# paramath

## * B2

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
Sat , November 06  2004
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** 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  .
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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 ( t , t )  in the general form :
v   = ( 1 + K* ) u
here   : arbitrary parameter . The solution of  u  can be represented with the Neumann series  :    .
The resolvent operator  *   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  )

Sokolov's method
As seen , the first approximation  :    We choose the initial adjustment quantity :   with       ;
adjustment quantity of the order i-th can be expresssed :
The coefficient
Compare with the initial function and we have the error estimated :

 > 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   (x) n is a Cauchy sequence in L2(T) as k -> .
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 || (x) -(x) ||  =  0   k ->

 > 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)  );

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

 > dothi(2,1,8,15);

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

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

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

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

RECURRING LOOP No

 > dothi(2,8,13,50);

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B2.COHONGTRAN-AVGAPPOX