paramath

* B2

Image 
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   
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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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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 ( 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      u[0](x) := 0 ; T[1](x) := u[1](x) 
adjustment quantity of the order i-th can be expresssed : alpha[i+1] := lambda*int(K*(T[i](y)-T[i-1](y)-alpha[i]), y = a .. b)*(b-a)/D 
The coefficient alpha[i+2] := min(alpha[i], lambda*int(K*(T[i](y)-T[i-1](y)-alpha[i]), y = a .. b)*(b-a)/D) 
Compare with the initial function and we have the error estimated :  
saiso[i] := sqrt(int(abs(u[i+1](x)-u[i](x))^2, x = a .. b)/(b-a)) 
 

> 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)  );
 


 

********************************************************************* 
                         THE VOLTERRA INTEGRAL EQUATION II ( second kind ) :
                         THE VOLTERRA INTEGRAL EQUATION II ( second kind ) :
                         THE VOLTERRA INTEGRAL EQUATION II ( second kind ) :
                         THE VOLTERRA INTEGRAL EQUATION II ( second kind ) :
 


RECURRING TASKS NUMBER
 

1&/.THE EQUATION OF 1st ADJUSTMENT QUANTITY :
1&/.THE EQUATION OF 1st ADJUSTMENT QUANTITY :
 
 THE 1st ADJUSTMENT QUANTITY : alpha[1]= 
 FUNCTION u[1](x)= 


 

 COMPARE WITH THE INITIAL FUNCTION 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

2,  
 FUNCTION  
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

3,  
 FUNCTION  
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

4,  
 FUNCTION  
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

5,  
 FUNCTION  
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

6,  
 FUNCTION  
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

7,  
 FUNCTION  
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

8,  
 FUNCTION  
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

9,  
 FUNCTION  
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

10,  
 FUNCTION  
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

11,  
 FUNCTION  
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  



 

--------------------------------CONCLUSION--------------------------------  


 

THE ESTIMATED ERRORS OF  
 order  
 order  
 order  
 order  
 order  
 order  
 order  
 order  
 order  
 order  



 

> dothi(2,1,8,15);
 

GRAPHIC : 
GRAPHIC : 
GRAPHIC : 
Plot 

> 1+Int(x*y*u(y),y = 0 .. x);
 

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;
 

1/18*x^7*y+1/18*x*y^7-1/9*x^4*y^4 

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


 

********************************************************************* 
                         THE VOLTERRA INTEGRAL EQUATION II ( second kind ) :
                         THE VOLTERRA INTEGRAL EQUATION II ( second kind ) :
                         THE VOLTERRA INTEGRAL EQUATION II ( second kind ) :
                         THE VOLTERRA INTEGRAL EQUATION II ( second kind ) :
 


RECURRING TASKS NUMBER
 

1&/.THE EQUATION OF 1st ADJUSTMENT QUANTITY :  
 THE 1st ADJUSTMENT QUANTITY : alpha[1]= 
 FUNCTION u[1](x)= 


 

 COMPARE WITH THE INITIAL FUNCTION 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

2,  
 FUNCTION
 FUNCTION
 
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION
 COMPARE WITH THE INITIAL FUNCTION
 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

3,  
 FUNCTION
 FUNCTION
 
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION
 COMPARE WITH THE INITIAL FUNCTION
 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

4,  
 FUNCTION
 FUNCTION
 
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION
 COMPARE WITH THE INITIAL FUNCTION
 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

5,  
 FUNCTION
 FUNCTION
 
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION
 COMPARE WITH THE INITIAL FUNCTION
 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

6,  
 FUNCTION
 FUNCTION
 
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION
 COMPARE WITH THE INITIAL FUNCTION
 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

7,  
 FUNCTION
 FUNCTION
 
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION
 COMPARE WITH THE INITIAL FUNCTION
 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

8,  
 FUNCTION
 FUNCTION
 
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION
 COMPARE WITH THE INITIAL FUNCTION
 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

9,  
 FUNCTION
 FUNCTION
 
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION
 COMPARE WITH THE INITIAL FUNCTION
 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

10,  
 FUNCTION
 FUNCTION
 
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION
 COMPARE WITH THE INITIAL FUNCTION
 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

11,  
 FUNCTION
 FUNCTION
 
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION
 COMPARE WITH THE INITIAL FUNCTION
 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

12,  
 FUNCTION
 FUNCTION
 
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION
 COMPARE WITH THE INITIAL FUNCTION
 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

13,  
 FUNCTION
 FUNCTION
 
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION
 COMPARE WITH THE INITIAL FUNCTION
 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

14,  
 FUNCTION
 FUNCTION
 
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION
 COMPARE WITH THE INITIAL FUNCTION
 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

15,  
 FUNCTION
 FUNCTION
 
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  


 

 COMPARE WITH THE INITIAL FUNCTION
 COMPARE WITH THE INITIAL FUNCTION
 


 

 ------------------------------------------------------------------------------------------- 


RECURRING LOOP No
 

16,  
 FUNCTION
 FUNCTION
 
 ALPHA COEFFICIENT 
 ERROR OF :  
 ERROR ESTIMATED =  



 

--------------------------------CONCLUSION--------------------------------  


 

THE ESTIMATED ERRORS OF  
 order  
 order  
 order  
 order  
 order  
 order  
 order  
 order  
 order  
 order  
 order  
 order  
 order  
 order  
 order  



 

> dothi(2,8,13,50);
 

GRAPHIC :
GRAPHIC :
 
GRAPHIC :
GRAPHIC :
 
GRAPHIC :
GRAPHIC :
 
Plot 

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