2 r dp dr, R 1 r R 2, pr 1 V 1 and pr 2. Approximate the potential for the case where R 1 2 inches, R 2 4 inches, and V 1 11 volts. 7