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MW

Thứ Hai, 2 tháng 7, 2007

THE SOKOLOV'S METHOD .

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THE  AVERAGE APPROXIMATING METHOD ON FUNCTIONAL ADJUSTMENT  QUANTITY   FOR SOLVING


The Volterra Integral Equation II 
                                                                                                                                                                                                                                              

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




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.  
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License
------------------------------------------------------------------------------------------- 
Toán học thuần túy, theo cách của riêng nó, là thi ca của tư duy logic. 
Pure mathematics is, in its way, the poetry of logical ideas.
 Albert Einstein . 

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Thank you .

Trần hồng Cơ .
Co.H.Tran
MMPC-VN
cohtran@mail.com
https://plus.google.com/+HongCoTranMMPC-VN/about

*******

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