This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License.
THE AVERAGE APPROXIMATING METHOD ON FUNCTIONAL ADJUSTMENT QUANTITY FOR SOLVING
Volterra Integral Equation of the second kind .
( 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 ;
adjustment quantity of the order i-th can be expresssed :
The coefficient
Compare with the initial function and we have the error estimated :
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 :
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 ->
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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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Co.H.Tran
MMPC-VN
cohtran@mail.com
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