{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "# Crude numerical solution s\n\n# filename = 02.6.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "# Example 1: Numerical solutions\nx:='x':y:='y': \n eq:=D(y)(x)=sin(2*x-y(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sol:=dsolve(\{eq,y(0)=.5\},y(x),numeric);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 7 "sol(1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "sol(1)[2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rhs(sol(1)[2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "with (plots):\n odeplot(sol,[x,y(x)],0..15);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "with(DEtools):\n DEplot(diff(y(x),x)=sin(2*x-y(x)), y(x),\n\011\011x=0..15,\{[0,1],[0,-1]\},stepsize=0.05,linecolor=BLACK, color=GRAY);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Example 2" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "inits:=\{seq([0,i/2],i=1..12 )\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "with(DEtools):\n \+ DEplot(diff(y(x),x)=sin(x*y(x)),y(x),x=0..7,inits,\n\011arrows=NONE,st epsize=0.1,linecolor=BLACK);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "A pplication: Modeling the Spread of a Disease" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "eq:=diff(i(t),t)+(gamma+mu-lambda)*i(t)=-lambda* i(t)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sol:=dsolve(\{eq ,i(0)=i0\},i(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simpli fy(sol);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "sigma:=subs(\{ lambda=0.5,\ngamma=0.75,\nmu=0.65\},lambda/(gamma+mu));\ntoplot:=subs( \{lambda=0.5,\ngamma=0.75,\nmu=0.65\},rhs(sol));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "i0vals:=seq(0.1*k,k=1..9):\nplot([seq(subs(i0 =k,toplot),k=i0vals)],t=0..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "sigma:=subs(\{lambda=1.5,gamma=0.75,mu=0.65\},lambda/(gamma+m u));\ntoplot:=subs(\{lambda=1.5,gamma=0.75,mu=0.65\},rhs(sol)):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "i0vals:=seq(0.1*k,k=1..9):\n plot([seq(subs(i0=k,toplot),k=i0vals)],t=0..20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "i0vals:=seq(0.01*k,k=1..9):\nplot([seq(subs(i 0=k,toplot),k=i0vals)],t=0..20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "lambda:=t->2-2.5*sin(6*t):\neq:=diff(i(t),t)=(lambda( t)-3)*i(t)-lambda(t)*i(t)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "inits:=\{seq([0,i/10],i=1..9)\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "with(DEtools):\n DEplot(eq,i(t),0..10,inits,stepsize =0.05,arrows=NONE,linecolor=BLACK,thickness=1);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 141 "eq:=diff(i(t),t)=(lambda(t)-2)*i(t)-lambda(t) *i(t)^2:\n DEplot(eq,i(t),0..10,inits,stepsize=0.05,arrows=NONE,\n \011\011linecolor=BLACK,thickness=1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "?dsolve/numeric" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Example 3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "exactsol:=d solve(\{diff(y(x),x)=x*y(x),y(0)=1\},y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "p1:=plot(rhs(exactsol),x=0..1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "f:=(x,y)->x*y:\n h:=0.1:\n x:=n->n*h: \n y:=proc(n) option remember;\n\011y(n-1)+h*f(x(n-1),y(n-1))\n\011en d:\n y(0):=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "first:=[s eq([x(n),y(n)],n=0..10)]:\narray(first);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "p2:=plot(first,style=POINT,color=BLACK):\nwith(plots) :\ndisplay(\{p1,p2\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 " x:='x':y:='y':\n h:=0.05:\n x:=n->n*h:\n y:=proc(n) option remember ;\n\011y(n-1)+h*f(x(n-1),y(n-1))\n\011end:\n y(0):=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "second:=[seq([x(n),y(n)],n=0..20)]: \narray(second);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "p3:=plo t(second,style=POINT,color=BLACK):\ndisplay(\{p1,p3\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Example 4: Forward Euler Method" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "x:='x':y:='y':\n f:=(x,y)->x*y:\n h:=0.1:\n x:=n- >n*h:\n y:=proc(n) option remember;\n\011y(n-1)+h/2*(f(x(n-1),y(n-1)) +\n\011\011f(x(n),y(n-1)+h*f(x(n-1),y(n-1))))\n\011end:\n y(0):=1:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "third:=[seq([x(n),y(n)],n= 0..10)]:\narray(third);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " p4:=plot(third,style=POINT,color=BLACK):\ndisplay(\{p1,p4\});" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Eample 5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "yrk:='yrk':\n f:=(x,y)->x*y:\n h:=0.1:\n x:=n- >n*h:\n yrk:=proc(n)\n\011local k1,k2;\n\011option remember;\n\011k1: =h*f(x(n-1),yrk(n-1));\n\011k2:=h*f(x(n-1)+h,yrk(n-1)+k1);\n\011yrk(n- 1)+1/2*(k1+k2)\n\011end:\n yrk(0):=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "rktable1:=[seq([x(n),yrk(n)],n=0..10)]:\narray(rktabl e1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "p2:=plot(rktable1,s tyle=POINT,color=BLACK):\ndisplay(\{p1,p2\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Example 6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "yrk4:='yrk4':\n f:=(x,y)->x*y:\n h:=0.1:\n x:=n->n*h:\n yrk4: =proc(n)\n\011local k1,k2,k3,k4;\n\011option remember;\n\011k1:=f(x(n- 1),yrk4(n-1));\n\011k2:=f(x(n-1)+h/2,yrk4(n-1)+h*k1/2);\n\011k3:=f(x(n -1)+h/2,yrk4(n-1)+h*k2/2);\n\011k4:=f(x(n),yrk4(n-1)+h*k3);\n\011yrk4( n-1)+h/6*(k1+2*k2+2*k3+k4)\n\011end:\n yrk4(0):=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "rktable2:=[seq([x(n),yrk4(n)],n=0..10)]: \n array(rktable2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "p3: =plot(rktable2,style=POINT,color=BLACK):\ndisplay(\{p1,p3\});" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 1 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }