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THE AVERAGE APPROXIMATING METHOD ON FUNCTIONAL ADJUSTMENT QUANTITY FOR SOLVING The Volterra Integral Equation II ( corrected for solving integral equations with Hereditary kernels ) by Co.H Tran , University of Natural Sciences , HCMC Vietnam - Institute of Applied Mechanics , HCMC - coth123@math.com & coth123@yahoo.com Copyright 2004 Sat , November 06 2004 ---------------------------------------------------------------------------------------------------- ** Abstract : Solving the Volterra's integral equation II with applying the Neumann series and the average approximating method on functional adjustment quantity . ** Subjects: Viscoelasticity Mechanics , The Integral equation . -----------------------------------------------------------------------------------------------------

Copyright Co.H Tran -- . The Average Approximating Method on Functional Adjustment Quantity ( Sokolov's method ) All rights reserved. No copying or transmitting of this material is allowed without the prior written permission of Co.H Tran

The Average Approximating Method on Functional Adjustment Quantity ( Sokolov's method ) In consideration of The Volterra Integral Equation II ( second kind ) , we find the explicit expression for the resolvent kernel NiMlJkdhbW1hRw== ( t , t ) in the general form : v = ( 1 + NiMlJ2xhbWJkYUc=K* ) u here NiMlJ2xhbWJkYUc= : arbitrary parameter . The solution of u can be represented with the Neumann series : . The resolvent operator NiMlJkdhbW1hRw==* is determined by a Neumann series : , then the kernel . The convergence of this series must be investigated in a connection with the Neumann series . The average approximating method on the functional adjustment quantity ( Sokolov's method ) makes increasing for the rate of convergence of this series . From the first approximation of the solution u , we find the adjustment quantity for the next and so on . We consider the following equation : ( 1 ) the first approximation : ( 2 ) by choosing the initial adjustment quantity : ( 3 ) From ( 2 ) and ( 3 ) we obtain : ( 4 ) with ( 5 ) the n-th approximation : ( 6 ) and the adjustment quantity of the n-th order can then be written as : ( 7 ) here ( 8 ) . From ( 6 ) , ( 7 ) and ( 8 ) we have : ( 9 ) Denoting the formulas ( 6 ) to ( 9 ) can be carried out by the computer programming language. We can show that the convergence condition of this method is ( 10 ) here : the project-operator from the Banach's space B into its space Bo ( the solution u B )

As seen , the first approximation : NiMvLSUjdTFHNiMlInRHLCYtJSJ2R0YmIiIiKiglJ2xhbWJkYUdGKyYlJmFscGhhRzYjRitGKy0lJGludEc2JC0lIktHNiRGJyUkdGF1Ry9GNzslImFHJSJiR0YrRis= We choose the initial adjustment quantity :NiMvJiUmYWxwaGFHNiMiIiIqJi0lJGludEc2JC0lI3UxRzYjJSJ0Ry9GLzslImFHJSJiR0YnLCZGM0YnRjIhIiJGNQ== with NiM+LSYlInVHNiMiIiE2IyUieEdGKA== ; NiM+LSYlIlRHNiMiIiI2IyUieEctJiUidUdGJ0Yp NiMlJFRoZUc= adjustment quantity of the order i-th can be expresssed : NiM+JiUmYWxwaGFHNiMsJiUiaUciIiJGKUYpKiolJ2xhbWJkYUdGKS0lJGludEc2JComJSJLR0YpLCgtJiUiVEc2I0YoNiMlInlHRiktJkY0NiMsJkYoRilGKSEiIkY2RjwmRiVGNUY8RikvRjc7JSJhRyUiYkdGKSwmRkFGKUZARjxGKSUiREdGPA== The coefficient NiM+JiUmYWxwaGFHNiMsJiUiaUciIiIiIiNGKS0lJG1pbkc2JCZGJTYjRigqKiUnbGFtYmRhR0YpLSUkaW50RzYkKiYlIktHRiksKC0mJSJUR0YvNiMlInlHRiktJkY6NiMsJkYoRilGKSEiIkY7RkFGLkZBRikvRjw7JSJhRyUiYkdGKSwmRkVGKUZERkFGKSUiREdGQQ== Compare with the initial function and we have the error estimated : NiM+JiUmc2Fpc29HNiMlImlHLSUlc3FydEc2IyomLSUkaW50RzYkKiQpLSUkYWJzRzYjLCYtJiUidUc2IywmRiciIiJGOkY6NiMlInhHRjotJkY3RiZGOyEiIiIiI0Y6L0Y8OyUiYUclImJHRjosJkZERjpGQ0Y/Rj8= restart; interface(warnlevel=0):

xapxi:=proc(f,lambda,a,b,n,K) local s,smax,smin,saiso,i,D,T,eq1,alpha,eq,alpha1,iv,dv; eq1:=u(x)=f+lambda*int(K*u(y), y=a..b);;printf("\n%s"," ");print("*********************************************************************");print(" THE VOLTERRA INTEGRAL EQUATION II ( second kind ) :",eq1,"*********************************************************************");u[1](x):=f+alpha1*lambda*int(K, y=a..b);alpha:=int(u[1](x), x=a..b);eq:=alpha1=alpha;printf("\n%s"," RECURRING TASKS NUMBER ");print("1&/.THE EQUATION OF 1st ADJUSTMENT QUANTITY : ", eq);alpha1:=Re(solve(eq,alpha1));print(" THE 1st ADJUSTMENT QUANTITY : alpha[1]=",evalf(alpha1,3));alpha[1]:=alpha1;u[1](x):=f+alpha[1]*lambda*int(K, y=a..b);print(" FUNCTION u[1](x)=",u[1](x));; T[0](x):=0: T[1](x):=x->u[1](x);D:=b-a-lambda*int(int(K, y=a..b), x=a..b); for i from 1 to n do u[0](x):=0:T[0](y):=0:T[1](x):=u[1](x);T[i](y):=subs(x=y,T[i](x));T[i-1](y):=subs(x=y,T[i-1](x));;printf("\n%s"," ");print(" COMPARE WITH THE INITIAL FUNCTION",u[i-1],"(x)=",T[i-1](x));;printf("\n%s"," ");print(" -------------------------------------------------------------------------------------------");printf("\n%s"," RECURRING LOOP No ");printf("\n%s"," "); ;print(i+1,"&/. ADJUSTMENT QUANTITY OF :",i+1," order WE OBTAIN : ");;alpha[i+1]:=(lambda/D)*int(int(K*(T[i](y)-T[i-1](y)-alpha[i]), y=a..b), x=a..b); alpha[i+1]:=evalf(alpha[i+1],5);;alpha[i+2]:=min(alpha[i+1],alpha[i]);;alpha[i+2]:= evalf(alpha[i+2],5); ;T[i+1](x):=f+lambda*int(K*(T[i](y)+alpha[i+1]), y=a..b);u[i+1](x):=T[i+1](x);;print(" FUNCTION ",u[i+1],"(x)=",T[i+1](x));print(" ALPHA COEFFICIENT",[i+1],"=" ,evalf(alpha[i+1],5));;;;saiso[i]:=sqrt((1/(b-a))*int((abs(u[i+1](x)-u[i](x))^2), x=a..b));;;;print(" ERROR OF : ",u[i+1],"(x)"," AND ",u[i],"(x)"," IS :");print( " ERROR ESTIMATED = ",evalf(saiso[i],5)); od:printf("\n%s"," ");printf("\n%s"," "); print("--------------------------------CONCLUSION-------------------------------- ");;printf("\n%s"," ");print( "THE ESTIMATED ERRORS OF ");for i from 1 to n do print( " order ",[i]," IS :", evalf(saiso[i],5));od:;printf("\n%s"," ");printf("\n%s"," "); end: It is easy to see that NiMpJSJ1RyUia0c=(x) n is a Cauchy sequence in L2(T) as k -> NiMlKWluZmluaXR5Rw==. It follows from the completeness of L2(T) that it converges in the L2 sense to a sum g in L2(T). That is, we have lim || NiMpJSJ1RyUia0c=(x) -NiMpJSJ1RywmJSJrRyIiIkYnISIi(x) || = 0 k -> NiMlKWluZmluaXR5Rw== dothi:=proc(k,m,n,h) local y,ym,yn; with(plottools):with(plots):y:=u[k](x);ym:=u[m](x);yn:=u[n](x);print("GRAPHIC :", u[k],"(x) = ",u[k](x)," RED ");;print("GRAPHIC :", u[m],"(x) = ",u[m](x)," YELLOW ");;print("GRAPHIC :", u[n],"(x) = ",u[n](x)," BLUE "); :plot([y(x),ym,yn],x=-h..h,color=[red,yellow,blue],thickness=[3,9,15],title='APPROXIMATEDGRAPHICS'); end: xapxi(-20.2*sqrt(x), 3 , 0 , 1 , 10 , sqrt(x)*(y+10) );

UWBvKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqNiI=

NiVRZ29+fn5+fn5+fn5+fn5+fn5+fn5+fn5+fn5+VEhFflZPTFRFUlJBfklOVEVHUkFMfkVRVUFUSU9OfklJfih+c2Vjb25kfmtpbmR+KX46NiIvLUkidUdGJDYjSSJ4R0YkLCYqJEYpIyIiIiIiIyQhJC0jISIiLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQqKEYpRiwsJkkieUdGJEYtIiM1Ri1GLS1GJzYjRjpGLS9GOjsiIiFGLSIiJFFgbyoqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKkYk

RECURRING TASKS NUMBER

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NiRRSX5USEV+MXN0fkFESlVTVE1FTlR+UVVBTlRJVFl+On5hbHBoYVsxXT02IiQiJHQnISIk

NiRRM35GVU5DVElPTn51WzFdKHgpPTYiLCQqJEkieEdGJCMiIiIiIiMkIiorKyssIiEiKQ==

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RECURRING LOOP No

NiYiIiNRPiYvLn5+QURKVVNUTUVOVH5RVUFOVElUWX5PRn46NiJGI1E0fm9yZGVyfldFfk9CVEFJTn46fkYl

NiZRK35GVU5DVElPTn42IiZJInVHRiQ2IyIiI1ElKHgpPUYkLCQqJEkieEdGJCMiIiJGKCQiKTAsKioqKiEiKQ==

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NiRRNH5FUlJPUn5FU1RJTUFURUR+PX42IiQiJjU5KCEiKA==

NiZRQ35DT01QQVJFfldJVEh+VEhFfklOSVRJQUx+RlVOQ1RJT042IiZJInVHRiQ2IyIiIlElKHgpPUYkLCQqJEkieEdGJCNGKCIiIyQiKisrKywiISIp

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RECURRING LOOP No

NiYiIiRRPiYvLn5+QURKVVNUTUVOVH5RVUFOVElUWX5PRn46NiJGI1E0fm9yZGVyfldFfk9CVEFJTn46fkYl

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UWdwfi0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS02Ig==

RECURRING LOOP No

NiYiIiVRPiYvLn5+QURKVVNUTUVOVH5RVUFOVElUWX5PRn46NiJGI1E0fm9yZGVyfldFfk9CVEFJTn46fkYl

NiZRK35GVU5DVElPTn42IiZJInVHRiQ2IyIiJVElKHgpPUYkLCQqJEkieEdGJCMiIiIiIiMkIikqKioqKioqKiEiKQ==

NiZRM35BTFBIQX5DT0VGRklDSUVOVDYiNyMiIiVRIj1GJCQhJiEqZichIzY=

NilRLX5FUlJPUn5PRn46fjYiJkkidUdGJDYjIiIlUSQoeClGJFEmfkFORH5GJCZGJjYjIiIkRilRJn5JU346RiQ=

NiRRNH5FUlJPUn5FU1RJTUFURUR+PX42IiQiJi8rKCEjNg==

NiZRQ35DT01QQVJFfldJVEh+VEhFfklOSVRJQUx+RlVOQ1RJT042IiZJInVHRiQ2IyIiJFElKHgpPUYkLCQqJEkieEdGJCMiIiIiIiMkIiopNCsrNSEiKQ==

UWdwfi0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS02Ig==

RECURRING LOOP No

NiYiIiZRPiYvLn5+QURKVVNUTUVOVH5RVUFOVElUWX5PRn46NiJGI1E0fm9yZGVyfldFfk9CVEFJTn46fkYl

NiZRK35GVU5DVElPTn42IiZJInVHRiQ2IyIiJlElKHgpPUYkLCQqJEkieEdGJCMiIiIiIiMkIiorKysrIiEiKQ==

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NilRLX5FUlJPUn5PRn46fjYiJkkidUdGJDYjIiImUSQoeClGJFEmfkFORH5GJCZGJjYjIiIlRilRJn5JU346RiQ=

NiRRNH5FUlJPUn5FU1RJTUFURUR+PX42IiQiJjYyKCEjOA==

NiZRQ35DT01QQVJFfldJVEh+VEhFfklOSVRJQUx+RlVOQ1RJT042IiZJInVHRiQ2IyIiJVElKHgpPUYkLCQqJEkieEdGJCMiIiIiIiMkIikqKioqKioqKiEiKQ==

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RECURRING LOOP No

NiYiIidRPiYvLn5+QURKVVNUTUVOVH5RVUFOVElUWX5PRn46NiJGI1E0fm9yZGVyfldFfk9CVEFJTn46fkYl

NiZRK35GVU5DVElPTn42IiZJInVHRiQ2IyIiJ1ElKHgpPUYkLCQqJEkieEdGJCMiIiIiIiMkIiorKysrIiEiKQ==

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NilRLX5FUlJPUn5PRn46fjYiJkkidUdGJDYjIiInUSQoeClGJFEmfkFORH5GJCZGJjYjIiImRilRJn5JU346RiQ=

NiRRNH5FUlJPUn5FU1RJTUFURUR+PX42IiQiIiFGJg==

NiZRQ35DT01QQVJFfldJVEh+VEhFfklOSVRJQUx+RlVOQ1RJT042IiZJInVHRiQ2IyIiJlElKHgpPUYkLCQqJEkieEdGJCMiIiIiIiMkIiorKysrIiEiKQ==

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RECURRING LOOP No

NiYiIihRPiYvLn5+QURKVVNUTUVOVH5RVUFOVElUWX5PRn46NiJGI1E0fm9yZGVyfldFfk9CVEFJTn46fkYl

NiZRK35GVU5DVElPTn42IiZJInVHRiQ2IyIiKFElKHgpPUYkLCQqJEkieEdGJCMiIiIiIiMkIiorKysrIiEiKQ==

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NilRLX5FUlJPUn5PRn46fjYiJkkidUdGJDYjIiIoUSQoeClGJFEmfkFORH5GJCZGJjYjIiInRilRJn5JU346RiQ=

NiRRNH5FUlJPUn5FU1RJTUFURUR+PX42IiQiIiFGJg==

NiZRQ35DT01QQVJFfldJVEh+VEhFfklOSVRJQUx+RlVOQ1RJT042IiZJInVHRiQ2IyIiJ1ElKHgpPUYkLCQqJEkieEdGJCMiIiIiIiMkIiorKysrIiEiKQ==

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RECURRING LOOP No

NiYiIilRPiYvLn5+QURKVVNUTUVOVH5RVUFOVElUWX5PRn46NiJGI1E0fm9yZGVyfldFfk9CVEFJTn46fkYl

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NiRRNH5FUlJPUn5FU1RJTUFURUR+PX42IiQiIiFGJg==

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RECURRING LOOP No

NiYiIipRPiYvLn5+QURKVVNUTUVOVH5RVUFOVElUWX5PRn46NiJGI1E0fm9yZGVyfldFfk9CVEFJTn46fkYl

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NiRRNH5FUlJPUn5FU1RJTUFURUR+PX42IiQiIiFGJg==

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RECURRING LOOP No

NiYiIzVRPiYvLn5+QURKVVNUTUVOVH5RVUFOVElUWX5PRn46NiJGI1E0fm9yZGVyfldFfk9CVEFJTn46fkYl

NiZRK35GVU5DVElPTn42IiZJInVHRiQ2IyIjNVElKHgpPUYkLCQqJEkieEdGJCMiIiIiIiMkIiorKysrIiEiKQ==

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NiRRNH5FUlJPUn5FU1RJTUFURUR+PX42IiQiIiFGJg==

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RECURRING LOOP No

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NiRRNH5FUlJPUn5FU1RJTUFURUR+PX42IiQiIiFGJg==

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UTlUSEV+RVNUSU1BVEVEfkVSUk9SU35PRn42Ig==

NiZRKH5vcmRlcn42IjcjIiIiUSZ+SVN+OkYkJCImNTkoISIo

NiZRKH5vcmRlcn42IjcjIiIjUSZ+SVN+OkYkJCImaDEoISIq

NiZRKH5vcmRlcn42IjcjIiIkUSZ+SVN+OkYkJCImLysoISM2

NiZRKH5vcmRlcn42IjcjIiIlUSZ+SVN+OkYkJCImNjIoISM4

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NiZRKH5vcmRlcn42IjcjIiInUSZ+SVN+OkYkJCIiIUYp

NiZRKH5vcmRlcn42IjcjIiIoUSZ+SVN+OkYkJCIiIUYp

NiZRKH5vcmRlcn42IjcjIiIpUSZ+SVN+OkYkJCIiIUYp

NiZRKH5vcmRlcn42IjcjIiIqUSZ+SVN+OkYkJCIiIUYp

NiZRKH5vcmRlcn42IjcjIiM1USZ+SVN+OkYkJCIiIUYp

dothi(2,1,8,15);

NidRKkdSQVBISUN+OjYiJkkidUdGJDYjIiIjUSgoeCl+PX5+RiQsJCokSSJ4R0YkIyIiIkYoJCIpMCwqKioqISIpUSZ+UkVEfkYk

NidRKkdSQVBISUN+OjYiJkkidUdGJDYjIiIiUSgoeCl+PX5+RiQsJCokSSJ4R0YkI0YoIiIjJCIqKysrLCIhIilRKX5ZRUxMT1d+RiQ=

NidRKkdSQVBISUN+OjYiJkkidUdGJDYjIiIpUSgoeCl+PX5+RiQsJCokSSJ4R0YkIyIiIiIiIyQiKisrKysiISIpUSd+QkxVRX5GJA==

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

1+Int(x*y*u(y),y = 0 .. x);

LCYiIiJGIy1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKihJInhHRilGI0kieUdGKUYjLUkidUdGKTYjRi1GIy9GLTsiIiFGLEYj

K:=1/18*x^7*y+1/18*x*y^7-1/9*x^4*y^4;

LCgqJkkieEc2IiIiKEkieUdGJSIiIiNGKCIjPSomRiRGKEYnRiZGKSomRiQiIiVGJ0YtIyEiIiIiKg==

xapxi( 1 , 1 , 0 , 1 , 15 , K );

UWBvKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqNiI=

NiVRZ29+fn5+fn5+fn5+fn5+fn5+fn5+fn5+fn5+VEhFflZPTFRFUlJBfklOVEVHUkFMfkVRVUFUSU9OfklJfih+c2Vjb25kfmtpbmR+KX46NiIvLUkidUdGJDYjSSJ4R0YkLCYiIiJGKy1JJGludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkKiYsKComRikiIihJInlHRiRGKyNGKyIjPSomRilGK0Y2RjVGNyomRikiIiVGNkY7IyEiIiIiKkYrLUYnNiNGNkYrL0Y2OyIiIUYrRitRYG8qKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKipGJA==

RECURRING TASKS NUMBER

NiRRTzEmLy5USEV+RVFVQVRJT05+T0Z+MXN0fkFESlVTVE1FTlR+UVVBTlRJVFl+On42Ii9JJ2FscGhhMUdGJCwmIiIiRihGJiNGKCIkKyU=

NiRRSX5USEV+MXN0fkFESlVTVE1FTlR+UVVBTlRJVFl+On5hbHBoYVsxXT02IiQiJCsiISIj

NiRRM35GVU5DVElPTn51WzFdKHgpPTYiLCoiIiJGJiokSSJ4R0YkIiIoIyIkKyIiJSJmJEYoIyIjREYsKiRGKCIiJSMhIyEpRiw=

NiZRQ35DT01QQVJFfldJVEh+VEhFfklOSVRJQUx+RlVOQ1RJT042IiZJInVHRiQ2IyIiIVElKHgpPUYkRig=

UWdwfi0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS02Ig==

RECURRING LOOP No

NiYiIiNRPiYvLn5+QURKVVNUTUVOVH5RVUFOVElUWX5PRn46NiJGI1E0fm9yZGVyfldFfk9CVEFJTn46fkYl

NiZRK35GVU5DVElPTn42IiZJInVHRiQ2IyIiI1ElKHgpPUYkLCoiIiJGK0kieEdGJCQiKyZcXnUpcCEjNyokRiwiIiUkISs7Z1NMQSEjNiokRiwiIigkIis1X0MoeSNGNA==

NiZRM35BTFBIQX5DT0VGRklDSUVOVDYiNyMiIiNRIj1GJCQiJllxJSEjNQ==

NilRLX5FUlJPUn5PRn46fjYiJkkidUdGJDYjIiIjUSQoeClGJFEmfkFORH5GJCZGJjYjIiIiRilRJn5JU346RiQ=

NiRRNH5FUlJPUn5FU1RJTUFURUR+PX42IiQiJiUpRychIzU=

NiZRQ35DT01QQVJFfldJVEh+VEhFfklOSVRJQUx+RlVOQ1RJT042IiZJInVHRiQ2IyIiIlElKHgpPUYkLCpGKEYoKiRJInhHRiQiIigjIiQrIiIlImYkRiwjIiNERjAqJEYsIiIlIyEjISlGMA==

UWdwfi0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS02Ig==

RECURRING LOOP No

NiYiIiRRPiYvLn5+QURKVVNUTUVOVH5RVUFOVElUWX5PRn46NiJGI1E0fm9yZGVyfldFfk9CVEFJTn46fkYl

NiZRK35GVU5DVElPTn42IiZJInVHRiQ2IyIiJFElKHgpPUYkLCoiIiJGK0kieEdGJCQiKy0oNHUpcCEjNyokRiwiIiUkIStbb1JMQSEjNiokRiwiIigkIis/SUMoeSNGNA==

NiZRM35BTFBIQX5DT0VGRklDSUVOVDYiNyMiIiRRIj1GJCQhJnJIJiEjOA==

NilRLX5FUlJPUn5PRn46fjYiJkkidUdGJDYjIiIkUSQoeClGJFEmfkFORH5GJCZGJjYjIiIjRilRJn5JU346RiQ=

NiRRNH5FUlJPUn5FU1RJTUFURUR+PX42IiQiJj09IiEjNw==

NiZRQ35DT01QQVJFfldJVEh+VEhFfklOSVRJQUx+RlVOQ1RJT042IiZJInVHRiQ2IyIiI1ElKHgpPUYkLCoiIiJGK0kieEdGJCQiKyZcXnUpcCEjNyokRiwiIiUkISs7Z1NMQSEjNiokRiwiIigkIis1X0MoeSNGNA==

UWdwfi0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS02Ig==

RECURRING LOOP No

NiYiIiVRPiYvLn5+QURKVVNUTUVOVH5RVUFOVElUWX5PRn46NiJGI1E0fm9yZGVyfldFfk9CVEFJTn46fkYl

NiZRK35GVU5DVElPTn42IiZJInVHRiQ2IyIiJVElKHgpPUYkLCoiIiJGK0kieEdGJCQiK0MpNHUpcCEjNyokRixGKCQhK3ZvUkxBISM2KiRGLCIiKCQiK0hJQyh5I0Yz

NiZRM35BTFBIQX5DT0VGRklDSUVOVDYiNyMiIiVRIj1GJCQiJjF1IiEjOg==

NilRLX5FUlJPUn5PRn46fjYiJkkidUdGJDYjIiIlUSQoeClGJFEmfkFORH5GJCZGJjYjIiIkRilRJn5JU346RiQ=

NiRRNH5FUlJPUn5FU1RJTUFURUR+PX42IiQiJiw9JCEjOg==

NiZRQ35DT01QQVJFfldJVEh+VEhFfklOSVRJQUx+RlVOQ1RJT042IiZJInVHRiQ2IyIiJFElKHgpPUYkLCoiIiJGK0kieEdGJCQiKy0oNHUpcCEjNyokRiwiIiUkIStbb1JMQSEjNiokRiwiIigkIis/SUMoeSNGNA==

UWdwfi0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS02Ig==

RECURRING LOOP No

NiYiIiZRPiYvLn5+QURKVVNUTUVOVH5RVUFOVElUWX5PRn46NiJGI1E0fm9yZGVyfldFfk9CVEFJTn46fkYl

NiZRK35GVU5DVElPTn42IiZJInVHRiQ2IyIiJlElKHgpPUYkLCoiIiJGK0kieEdGJCQiK0MpNHUpcCEjNyokRiwiIiUkISt2b1JMQSEjNiokRiwiIigkIitISUMoeSNGNA==

NiZRM35BTFBIQX5DT0VGRklDSUVOVDYiNyMiIiZRIj1GJCQhJit6JCEjPQ==

NilRLX5FUlJPUn5PRn46fjYiJkkidUdGJDYjIiImUSQoeClGJFEmfkFORH5GJCZGJjYjIiIlRilRJn5JU346RiQ=

NiRRNH5FUlJPUn5FU1RJTUFURUR+PX42IiQiIiFGJg==

NiZRQ35DT01QQVJFfldJVEh+VEhFfklOSVRJQUx+RlVOQ1RJT042IiZJInVHRiQ2IyIiJVElKHgpPUYkLCoiIiJGK0kieEdGJCQiK0MpNHUpcCEjNyokRixGKCQhK3ZvUkxBISM2KiRGLCIiKCQiK0hJQyh5I0Yz

UWdwfi0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS02Ig==

RECURRING LOOP No

NiYiIidRPiYvLn5+QURKVVNUTUVOVH5RVUFOVElUWX5PRn46NiJGI1E0fm9yZGVyfldFfk9CVEFJTn46fkYl

NiZRK35GVU5DVElPTn42IiZJInVHRiQ2IyIiJ1ElKHgpPUYkLCoiIiJGK0kieEdGJCQiK0MpNHUpcCEjNyokRiwiIiUkISt2b1JMQSEjNiokRiwiIigkIitISUMoeSNGNA==

NiZRM35BTFBIQX5DT0VGRklDSUVOVDYiNyMiIidRIj1GJCQiJigpXCohI0A=

NilRLX5FUlJPUn5PRn46fjYiJkkidUdGJDYjIiInUSQoeClGJFEmfkFORH5GJCZGJjYjIiImRilRJn5JU346RiQ=

NiRRNH5FUlJPUn5FU1RJTUFURUR+PX42IiQiIiFGJg==

NiZRQ35DT01QQVJFfldJVEh+VEhFfklOSVRJQUx+RlVOQ1RJT042IiZJInVHRiQ2IyIiJlElKHgpPUYkLCoiIiJGK0kieEdGJCQiK0MpNHUpcCEjNyokRiwiIiUkISt2b1JMQSEjNiokRiwiIigkIitISUMoeSNGNA==

UWdwfi0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS02Ig==

RECURRING LOOP No

NiYiIihRPiYvLn5+QURKVVNUTUVOVH5RVUFOVElUWX5PRn46NiJGI1E0fm9yZGVyfldFfk9CVEFJTn46fkYl

NiZRK35GVU5DVElPTn42IiZJInVHRiQ2IyIiKFElKHgpPUYkLCoiIiJGK0kieEdGJCQiK0MpNHUpcCEjNyokRiwiIiUkISt2b1JMQSEjNiokRixGKCQiK0hJQyh5I0Y0

NiZRM35BTFBIQX5DT0VGRklDSUVOVDYiNyMiIihRIj1GJCQhJjFRIyEjQg==

NilRLX5FUlJPUn5PRn46fjYiJkkidUdGJDYjIiIoUSQoeClGJFEmfkFORH5GJCZGJjYjIiInRilRJn5JU346RiQ=

NiRRNH5FUlJPUn5FU1RJTUFURUR+PX42IiQiIiFGJg==

NiZRQ35DT01QQVJFfldJVEh+VEhFfklOSVRJQUx+RlVOQ1RJT042IiZJInVHRiQ2IyIiJ1ElKHgpPUYkLCoiIiJGK0kieEdGJCQiK0MpNHUpcCEjNyokRiwiIiUkISt2b1JMQSEjNiokRiwiIigkIitISUMoeSNGNA==

UWdwfi0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS0tLS02Ig==

RECURRING LOOP No

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RECURRING LOOP No

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RECURRING LOOP No

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RECURRING LOOP No

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RECURRING LOOP No

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RECURRING LOOP No

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RECURRING LOOP No

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RECURRING LOOP No

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RECURRING LOOP No

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dothi(2,8,13,50);

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Legal Notice: The copyright for this application is owned by Maplesoft. The application is intended to demonstrate the use of Maple to solve a particular problem. It has been made available for product evaluation purposes only and may not be used in any other context without the express permission of Maplesoft. 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