Jeremy Knight Math 639 5/3/11 Algorithm 12.2: Heat Equation Backward-Difference restart;
<Text-field style="Heading 1" layout="Heading 1">HeatEqBackwardDiff(f,alpha,l,T,m::integer,N::integer,display)</Text-field> To approximate the solution to the parabolic partial differential equation 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. subject to boundary conditions 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, and the initial conditions 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. HeatEqBackwardDiff:=proc(f,l,T,alpha,m::integer,N::integer,interval,DISPLAY) #L=endpoint #T= maximum time #alpha = constant #m = subintervals of x #N = subintervals of t #display=0 only prints final time values local i,j,h,k,L,w,t,u,z,x,lambda: #option trace: h:=l/m: k:=T/N: lambda:=(alpha^2*k)/h^2: w:=Array(1..m-1): u:=Array(1..m-2): for i from 1 to m-1 do w[i]:=f(i*h): end do: #Now solve a tridiagonal linear system using Algoritm 6.7 L:=Array(1..m-1): L[1]:=1.0+2.0*lambda: u[1]:=-lambda/L[1]: for i from 2 to m-2 do L[i]:=1.0+2.0*lambda+lambda*u[i-1]: u[i]:=-lambda/L[i]: end do: L[m-1]:=1.0+2.0*lambda+lambda*u[m-2]: for j from 1 to N do #Curent t_j t:=j*k: z[1]:=w[1]/L[1]: for i from 2 to m-1 do z[i]:=(w[i]+lambda*z[i-1])/L[i]: end do: w[m-1]=z[m-1]: for i from m-2 to 1 by -1 do w[i]:=z[i]-u[i]*w[i+1]: end do: if j=N or DISPLAY<>0 then print(t): for i from 1 to m-1 by interval do x:=i*h: print(x,evalf(w[i])) #w[i] = w[i,j] end do: end if: end do: end proc:
<Text-field style="Heading 1" layout="Heading 1">HeatEqBackwardDiff2(f,alpha,l,T,h,k,display)</Text-field> To approximate the solution to the parabolic partial differential equation 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. subject to boundary conditions 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, and the initial conditions 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. HeatEqBackwardDiff2:=proc(f,l,T,alpha,h,k,interval,DISPLAY) #L=endpoint #T= maximum time #alpha = constant #m = subintervals of x #N = subintervals of t #display=0 only prints final time values local i,j,L,w,t,u,z,x,lambda: #option trace: lambda:=(alpha^2*k)/h^2: w:=Array(1..m-1): u:=Array(1..m-2): for i from 1 to m-1 do w[i]:=f(i*h): end do: #Now solve a tridiagonal linear system using Algoritm 6.7 L:=Array(1..m-1): L[1]:=1.0+2.0*lambda: u[1]:=-lambda/L[1]: for i from 2 to m-2 do L[i]:=1.0+2.0*lambda+lambda*u[i-1]: u[i]:=-lambda/L[i]: end do: L[m-1]:=1.0+2.0*lambda+lambda*u[m-2]: for j from 1 to N do #Curent t_j t:=j*k: z[1]:=w[1]/L[1]: for i from 2 to m-1 do z[i]:=(w[i]+lambda*z[i-1])/L[i]: end do: w[m-1]=z[m-1]: for i from m-2 to 1 by -1 do w[i]:=z[i]-u[i]*w[i+1]: end do: if j=N or DISPLAY<>0 then print(t): for i from 1 to m-1 by interval do x:=i*h: print(x,evalf(w[i])) #w[i] = w[i,j] end do: end if: end do: end proc:
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Example 2 pg 731</Text-field> JSFH JSFH interface(prettyprint=2); f:=x->(evalf(sin(Pi*x))); IiIj Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSRzaW5HNiRGK0koX3N5c2xpYkdGJTYjKiZJI1BpR0YrIiIiOSRGNEYlRiVGJQ== #HeatEqBackwarDiff:=proc(f,l,T,alpha,m::integer,N::integer) HeatEqBackwardDiff(f,1.0,0.5,1.0,1000,10,100,0); JCInKytdISIn NiQkIicrKzUhIikkIidqL1ohIzU= NiQkIicrNTUhIickIidAcVkhIik= NiQkIicrNT8hIickIicneSIpKSEiKQ== NiQkIicrNUkhIickIidfMzchIig= NiQkIicrNVMhIickIidGOzkhIig= NiQkIicrNV0hIickIidVJ1siISIo NiQkIicrNWchIickIic7SzkhIig= NiQkIicrNXEhIickIicmUUUiISIo NiQkIicrNSEpISInJCInWis1ISIo NiQkIicrNSEqISInJCInKCoqcCchIik=
JSFH
<Text-field style="Heading 1" layout="Heading 1">Exercise Set 12.2 </Text-field> 1. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NDU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjFGJ0YyRjI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYzLUkjbW9HRiQ2LVEiLEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYtUSJ+RidGL0YyL0Y2RjRGOEY6RjxGPkZARkIvRkZGRC1JI21pR0YkNiVRIm1GJy8lJ2l0YWxpY0dGNy9GMFEnaXRhbGljRictRiw2LVEiPUYnRi9GMkZLRjhGOkY8Rj5GQC9GQ1EsMC4yNzc3Nzc4ZW1GJy9GRkZZLUkjbW5HRiQ2JFEiNEYnRi9GK0ZILUZONiVRIlRGJ0ZRRlNGVS1GZm42JFEjMC5GJ0YvRkgtRmZuNiRRIjFGJ0YvRistRk42JVEiTkYnRlFGU0ZVLUZmbjYkUSIyRidGL0Yv 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 Digits:=10; alpha:=1.0; l:=2.0; m:=4; N:=2; T:=0.1; f:=x->evalf(sin(x*Pi/2)); IiM1 JCIjNSEiIg== JCIjPyEiIg== IiIl IiIj JCIiIiEiIg== Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSRzaW5HNiRGK0koX3N5c2xpYkdGJTYjLCQqJkkjUGlHRisiIiI5JEY1I0Y1IiIjRiVGJUYl HeatEqBackwardDiff(f,l,T,alpha,m,N,1,1); JCIrKysrK10hIzY= NiQkIisrKysrXSEjNSQiK0Z2J1xNJ0Yl NiQkIisrKysrNSEiKiQiKz9PVmYhKiEjNQ== NiQkIisrKysrOiEiKiQiKzh5MXJxISM1 JCIrKysrKzUhIzU= NiQkIisrKysrXSEjNSQiK1BQYTxkRiU= NiQkIisrKysrNSEiKiQiK0Mmb3pIKSEjNQ== NiQkIisrKysrOiEiKiQiKzh5MXJxISM1 3. alpha:=1; l:=2.0; T:=0.5; h:=evalf(Pi/10); k:=0.05; m:=5; N:=10; HeatEqBackwardDiff(f,l,T,alpha,m,N,1,1); IiIi JCIjPyEiIg== JCIiJiEiIg== JCIrYUVmVEohIzU= JCIiJiEiIw== IiIm IiM1 JCIrKysrK10hIzY= NiQkIisrKysrUyEjNSQiK1trKGVEJkYl NiQkIisrKysrISkhIzUkIithJUg5XylGJQ== NiQkIisrKysrNyEiKiQiK3YiWzxpKSEjNQ== NiQkIisrKysrOyEiKiQiK0FEJnkoZSEjNQ== JCIrKysrKzUhIzU= NiQkIisrKysrUyEjNSQiK3VhZTRaRiU= NiQkIisrKysrISkhIzUkIitOKVI1bihGJQ== NiQkIisrKysrNyEiKiQiKzN1QzZ6ISM1 NiQkIisrKysrOyEiKiQiK0FEJnkoZSEjNQ== JCIrKysrKzohIzU= NiQkIisrKysrUyEjNSQiK0tNJVFCJUYl NiQkIisrKysrISkhIzUkIitEQkpYcEYl NiQkIisrKysrNyEiKiQiKyQ9YFdMKCEjNQ== NiQkIisrKysrOyEiKiQiK0FEJnkoZSEjNQ== JCIrKysrKz8hIzU= NiQkIisrKysrUyEjNSQiK3FyWUFRRiU= NiQkIisrKysrISkhIzUkIisvLmBHakYl NiQkIisrKysrNyEiKiQiK2s0KjMnbyEjNQ== NiQkIisrKysrOyEiKiQiK0FEJnkoZSEjNQ== JCIrKysrK0QhIzU= NiQkIisrKysrUyEjNSQiK2dBdW9NRiU= NiQkIisrKysrISkhIzUkIis0W2MwZUYl NiQkIisrKysrNyEiKiQiK3R4KilvayEjNQ== NiQkIisrKysrOyEiKiQiK0FEJnkoZSEjNQ== JCIrKysrK0khIzU= NiQkIisrKysrUyEjNSQiK0ZmIWY7JEYl NiQkIisrKysrISkhIzUkIismZU5GTyZGJQ== NiQkIisrKysrNyEiKiQiK0Y8XlVoISM1 NiQkIisrKysrOyEiKiQiK0FEJnkoZSEjNQ== JCIrKysrK04hIzU= NiQkIisrKysrUyEjNSQiKy4oKlsySEYl NiQkIisrKysrISkhIzUkIishXFohKSlcRiU= NiQkIisrKysrNyEiKiQiK3RMZ3BlISM1 NiQkIisrKysrOyEiKiQiK0FEJnkoZSEjNQ== JCIrKysrK1MhIzU= NiQkIisrKysrUyEjNSQiKzFJXyhvI0Yl NiQkIisrKysrISkhIzUkIishZV82biVGJQ== NiQkIisrKysrNyEiKiQiK1UlPTJrJiEjNQ== NiQkIisrKysrOyEiKiQiK0FEJnkoZSEjNQ== JCIrKysrK1ghIzU= NiQkIisrKysrUyEjNSQiK0UsaitERiU= NiQkIisrKysrISkhIzUkIitSST8uV0Yl NiQkIisrKysrNyEiKiQiKz51TFthISM1 NiQkIisrKysrOyEiKiQiK0FEJnkoZSEjNQ== JCIrKysrK10hIzU= NiQkIisrKysrUyEjNSQiKzZgMFVCRiU= NiQkIisrKysrISkhIzUkIis8N253VEYl NiQkIisrKysrNyEiKiQiKyVHJlEnRyYhIzU= NiQkIisrKysrOyEiKiQiK0FEJnkoZSEjNQ==
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