? ? ? Email: yqyang ? August 3, 2015 Yu-Qian Yang?1/ 108 ? CONTENTS 1 ? 2 ? Yu-Qian Yang?2/ 108 ? ? CONTENTS 1 ? 2 ? ? ?Euler? ? ? ? ? Matlab? Yu-Qian Yang?3/ 108 ? ? ? ? ? ? ? ?dy dx = y2+x2? Yu-Qian Yang?4/ 108 ? ? ? ?(1)? ? ? dy dx = f(x,y)a x b y(a) = y0 ?f(x,y)?y?(Lipschitz)? ?L? |f(x,y) f(x, y)| L|y y| ?(1)? back Yu-Qian Yang?5/ 108 ? ? ? ?y(x),? a = x0 min 1im |Rei| ?Stiff? s = max 1im |Rei|/ min 1im |Rei| ? Yu-Qian Yang?36/ 108 ? ? ? ? ? ? ? ? Yu-Qian Yang?37/ 108 ? ? Matlab? MATLAB? 1 Matlab? ? Matlab? ?ode45?ode23?ode113?ode45? ?RK?ode23? ?RK?ode113? ?ode45? Matlab?Euler? Yu-Qian Yang?38/ 108 ? ? Matlab? ?I? ? y= f(x,y) y(x0) = y0 ?Euler? yp= yn+ hf(xn,yn) yq= yn+ hf(xn+ h,yp) yn+1= 1 2(yp + yq) back Yu-Qian Yang?39/ 108 ? ? Matlab? ?Euler?eulerpro.m? function x,y=eulerpro(fun,x0,xfi nal,y0,n); if nargin 0? ?i?y1= y,y2= y? ? y1= y2 y2= (1 y2 1)y2 y1 back Yu-Qian Yang?48/ 108 ? ? Matlab? ?ii?M?vdp1.m: functiondy = vdp1(t,y); dy = y(2);(1 y(1)2) y(2) y(1); ?iii?Matlab? T,Y = ode45(vdp1,0 20,2;0); back Yu-Qian Yang?49/ 108 ? ? Matlab? ?iv? plot(T,Y(:,1),-,T,Y(:,2),) title(Solution of van der Pol Equation,mu=1); xlabel(time t); ylabel(solution y); legend(y1,y2); 05101520 3 2 1 0 1 2 3 Solution of van der Pol Equation,mu=1 time t solution y y1 y2 back Yu-Qian Yang?50/ 108 ? ? Matlab? 2)? ?Matlab? ?dsolve?Matlab? ? ?D?Dy?y? ?n?Dn?D4y?y4? ? ?y+2y= y?Matlab?D2y+2 Dy = y bac Yu-Qian Yang?51/ 108 ? ? Matlab? ? ?,? dsolve(diff equation) dsolve(diff equation,var) diff equation?1?t? ?2?var?6? ? x2+ y + (x 2y)y= 0 ? symsx y diff equ =x2+ y + (x 2 y) Dy = 0; dsolve(diff equ,x) back Yu-Qian Yang?52/ 108 ? ? Matlab? ? ? dsolve(diff equation,condition1,condition2,var) ?condition1?condition2? ?7? y y= x?y(1) = 8,y(1) = 7,y(2) = 4? ? y = dsolve(D3y D2y = x,y(1) = 8,Dy(1) = 7,D2y(2) = 4,x) back Yu-Qian Yang?53/ 108 ? ? Matlab? ? ? dsolve(diff equ1diff equ2,var) dsolve(diff equ1diff equ2,condition1condition2,var) ?1?2? ? Yu-Qian Yang?54/ 108 ? ? Matlab? ?8? ? f + 3g = sinx g+ f = cosx ?f (2) = 0,f(3) = 3,g(5) = 1 ? ? ?clc,clear equ1 =D2f + 3 g = sin(x); equ2 =Dg + Df = cos(x); general f,general g = dsolve(equ1,equ2,x) f,g = dsolve(equ1,equ2,Df(2) = 0,f(3) = 3,g(5) = 1,x) back Yu-Qian Yang?55/ 108 ? ? CONTENTS 1 ? 2 ? ? ? ?Matlab? Yu-Qian Yang?56/ 108 ? ? ? ? ? ? ? ? back Yu-Qian Yang?57/ 108 ? ? ? ? ? ? back Yu-Qian Yang?58/ 108 ? ? ? ? ? ?(Poisson)? ? u = 2u x2 + 2u y2 = f(x,y) ?f(x,y) 0?(Laplace)? ? u = 2u x2 + 2u y2 = 0 back Yu-Qian Yang?59/ 108 ? ? ? Poisson? ? 2u x2 + 2u y2 = f(x,y)(x,y) u(x,y)|(x,y)= (x,y) = ? ?f(x,y),(x,y)?,? ? back Yu-Qian Yang?60/ 108 ? ? ? ? ?u n + u ? ? ?(x,y) = (x,y) ?n? = 0? ? 6= 0? back Yu-Qian Yang?61/ 108 ? ? ? ? ? ? u t a 2u x2 = 0(a 0) ? ?Cauchy? ? u t a 2u x2 = 0t 0, 0 x + u(x,0) = (x) x 0.01, p,e,t = refi nemesh(g,p,e,t); u = assempde(b,p,e,t,c,a,f); exact = (1 p(1,:).2 p(2,:).2)/4; err = norm(u exact,inf); error = errorerr; end subplot(2,