paramath - * B2

* B2

THE AVERAGE APPROXIMATING METHOD ON FUNCTIONAL ADJUSTMENT QUANTITY FOR SOLVING

( 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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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

;

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: